Debye–Hückel theory

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measurement of electric potential in cells (direct method). Going towards high dilutions good results have been found using liquid membrane cells, it has been possible to investigate aqueous media 10 M and it has been found that for 1:1 electrolytes (as NaCl or KCl) the Debye–Hückel equation is totally correct, but for 2:2 or 3:2 electrolytes it is possible to find negative deviation from the Debye–Hückel limit law: this strange behavior can be observed only in the very dilute area, and in more concentrate regions the deviation becomes positive. It is possible that Debye–Hückel equation is not able to foresee this behavior because of the linearization of the Poisson–Boltzmann equation, or maybe not: studies about this have been started only during the last years of the 20th century because before it wasn't possible to investigate the 10 M region, so it is possible that during the next years new theories will be born.

9914: 9479: 445: 5895: 9909:{\displaystyle -\sum _{i}{\frac {z_{i}qn_{i}^{0}}{\varepsilon _{r}\varepsilon _{0}}}e^{-{\frac {z_{i}q\varphi }{k_{\text{B}}T}}}\approx -\sum _{i}{\frac {z_{i}qn_{i}^{0}}{\varepsilon _{r}\varepsilon _{0}}}\left(1-{\frac {z_{i}q\varphi }{k_{\text{B}}T}}\right)=-\left(\sum _{i}{\frac {z_{i}qn_{i}^{0}}{\varepsilon _{r}\varepsilon _{0}}}-\sum _{i}{\frac {z_{i}^{2}q^{2}n_{i}^{0}\varphi }{\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}\right).} 4341: 10717: 3436: 6148:. An English translation of the article is included in a book of collected papers presented to Debye by "his pupils, friends, and the publishers on the occasion of his seventieth birthday on March 24, 1954". Another English translation was completed in 2019. The article deals with the calculation of properties of electrolyte solutions that are under the influence of ion-induced electric fields, thus it deals with 12603: 5456: 10961: 2329: 4036: 10492: 3657: 4025: 3141: 11968: 1554: 9382: 12588: 5804:. The effective hydrated radius of the ion, a is the radius of the ion and its closely bound water molecules. Large ions and less highly charged ions bind water less tightly and have smaller hydrated radii than smaller, more highly charged ions. Typical values are 3Å for ions such as H, Cl, CN, and HCOO. The effective diameter for the 10284: 11733: 1362: 1153: 9476:) to create a linear differential equation. D&H say that this approximation holds at large distances between ions, which is the same as saying that the concentration is low. Lastly, they claim without proof that the addition of more terms in the expansion has little effect on the final solution. Thus 3451: 10744: 3838: 12642:

To verify the validity of the Debye–Hückel theory, many experimental ways have been tried, measuring the activity coefficients: the problem is that we need to go towards very high dilutions. Typical examples are: measurements of vapour pressure, freezing point, osmotic pressure (indirect methods) and

6133:

are constants that depend only on known quantities such as temperature, the charges on the ions and the dielectric constant and viscosity of the solvent. This is known as the Debye–Hückel–Onsager equation. However, this equation only applies to very dilute solutions and has been largely superseded by

5507:

gives satisfactory agreement with experimental measurements for low electrolyte concentrations, typically less than 10 mol/L. Deviations from the theory occur at higher concentrations and with electrolytes that produce ions of higher charges, particularly unsymmetrical electrolytes. Essentially these

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In an ideal electrolyte solution the activity coefficients for all the ions are equal to one. Ideality of an electrolyte solution can be achieved only in very dilute solutions. Non-ideality of more concentrated solutions arises principally (but not exclusively) because ions of opposite charge attract

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of a solution is lowered by the electrical interaction of its ions, but that this effect can't be determined by using the crystallographic approximation for distances between dissimilar atoms (the cube root of the ratio of total volume to the number of particles in the volume). This is because there

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is a target for a zero order of magnitude approximation such as perturbation analysis. Unfortunately, because of the boundary condition at infinity, regular perturbation does not work. The same boundary condition prevents us from finding the exact solution to the equations. Singular perturbation may

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In the same year they first published this article, Debye and Hückel, hereinafter D&H, also released an article that covered their initial characterization of solutions under the influence of electric fields called "On the Theory of Electrolytes. II. Limiting Law for Electric Conductivity", but

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A snapshot of a 2-dimensional section of an idealized electrolyte solution is shown in the picture. The ions are shown as spheres with unit electrical charge. The solvent (pale blue) is shown as a uniform medium, without structure. On average, each ion is surrounded more closely by ions of opposite

429:

Activity coefficients are themselves functions of concentration as the amount of inter-ionic interaction increases as the concentration of the electrolyte increases. Debye and Hückel developed a theory with which single ion activity coefficients could be calculated. By calculating the mean activity

11091:, beyond which no ions in the atmosphere may approach the (charge) center of the singled out ion. This radius may be due to the physical size of the ion itself, the sizes of the ions in the cloud, and any water molecules that surround the ions. Mathematically, they treat the singled out ion as a 2339: 2023: 3135:

rather than the electrolyte concentration. For very low values of the ionic strength the value of the denominator in the expression above becomes nearly equal to one. In this situation the mean activity coefficient is proportional to the square root of the ionic strength. This is known as the

6159:

In the following summary (as yet incomplete and unchecked), modern notation and terminology are used, from both chemistry and mathematics, in order to prevent confusion. Also, with a few exceptions to improve clarity, the subsections in this summary are (very) condensed versions of the same

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in 1927. All the postulates of the original theory were retained. In addition it was assumed that the electric field causes the charge cloud to be distorted away from spherical symmetry. After taking this into account, together with the specific requirements of moving ions, such as

4336:{\displaystyle {\begin{aligned}\pi _{1}&={\frac {q\varphi (r)}{k_{\text{B}}T}}=\Phi (R(r))\\\pi _{2}&=\varepsilon _{r}\\\pi _{3}&={\frac {ak_{\text{B}}T\varepsilon _{0}}{q^{2}}}\\\pi _{4}&=a^{3}I\\\pi _{5}&=z_{0}\\\pi _{6}&={\frac {r}{a}}=R(r).\end{aligned}}} 9029: 12373: 4619: 10712:{\displaystyle \nabla ^{2}\varphi ={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial \varphi (r)}{\partial r}}\right)={\frac {\partial ^{2}\varphi (r)}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial \varphi (r)}{\partial r}}=\kappa ^{2}\varphi (r).} 10046: 12363: 4757: 3431:{\displaystyle \ln(\gamma _{i})=-{\frac {z_{i}^{2}q^{2}\kappa }{8\pi \varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}=-{\frac {z_{i}^{2}q^{3}N_{\text{A}}^{1/2}}{4\pi (\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T)^{3/2}}}{\sqrt {10^{3}{\frac {I}{2}}}}=-Az_{i}^{2}{\sqrt {I}},} 5859: 10105: 2334:

This is the potential energy of a single ion in a solution. The multiple-charge generalization from electrostatics gives an expression for the potential energy of the entire solution. The mean activity coefficient is given by the logarithm of this quantity as follows

5585:. Most extensions to Debye–Hückel theory are empirical in nature. They usually allow the Debye–Hückel equation to be followed at low concentration and add further terms in some power of the ionic strength to fit experimental observations. The main extensions are the 11540: 12092: 1378: 3822: 12205: 9185: 11222: 1183: 956: 305: 8397:

or cloud. Like the crystal lattice, each ion still attempts to surround itself with oppositely charged ions, but in a more free-form manner; at small distances away from positive ions, one is more likely to find negative ions and vice versa.

1841: 2678: 918: 596: 11431:. The shell theorem says that no force is exerted on charged particles inside a sphere (of arbitrary charge). Since the ion atmosphere is assumed to be (time-averaged) spherically symmetric, with charge varying as a function of radius 2478: 770: 2572: 11229: 2324:{\displaystyle u_{j}=z_{j}e{\Big (}\psi _{j}(a_{0})-{\frac {z_{j}e}{4\pi \varepsilon _{0}\varepsilon _{r}}}{\frac {1}{a_{0}}}{\Big )}=-{\frac {z_{j}^{2}e^{2}}{4\pi \varepsilon _{0}\varepsilon _{r}}}{\frac {\kappa }{1+\kappa a_{0}}}} 5710: 424: 7470: 5209: 8872: 1372:. This equation applies to electrolytes with equal numbers of ions of each charge. Nonsymmetrical electrolytes require another term with ψ. For symmetrical electrolytes, this reduces to the modified spherical Bessel equation 4382:

is called the reduced radius. The existing groups may be recombined to form two other dimensionless groups for substitution into the differential equation. The first is what could be called the square of the reduced inverse

7631: 4482: 1874: 8045: 6120: 9921: 5171: 3652:{\displaystyle P^{\text{ex}}=-{\frac {k_{\text{B}}T\kappa _{\text{cgs}}^{3}}{24\pi }}=-{\frac {k_{\text{B}}T\left({\frac {4\pi \sum _{j}c_{j}q_{j}}{\varepsilon _{0}\varepsilon _{r}k_{\text{B}}T}}\right)^{3/2}}{24\pi }}.} 12216: 10956:{\displaystyle \varphi (r)=A{\frac {e^{-{\sqrt {\kappa ^{2}}}r}}{r}}+A'{\frac {e^{{\sqrt {\kappa ^{2}}}r}}{2r{\sqrt {\kappa ^{2}}}}}=A{\frac {e^{-\kappa r}}{r}}+A''{\frac {e^{\kappa r}}{r}}=A{\frac {e^{-\kappa r}}{r}},} 4623: 4020:{\displaystyle {\frac {\partial ^{2}\varphi (r)}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial \varphi (r)}{\partial r}}={\frac {Iq\varphi (r)}{\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}=\kappa ^{2}\varphi (r).} 8393:

is more thermal motion in a liquid solution than in a crystal. The thermal motion tends to smear out the natural lattice that would otherwise be constructed by the ions. Instead, D&H introduce the concept of an

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The deviation from ideality is taken to be a function of the potential energy resulting from the electrostatic interactions between ions and their surrounding clouds. To calculate this energy two steps are needed.

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The last assumption means that each cation is surrounded by a spherically symmetric cloud of other ions. The cloud has a net negative charge. Similarly each anion is surrounded by a cloud with net positive charge.

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Activity coefficients of single ions cannot be measured experimentally because an electrolyte solution must contain both positively charged ions and negatively charged ions. Instead, a mean activity coefficient,

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amounts to 20 atmospheres. Furthermore, they note that this level of pressure gives a relative volume change of 0.001. Therefore, they neglect change in volume of water due to electric pressure, writing

11963:{\displaystyle \varphi '(a_{i})=-{\frac {A_{i}e^{-\kappa a_{i}}(1+\kappa a_{i})}{a_{i}^{2}}}=-{\frac {1}{4\pi \varepsilon _{r}\varepsilon _{0}}}{\frac {z_{i}q}{a_{i}^{2}}}=\varphi _{\text{sp}}'(a_{i}),} 3729: 9377:{\displaystyle \nabla ^{2}\varphi =-{\frac {\rho }{\varepsilon _{r}\varepsilon _{0}}}=-\sum _{i}{\frac {z_{i}qn_{i}^{0}}{\varepsilon _{r}\varepsilon _{0}}}e^{-{\frac {z_{i}q\varphi }{k_{\text{B}}T}}}.} 7371: 5990: 2925: 2832: 8334: 12096: 10414: 6142:

The English title of the article is "On the Theory of Electrolytes. I. Freezing Point Depression and Related Phenomena". It was originally published in 1923 in volume 24 of a German-language journal

9043: 6754: 3720: 11128: 12583:{\displaystyle U_{e}={\frac {1}{2}}\sum _{i=1}^{s}N_{i}u_{i}=-\sum _{i=1}^{s}{\frac {N_{i}z_{i}^{2}}{2}}{\frac {q^{2}\kappa }{4\pi \varepsilon _{r}\varepsilon _{0}}}{\frac {1}{1+\kappa a_{i}}}.} 6239: 4041: 1549:{\displaystyle (\partial _{r}^{2}+{\frac {2}{r}}\partial _{r}-\kappa ^{2})\psi _{j}=0\qquad {\text{with solutions}}\qquad \psi _{j}(r)=A'{\frac {e^{-\kappa r}}{r}}+A''{\frac {e^{\kappa r}}{r}}} 8471: 3098: 3039: 7302: 8677:. If electric forces were the only factor in play, the minimal-energy configuration of all the ions would be achieved in a close-packed lattice configuration. However, the ions are in 10279:{\displaystyle \kappa ^{2}=\sum _{i}{\frac {z_{i}^{2}q^{2}n_{i}^{0}}{\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}={\frac {2Iq^{2}}{\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}},} 7848: 11728:{\displaystyle \varphi (a_{i})=A_{i}{\frac {e^{-\kappa a_{i}}}{a_{i}}}={\frac {1}{4\pi \varepsilon _{r}\varepsilon _{0}}}{\frac {z_{i}q}{a_{i}}}+B_{i}=\varphi _{\text{sp}}(a_{i}),} 1357:{\displaystyle \nabla ^{2}\psi _{j}(r)=\kappa ^{2}\psi _{j}(r)\qquad {\text{with}}\qquad \kappa ^{2}={\frac {e^{2}}{\epsilon _{0}\epsilon _{r}k_{\rm {B}}T}}\sum _{i}n_{i}z_{i}^{2}} 1633: 5505: 2373: 1148:{\displaystyle \nabla ^{2}\psi _{j}(r)=-{\frac {1}{\epsilon _{0}\epsilon _{r}}}\sum _{i}\left\{n_{i}(z_{i}e)\exp \left({\frac {-z_{i}e\psi _{j}(r)}{k_{\rm {B}}T}}\right)\right\}} 7512: 4939: 9442: 6024: 5205: 164: 5603: 1684: 211: 8607: 5444: 5381: 5035: 4421: 203: 11059: 11031: 7378: 7204: 1607: 452:

A description of Debye–Hückel theory includes a very detailed discussion of the assumptions and their limitations as well as the mathematical development and applications.

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Individual ions surrounding a "central" ion can be represented by a statistically averaged cloud of continuous charge density, with a minimum distance of closest approach.

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forces, while ions of the same charge repel each other. In consequence ions are not randomly distributed throughout the solution, as they would be in an ideal solution.

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being a constant is given. However, one can see that this is the case by considering that any spherical static charge distribution is subject to the mathematics of the

8749: 8651: 8363: 8253: 8152: 6477: 6396: 11006: 10739: 6524: 6497: 5730: 1866: 1722: 1717: 1582: 787:) is the averaged charge density of the surrounding cloud at that distance. To apply this formula it is essential that the cloud has spherical symmetry, that is, the 117: 11481:

In a combination of the continuously distributed model which gave the Poisson–Boltzmann equation and the model of the point charge, it is assumed that at the radius

7543: 3125: 2952: 2858: 11506: 11476: 11425: 11398: 11120: 11089: 8717: 8534: 8501: 8218: 8119: 7935: 7798: 6913: 6777: 6672: 6343: 5408: 5089: 5062: 4475: 4448: 4360: 2578: 2057: 1653: 502: 7678: 7538: 6692: 5539:. The dissociation constant can be used to calculate the extent of dissociation and hence, make the necessary correction needed to calculate activity coefficients. 5094: 4875: 4779: 10487: 6262: 2980: 2382: 2083: 675: 2486: 11449: 10981: 10096: 8867: 8675: 8627: 8570: 8390: 8090: 8068: 7651: 7249: 7227: 7035: 7009: 6881: 6855: 6829: 6803: 6454: 6420: 6367: 6314: 6288: 5846: 5826: 5794: 5774: 5754: 4999: 4979: 4959: 4895: 4380: 3059: 3000: 2766: 2746: 2726: 5858: 808: 7100: 324: 6556: 10452:. D&H recognize the importance of the parameter in their article and characterize it as a measure of the thickness of the ion atmosphere, which is an 7683: 7048: 9390:

for the relationship between the number of charges and the strength of the potential field. It has been solved analyticallt by the Swedish mathematician

7853: 2018:{\displaystyle \psi _{j}(r)={\frac {z_{j}e}{4\pi \varepsilon _{0}\varepsilon _{r}}}{\frac {e^{\kappa a_{0}}}{1+\kappa a_{0}}}{\frac {e^{-\kappa r}}{r}}} 13095: 6918: 12370:

To calculate the potential energy of the entire electrolyte solution, one must use the multiple-charge generalization for electric potential energy:

6040: 12911: 11366:{\displaystyle \varphi _{\text{sp}}(r)=\varphi _{\text{pc}}(r)+B_{i}={\frac {1}{4\pi \varepsilon _{r}\varepsilon _{0}}}{\frac {z_{i}q}{r}}+B_{i},} 10288: 2748:

is expressed in terms of molality, instead of molarity (as in the equation above and in the rest of this article), then an experimental value for

1719:, which is the distance of the closest approach of ions, the force exerted by the charge should be balanced by the force of other ions, imposing 47:

for ions in dilute solution. The Debye–Hückel equation provides a starting point for modern treatments of non-ideality of electrolyte solutions.

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D&H say that, due to the "mutual electrostatic forces between the ions", it is necessary to modify the Guldberg–Waage equation by replacing

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coefficients from them the theory could be tested against experimental data. It was found to give excellent agreement for "dilute" solutions.

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Role of the solvent. The solvent is not a structureless medium but is made up of molecules. The water molecules in aqueous solution are both

5338:{\displaystyle {\frac {\partial ^{2}\Phi (R)}{\partial R^{2}}}+{\frac {2}{R}}{\frac {\partial \Phi (R)}{\partial R}}=(\kappa a)^{2}\Phi (R).} 480:

can be any integer. The principal assumption is that departure from ideality is due to electrostatic interactions between ions, mediated by

9024:{\displaystyle n_{i}={\frac {N_{i}}{V}}e^{-{\frac {z_{i}q\varphi }{k_{\text{B}}T}}}=n_{i}^{0}e^{-{\frac {z_{i}q\varphi }{k_{\text{B}}T}}}.} 6175: 5910:

voltage to electrodes immersed in the solution. Debye and Hückel modified their theory in 1926 and their theory was further modified by

4614:{\displaystyle {\frac {\pi _{4}}{\pi _{2}\pi _{3}}}={\frac {a^{2}q^{2}I}{\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}=(\kappa a)^{2}} 13126: 10041:{\displaystyle \nabla ^{2}\varphi =\sum _{i}{\frac {z_{i}^{2}q^{2}n_{i}^{0}\varphi }{\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}},} 9033:

Note that in the infinite temperature limit, all ions are distributed uniformly, with no regard for their electrostatic interactions.

12358:{\displaystyle u_{i}=z_{i}qB_{i}=-{\frac {z_{i}^{2}q^{2}\kappa }{4\pi \varepsilon _{r}\varepsilon _{0}}}{\frac {1}{1+\kappa a_{i}}}.} 4752:{\displaystyle {\frac {\pi _{5}}{\pi _{2}\pi _{3}}}={\frac {z_{0}q^{2}}{4\pi a\varepsilon _{r}\varepsilon _{0}k_{\text{B}}T}}=Z_{0}} 8757: 8409: 6694:

to express the effect of electrostatic forces in an electrolyte on its thermodynamic state. Specifically, they split most of the

7258: 5937: 2867: 2774: 5512:, so there is little to be gained making small adjustments to the model. The individual assumptions can be challenged in turn. 13070: 12766: 12733: 7804: 8255:

is as follows, where lower-case letters indicate the particle specific versions of the corresponding extensive properties:

7474:

Without mentioning it specifically, D&H later give what might be the required (above) justification while arguing that

3662: 43:, which assumes an extremely simplified model of electrolyte solution but nevertheless gave accurate predictions of mean 12087:{\displaystyle A_{i}={\frac {z_{i}q}{4\pi \varepsilon _{r}\varepsilon _{0}}}{\frac {e^{\kappa a_{i}}}{1+\kappa a_{i}}},} 6034:

is the electrolyte concentration. Limiting here means "at the limit of the infinite dilution"). Onsager's expression is

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The extended Debye–Hückel equation provides accurate results for μ ≤ 0.1. For solutions of greater ionic strengths, the

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Notice that this only requires knowledge of the charge of the singled out ion and the potential of all the other ions.

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is already dimensionless, without the substitution given below, the differential equation would still be dimensional.

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charge than by ions of like charge. These concepts were developed into a quantitative theory involving ions of charge

12895: 6503:(a separate activity coefficient associated with each species)—which is what is used in modern chemistry as of 2007. 3817:{\displaystyle \phi ={\frac {P^{\text{id}}+P^{\text{ex}}}{P^{\text{id}}}}=1+{\frac {P^{\text{ex}}}{P^{\text{id}}}}.} 7163: 3064: 3005: 12712: 12200:{\displaystyle B_{i}=-{\frac {z_{i}q\kappa }{4\pi \varepsilon _{r}\varepsilon _{0}}}{\frac {1}{1+\kappa a_{i}}}.} 5594: 3131:

The most significant aspect of this result is the prediction that the mean activity coefficient is a function of

434: 12617: 9180:{\displaystyle \rho =\sum _{i}z_{i}qn_{i}=\sum _{i}z_{i}qn_{i}^{0}e^{-{\frac {z_{i}q\varphi }{k_{\text{B}}T}}}.} 2029: 1158:

Solution of this equation is far from straightforward. Debye and Hückel expanded the exponential as a truncated

13243: 9194: 947: 791:

is a function only of distance from the central ion as this allows the Poisson equation to be cast in terms of

40: 11478:, the ion atmosphere exerts no force. If the force is zero, then the potential is a constant (by definition). 11217:{\displaystyle \varphi _{\text{pc}}(r)={\frac {1}{4\pi \varepsilon _{r}\varepsilon _{0}}}{\frac {z_{i}q}{r}}.} 79:. Real solutions may show departures from this kind of ideality. In order to accommodate these effects in the 13233: 2698:

is a parameter that represents the distance of closest approach of ions. For aqueous solutions at 25 °C

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effects, Onsager was able to derive a theoretical expression to account for the empirical relation known as

5874:

influences on the electrochemical properties of the system (even rough judgement overestimates γ for ~50%).

6892: 1612: 12682: 5903: 5477: 2345: 11451:, it may be represented as an infinite series of concentric charge shells. Therefore, inside the radius 7477: 5528:

is the separation at which the electrostatic interaction between two ions is comparable in magnitude to

4900: 300:{\displaystyle \gamma _{\pm }=\left(\gamma _{\mathrm {Na^{+}} }\gamma _{\mathrm {Cl^{-}} }\right)^{1/2}} 12667: 6399: 5855:

should be used. In these solutions the activity coefficient may actually increase with ionic strength.

9408: 5997: 125: 13238: 13075: 12738: 12210: 9395: 5883: 8579: 5413: 5350: 5004: 4390: 181: 13199: 6265: 6137: 13213: 10421: 8821: 8163: 12924: 12692: 11511: 10453: 10054: 9447: 9402: 6695: 6529: 4813: 4784: 943: 11400:

is a constant that represents the potential added by the ionic atmosphere. No justification for

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is measured the system is subject to an oscillating external field due to the application of an

4030: 1836:{\displaystyle \partial _{r}\psi _{j}(a_{0})=-z_{j}e/(4\pi \epsilon _{0}\epsilon _{r}a_{0}^{2})} 8722: 8686: 3659:

Therefore, the total pressure is the sum of the excess osmotic pressure and the ideal pressure

799: 68: 8636: 8341: 8231: 8130: 6459: 6374: 2673:{\displaystyle B=\left({\frac {2e^{2}N}{\epsilon _{0}\epsilon _{r}k_{\rm {B}}T}}\right)^{1/2}} 591:{\displaystyle {\text{force}}={\frac {z_{1}z_{2}e^{2}}{4\pi \epsilon _{0}\epsilon _{r}r^{2}}}} 87:

was introduced: the properties are then proportional to the activities of the ions. Activity,

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and his collaborators physical chemists V. K. La Mer and Karl Sandved in a 1928 article from

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may be derived from the functional dependence of the chemical potential of a component of an

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Weak electrolytes. A weak electrolyte is one that is not fully dissociated. As such it has a

1689: 792: 667: 489: 102: 13123: 3103: 2930: 2837: 2473:{\displaystyle \log _{10}\gamma _{\pm }=-Az_{j}^{2}{\frac {\sqrt {I}}{1+Ba_{0}{\sqrt {I}}}}} 765:{\displaystyle \nabla ^{2}\psi _{j}(r)=-{\frac {1}{\epsilon _{0}\epsilon _{r}}}\rho _{j}(r)} 13248: 13173: 12657: 11484: 11454: 11403: 11376: 11098: 11067: 9387: 8695: 8682: 8512: 8479: 8196: 8097: 7776: 6898: 6762: 6657: 6500: 6429: 6321: 6169: 5902:

The treatment given so far is for a system not subject to an external electric field. When

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may take place, particularly with ions of higher charge. This was followed up in detail by

5386: 5067: 5040: 4453: 4426: 4345: 2567:{\displaystyle A={\frac {e^{2}B}{2.303\times 8\pi \epsilon _{0}\epsilon _{r}k_{\rm {B}}T}}} 2035: 1658: 1638: 1163: 631:

The solvent plays no role other than providing a medium of constant relative permittivity (

96: 84: 44: 7663: 7523: 6677: 4851: 4764: 8: 11039: 11011: 10466: 8678: 5924: 5907: 3723: 2957: 2062: 1587: 1369: 913:{\displaystyle n'_{i}=n_{i}\exp \left({\frac {-z_{i}e\psi _{j}(r)}{k_{\rm {B}}T}}\right)} 632: 310:

In general, the mean activity coefficient of a fully dissociated electrolyte of formula A

12616:

Please expand the article to include this information. Further details may exist on the

10986: 6247: 5705:{\displaystyle -\log _{10}(\gamma )={\frac {A|z_{+}z_{-}|{\sqrt {I}}}{1+Ba{\sqrt {I}}}}} 1846: 1562: 419:{\displaystyle \gamma _{\pm }=\left({\gamma _{A}}^{n}{\gamma _{B}}^{m}\right)^{1/(n+m)}} 13071:"Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen" 12905: 12734:"Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen" 11434: 10966: 10364: 10081: 8852: 8846: 8660: 8654: 8612: 8573: 8555: 8375: 8188: 8075: 8053: 7932:

D&H say that, according to Planck, the classical part of the Gibbs free entropy is

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To obtain the nondimensionalized differential equation and initial conditions, use the

4365: 3044: 2985: 2751: 2731: 2711: 1175: 931: 610: 7465:{\displaystyle \Phi _{e}=\int {\frac {P_{e}}{T}}\,dV+\int {\frac {U_{e}}{T^{2}}}\,dT.} 5577:

Moreover, ionic radius is assumed to be negligible, but at higher concentrations, the

1174:= 0), which leaves us with only the first order term. The result has the form of the 16:

Model describing the departures from ideality in solutions of electrolytes and plasmas

12891: 12762: 8630: 7654: 5848:

are constants with values of respectively 0.5085 and 0.3281 at 25 °C in water .

13164: 13147: 12213:, the potential energy associated with the singled out ion in the ion atmosphere is 13159: 12687: 9190: 8752: 8394: 5852: 5733: 5582: 3441: 639: 628:

of ions is ignored except insofar as it determines the effective sizes of the ions.

481: 36: 1162:

to first order. The zeroth order term vanishes because the solution is on average

484:: the force of interaction between two electric charges, separated by a distance, 13169: 13130: 12707: 12702: 12697: 10051:

Factor out the scalar potential and assign the leftovers, which are constant, to

6832: 5920: 5590: 5586: 5566: 5517: 6138:

Summary of Debye and Hückel's first article on the theory of dilute electrolytes

2834:

at 25 °C. It is common to use a base-10 logarithm, in which case we factor

28: 12677: 12662: 10099: 9037: 8690: 7626:{\displaystyle \Xi =S-{\frac {U+PV}{T}}=\Phi -{\frac {PV}{T}}=-{\frac {G}{T}},} 6149: 5862:

The Debye–Hückel plot with different values for ion charge Z and ion diameter a

5547: 5525: 2688: 788: 621: 617: 80: 64: 12592: 8040:{\displaystyle \Xi _{k}=\sum _{i=0}^{s}N_{i}(\xi _{i}-k_{\text{B}}ln(x_{i})),} 6115:{\displaystyle \Lambda _{m}=\Lambda _{m}^{0}-(A+B\Lambda _{m}^{0}){\sqrt {c}}} 13227: 11428: 8537: 8366: 6346: 5805: 5521: 5166:{\displaystyle {\frac {\partial \Phi (R)}{\partial R}}{\bigg |}_{R=1}=-Z_{0}} 2375:

values for KBr at 25°C (points) and Debye–Hückel limiting law (coloured line)

1159: 444: 171: 72: 12672: 11092: 10449: 7518: 6780: 6652: 5911: 5578: 5543: 4384: 7769:

At this point D&H note that, for water containing 1 mole per liter of

5882:

The theory can be applied also to dilute solutions of mixed electrolytes.

3136:

Debye–Hückel limiting law. In this limit the equation is given as follows

31:

as a theoretical explanation for departures from ideality in solutions of

13187:

http://guava.physics.uiuc.edu/~nigel/courses/569/Essays_2004/files/lu.pdf

13143: 12652: 9386:

This equation is difficult to solve and does not follow the principle of

6858: 606: 60: 32: 24: 7150:{\displaystyle {\frac {U}{T^{2}}}={\frac {\partial \Phi }{\partial T}},} 6168:

D&H note that the Guldberg–Waage formula for electrolyte species in

5466:

dissolved in water. The oxygen atoms are arranged at the vertices of an

7773:(nominal pressure and temperature aren't given), the electric pressure 6637:{\displaystyle \log(\gamma )=\sum _{i=1}^{s}\nu _{i}\log(\gamma _{i}).} 5867: 5467: 7255:

Similarly, the Helmholtz free entropy is also divided into two parts:

938:

is the temperature. This distribution also depends on the potential ψ(

7160:

As stated previously, the internal energy is divided into two parts:

5916: 5570: 3445: 625: 56: 9401:

However, for sufficiently low concentrations of ions, a first-order

8401: 7760:{\displaystyle d\Xi =-{\frac {V}{T}}\,dP+{\frac {U+PV}{T^{2}}}\,dT.} 7091:{\displaystyle {\frac {P}{T}}={\frac {\partial \Phi }{\partial V}},} 5347:

For table salt in 0.01 M solution at 25 °C, a typical value of

12961: 10721:

The equation has the following general solution (keep in mind that

7923:{\displaystyle \Xi _{e}=\Phi _{e}=\int {\frac {U_{e}}{T^{2}}}\,dT.} 7012: 6134:

other equations due to Fuoss and Onsager, 1932 and 1957 and later.

5898:

Forces affecting a charged ion or cloud moving in an electric field

5894: 5801: 3835:(as stated above, the equation only holds for low concentrations): 2338: 6981:{\displaystyle d\Phi ={\frac {P}{T}}\,dV+{\frac {U}{T^{2}}}\,dT,} 6806: 5871: 5562: 5551: 946:. Nevertheless, the two equations can be combined to produce the 798:

The second step is to calculate the charge density by means of a

662:

The first step is to specify the electrostatic potential for ion

12602: 11064:

In the next step, D&H assume that there is a certain radius

8681:

with each other and are relatively free to move. Thus they obey

7038: 5565:

and polarizable. Both cations and anions have a strong primary

5460: 4033:

on this problem results in the following dimensionless groups:

76: 12925:

http://homepages.rpi.edu/~keblip/THERMO/chapters/Chapter33.pdf

12885: 10048:

because the first summation is zero due to electroneutrality.

448:

An idealized representation of a solution of a 1:1 electrolyte

11036:

The electric potential is zero at infinity by definition, so

7045:

By the definition of the total differential, this means that

5866:

The Debye–Hückel equation cannot be used in the solutions of

5410:

is 7.017, highlighting the fact that, in low concentrations,

4423:. The second could be called the reduced central ion charge, 10354:{\displaystyle I={\frac {1}{2}}\sum _{i}z_{i}^{2}n_{i}^{0}.} 8811:{\displaystyle e^{-{\frac {z_{i}q\varphi }{k_{\text{B}}T}}}} 12593:

The additional electric term to the thermodynamic potential

9189:

When combining this result for the charge density with the

5756:

is the integer charge of the ion (1 for H, 2 for Mg etc.),

5455: 942:) and this introduces a serious difficulty in terms of the 12637: 11226:

D&H say that the total potential inside the sphere is

7366:{\displaystyle \Phi _{e}=\int {\frac {U_{e}}{T^{2}}}\,dT.} 5985:{\displaystyle \Lambda _{m}=\Lambda _{m}^{0}-K{\sqrt {c}}} 5558:, are not spherical. Polyatomic ions are also polarizable. 2920:{\displaystyle 0.509{\text{ mol}}^{-1/2}{\text{kg}}^{1/2}} 2827:{\displaystyle 1.172{\text{ mol}}^{-1/2}{\text{kg}}^{1/2}} 13124:

http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html

13058:. New York, New York: Interscience Publishers, Inc. 1954. 10363:

So, the fundamental equation is reduced to a form of the

8329:{\displaystyle \xi _{i}=s_{i}-{\frac {u_{i}+Pv_{i}}{T}}.} 5463: 10409:{\displaystyle \nabla ^{2}\varphi =\kappa ^{2}\varphi .} 9405:

approximation for the exponential function may be used (

8154:

is the particle specific Gibbs free entropy of species

6749:{\displaystyle \Phi =S-{\frac {U}{T}}=-{\frac {A}{T}},} 6316:

is the number of species participating in the reaction,

4362:

is called the reduced scalar electric potential field.

2982:

in the equation is for the case when the dimensions of

13068: 12731: 8092:

is the number of different particle types in solution,

6250: 3715:{\textstyle P^{\text{id}}=k_{\text{B}}T\sum _{i}c_{i}} 3665: 13096:"Mixtures; Partial Molar Quantities; Ideal Solutions" 12376: 12219: 12099: 11975: 11740: 11543: 11514: 11487: 11457: 11437: 11406: 11379: 11232: 11131: 11101: 11070: 11042: 11014: 10989: 10969: 10747: 10727: 10495: 10469: 10424: 10373: 10291: 10108: 10084: 10057: 9924: 9482: 9450: 9411: 9203: 9046: 8875: 8869:

is the temperature. Thus at every point in the cloud

8855: 8824: 8760: 8725: 8719:

are altered from their bulk (overall average) values

8698: 8663: 8639: 8615: 8582: 8558: 8515: 8482: 8412: 8378: 8344: 8261: 8234: 8199: 8166: 8133: 8100: 8078: 8056: 7938: 7856: 7807: 7779: 7686: 7666: 7639: 7546: 7526: 7480: 7381: 7312: 7261: 7237: 7215: 7166: 7103: 7051: 7023: 6997: 6921: 6901: 6869: 6843: 6817: 6791: 6765: 6704: 6680: 6660: 6559: 6532: 6512: 6485: 6462: 6442: 6408: 6377: 6355: 6324: 6302: 6276: 6178: 6043: 6000: 5940: 5834: 5814: 5782: 5762: 5742: 5718: 5606: 5600:

One such extended Debye–Hückel equation is given by:

5480: 5416: 5389: 5353: 5212: 5178: 5097: 5070: 5043: 5007: 4987: 4967: 4947: 4903: 4883: 4854: 4816: 4787: 4767: 4626: 4485: 4456: 4429: 4393: 4368: 4348: 4039: 3841: 3732: 3454: 3144: 3106: 3067: 3047: 3008: 2988: 2960: 2933: 2870: 2840: 2777: 2754: 2734: 2714: 2581: 2489: 2385: 2348: 2094: 2065: 2038: 1877: 1849: 1725: 1692: 1661: 1641: 1615: 1590: 1565: 1381: 1186: 959: 811: 678: 505: 327: 214: 184: 128: 105: 7514:, an assumption that the solvent is incompressible. 6499:

is an overall activity coefficient, not a "special"

6234:{\displaystyle \prod _{i=1}^{s}x_{i}^{\nu _{i}}=K,} 6156:that subsequent article is not (yet) covered here. 5776:is the ionic strength of the aqueous solution, and 4897:while carrying out the chain rule and substituting 205:is defined. For example, with the electrolyte NaCl 12582: 12357: 12199: 12086: 11962: 11727: 11529: 11500: 11470: 11443: 11419: 11392: 11365: 11216: 11114: 11083: 11053: 11025: 11000: 10975: 10955: 10733: 10711: 10481: 10440: 10408: 10353: 10278: 10090: 10070: 10040: 9908: 9468: 9436: 9376: 9179: 9023: 8861: 8837: 8810: 8743: 8711: 8669: 8645: 8621: 8601: 8564: 8528: 8495: 8465: 8384: 8357: 8338:D&H don't say so, but the functional form for 8328: 8247: 8212: 8179: 8146: 8113: 8084: 8062: 8039: 7922: 7842: 7792: 7759: 7672: 7645: 7625: 7532: 7506: 7464: 7365: 7296: 7243: 7221: 7198: 7149: 7090: 7029: 7003: 6980: 6907: 6875: 6849: 6823: 6797: 6771: 6748: 6686: 6666: 6636: 6545: 6518: 6491: 6471: 6448: 6414: 6390: 6361: 6337: 6308: 6282: 6256: 6233: 6114: 6018: 5984: 5840: 5820: 5788: 5768: 5748: 5724: 5704: 5499: 5438: 5402: 5375: 5337: 5199: 5165: 5083: 5056: 5029: 4993: 4973: 4953: 4933: 4889: 4869: 4840: 4802: 4773: 4751: 4613: 4469: 4442: 4415: 4374: 4354: 4335: 4019: 3816: 3714: 3651: 3430: 3119: 3092: 3053: 3033: 2994: 2974: 2946: 2919: 2852: 2826: 2760: 2740: 2720: 2672: 2566: 2472: 2367: 2323: 2077: 2051: 2017: 1860: 1835: 1711: 1678: 1647: 1627: 1601: 1576: 1548: 1356: 1147: 912: 764: 590: 418: 299: 197: 158: 111: 9918:The Poisson–Boltzmann equation is transformed to 8402:The potential energy of an arbitrary ion solution 5130: 2223: 2123: 13225: 12759:An Introduction to Aqueous Electrolyte Solutions 11095:to which one may not approach within the radius 9398:dealing with extensions to Debye–Huckel theory. 8228:Species zero is the solvent. The definition of 7375:It would seem that, without some justification, 8406:Electroneutrality of a solution requires that 7306:D&H state, without giving the logic, that 2728:is a constant that depends on temperature. If 95:. The proportionality constant is known as an 13050: 13048: 13046: 13044: 13042: 13040: 13038: 13036: 13034: 13032: 13030: 13028: 13026: 13024: 13022: 13020: 13018: 13016: 13014: 13012: 13010: 13008: 11125:The potential of a point charge by itself is 6026:is known as the limiting molar conductivity, 5450: 13142: 13006: 13004: 13002: 13000: 12998: 12996: 12994: 12992: 12990: 12988: 12910:: CS1 maint: multiple names: authors list ( 12756: 8466:{\displaystyle \sum _{i=1}^{s}N_{i}z_{i}=0,} 6143: 5886:measurements has been used to this purpose. 3093:{\displaystyle {\text{mole}}/{\text{m}}^{3}} 3034:{\displaystyle {\text{mol}}/{\text{dm}}^{3}} 13136: 13087: 653: 7297:{\displaystyle \Phi =\Phi _{k}+\Phi _{e}.} 5091:. The resulting equations are as follows: 1609:are fixed by the boundary conditions. As 13163: 13056:The Collected Papers of Peter J. W. Debye 12985: 12881: 12879: 12877: 7910: 7747: 7709: 7452: 7415: 7353: 6968: 6941: 13214:"Electric potential of a charged sphere" 6698:into classical and electrostatic terms: 5893: 5877: 5581:becomes comparable to the radius of the 5569:and a weaker secondary solvation shell. 5454: 3826: 3444:obtained from Debye–Hückel theory is in 2337: 779:) is the total potential at a distance, 443: 13102:. University of Arizona. Archived from 13093: 12950:. London: Butterworths. pp. 37–53. 12886:Hamann, Hamnett, and Vielstich (1998). 12638:Experimental verification of the theory 8503:is the total number of ions of species 7843:{\displaystyle \Xi =\Xi _{k}+\Xi _{e},} 7660:D&H give the total differential of 5383:is 0.0005636, while a typical value of 4981:(no chain rule needed), then eliminate 13226: 12945: 12874: 8372:D&H note that the internal energy 8121:is the number of particles of species 6526:and the special activity coefficients 5508:deviations occur because the model is 4450:(with a capital Z). Note that, though 3831:Taking the differential equation from 12862:Data from Wright, Worked problem 8.17 6160:subsections of the original article. 12596: 1628:{\displaystyle r\rightarrow \infty } 91:, is proportional to concentration, 13200:"LaPlace's and Poisson's Equations" 13148:"Thomas Hakon Gronwall—In memoriam" 12890:. Weinheim: Wiley-VCH Verlag GmbH. 9193:from electrostatics, a form of the 5573:are ignored in Debye–Hückel theory. 5500:{\displaystyle \log \gamma _{\pm }} 3127:must be dropped from the equation 2368:{\displaystyle \log \gamma _{\pm }} 609:is completely dissociated; it is a 13: 13094:Salzman, William R. (2001-08-21). 10672: 10655: 10623: 10600: 10579: 10562: 10535: 10531: 10497: 10375: 9926: 9205: 9040:is related to the number density: 8346: 7940: 7871: 7858: 7828: 7815: 7808: 7690: 7667: 7583: 7547: 7527: 7507:{\displaystyle \Phi _{e}=\Xi _{e}} 7495: 7482: 7383: 7314: 7282: 7269: 7262: 7135: 7130: 7127: 7076: 7071: 7068: 6925: 6902: 6766: 6705: 6681: 6661: 6653:Helmholtz and Gibbs free entropies 6088: 6058: 6045: 6002: 5955: 5942: 5857: 5320: 5289: 5275: 5272: 5240: 5226: 5217: 5185: 5179: 5118: 5104: 5101: 4934:{\displaystyle {R^{\prime }}(r)=a} 4910: 4817: 4349: 4100: 3918: 3901: 3869: 3846: 2640: 2552: 1727: 1622: 1414: 1386: 1307: 1188: 1124: 961: 894: 680: 266: 262: 244: 240: 14: 13260: 10459:The equation may be expressed in 8552:, initially far away, to a point 5470:with the sodium ion at its centre 12601: 9437:{\displaystyle e^{x}\approx 1+x} 8220:is the mole fraction of species 6019:{\displaystyle \Lambda _{m}^{0}} 5200:{\displaystyle \Phi (\infty )=0} 159:{\displaystyle a=\gamma c/c^{0}} 13206: 13192: 13180: 13165:10.1090/s0002-9904-1932-05492-1 13117: 13069:P. Debye and E. Hückel (1923). 13062: 12982:Wright, sections 12.10 to 12.17 12976: 12967: 12954: 12939: 12930: 12918: 12865: 12856: 12847: 12838: 12732:P. Debye and E. Hückel (1923). 12713:Specific ion Interaction Theory 11537:and its first derivative. Thus 6646: 6163: 5927:, for the molar conductivity, Λ 5889: 5595:specific ion interaction theory 5459:The first solvation shell of a 1461: 1455: 1254: 1248: 616:Ions are spherical and are not 12829: 12820: 12811: 12802: 12793: 12784: 12775: 12750: 12725: 11954: 11941: 11829: 11807: 11762: 11749: 11719: 11706: 11560: 11547: 11524: 11518: 11271: 11265: 11249: 11243: 11148: 11142: 10757: 10751: 10703: 10697: 10667: 10661: 10618: 10612: 10574: 10568: 8602:{\displaystyle z_{i}q\varphi } 8572:within the ion cloud requires 8031: 8028: 8015: 7983: 6628: 6615: 6572: 6566: 6102: 6075: 5667: 5642: 5629: 5623: 5439:{\displaystyle (\kappa a)^{2}} 5427: 5417: 5376:{\displaystyle (\kappa a)^{2}} 5364: 5354: 5329: 5323: 5311: 5301: 5284: 5278: 5235: 5229: 5188: 5182: 5113: 5107: 5030:{\displaystyle (\kappa a)^{2}} 5018: 5008: 4922: 4916: 4864: 4858: 4835: 4832: 4826: 4820: 4797: 4791: 4602: 4592: 4416:{\displaystyle (\kappa a)^{2}} 4404: 4394: 4323: 4317: 4118: 4115: 4109: 4103: 4076: 4070: 4011: 4005: 3948: 3942: 3913: 3907: 3864: 3858: 3350: 3313: 3164: 3151: 2151: 2138: 2030:electrostatic potential energy 1894: 1888: 1830: 1786: 1759: 1746: 1619: 1478: 1472: 1436: 1382: 1245: 1239: 1213: 1207: 1113: 1107: 1065: 1049: 986: 980: 883: 877: 759: 753: 705: 699: 411: 399: 198:{\displaystyle \gamma _{\pm }} 1: 12718: 7229:indicates the classical part, 7199:{\displaystyle U=U_{k}+U_{e}} 7157:which are useful further on. 6170:chemical reaction equilibrium 6030:is an empirical constant and 5736:is the activity coefficient, 783:, from the central ion and ρ( 83:of solutions, the concept of 10441:{\displaystyle \kappa ^{-1}} 8838:{\displaystyle k_{\text{B}}} 8180:{\displaystyle k_{\text{B}}} 7251:indicates the electric part. 795:with no angular dependence. 439: 7: 12683:Bates-Guggenheim Convention 12646: 11530:{\displaystyle \varphi (r)} 11508:, there is a continuity of 11033:are undetermined constants 10071:{\displaystyle \kappa ^{2}} 9469:{\displaystyle 0<x\ll 1} 8653:is the value of the scalar 8548:To bring an ion of species 6546:{\displaystyle \gamma _{i}} 4841:{\displaystyle \Phi (R(r))} 4803:{\displaystyle \varphi (r)} 435:Bates–Guggenheim Convention 50: 10: 13265: 12668:Poisson-Boltzmann equation 10456:of the Gouy–Chapman type. 9195:Poisson–Boltzmann equation 6400:stoichiometric coefficient 5451:Limitations and extensions 948:Poisson–Boltzmann equation 432: 13076:Physikalische Zeitschrift 12739:Physikalische Zeitschrift 12211:electric potential energy 10741:is a positive constant): 9396:Physikalische Zeitschrift 8744:{\displaystyle n_{i}^{0}} 6506:The relationship between 6145:Physikalische Zeitschrift 5884:Freezing point depression 3041:. When the dimensions of 8646:{\displaystyle \varphi } 8369:upon its mole fraction. 8358:{\displaystyle \Xi _{k}} 8248:{\displaystyle \xi _{i}} 8147:{\displaystyle \xi _{i}} 6696:thermodynamic potentials 6472:{\displaystyle \gamma K} 6391:{\displaystyle \nu _{i}} 5571:Ion–solvent interactions 5550:. Many ions such as the 5542:Ions are spherical, not 654:Mathematical development 601:It is also assumed that 71:are proportional to the 13100:Chemical Thermodynamics 12853:Wright, section 10.6.15 12844:Wright, section 10.6.10 12693:Electrical double layer 10734:{\displaystyle \kappa } 10489:at some arbitrary ion: 10454:electrical double layer 9403:Taylor series expansion 6519:{\displaystyle \gamma } 6492:{\displaystyle \gamma } 6294:indicating the species, 5725:{\displaystyle \gamma } 5516:Complete dissociation. 1712:{\displaystyle r=a_{0}} 944:superposition principle 112:{\displaystyle \gamma } 41:Poisson–Boltzmann model 12946:Davies, C. W. (1962). 12835:Wright, section 10.6.8 12826:Wright, section 10.6.3 12612:is missing information 12584: 12467: 12420: 12359: 12201: 12088: 11964: 11729: 11531: 11502: 11472: 11445: 11421: 11394: 11367: 11218: 11116: 11085: 11055: 11027: 11002: 10977: 10957: 10735: 10713: 10483: 10450:Debye screening length 10442: 10410: 10355: 10280: 10092: 10072: 10042: 9910: 9470: 9438: 9378: 9181: 9025: 8863: 8839: 8812: 8745: 8713: 8687:Boltzmann distribution 8671: 8647: 8623: 8603: 8566: 8530: 8497: 8467: 8433: 8386: 8359: 8330: 8249: 8214: 8181: 8148: 8115: 8086: 8064: 8041: 7972: 7924: 7844: 7794: 7761: 7674: 7647: 7627: 7534: 7517:The definition of the 7508: 7466: 7367: 7298: 7245: 7223: 7200: 7151: 7092: 7031: 7005: 6982: 6909: 6877: 6851: 6825: 6799: 6781:Helmholtz free entropy 6773: 6750: 6688: 6668: 6638: 6598: 6547: 6520: 6493: 6473: 6450: 6416: 6392: 6363: 6339: 6310: 6284: 6258: 6235: 6199: 6144: 6116: 6020: 5986: 5899: 5870:where the presence of 5863: 5842: 5822: 5790: 5770: 5750: 5726: 5706: 5501: 5471: 5440: 5404: 5377: 5339: 5201: 5167: 5085: 5058: 5031: 4995: 4975: 4955: 4935: 4891: 4871: 4842: 4804: 4775: 4753: 4615: 4471: 4444: 4417: 4376: 4356: 4337: 4021: 3818: 3716: 3653: 3432: 3121: 3120:{\displaystyle 10^{3}} 3094: 3055: 3035: 2996: 2976: 2948: 2947:{\displaystyle 10^{3}} 2921: 2854: 2853:{\displaystyle \ln 10} 2828: 2762: 2742: 2722: 2702:= 0.51 moldm and 2674: 2568: 2474: 2376: 2369: 2325: 2079: 2053: 2019: 1862: 1837: 1713: 1680: 1649: 1629: 1603: 1578: 1550: 1358: 1149: 914: 800:Boltzmann distribution 766: 592: 449: 420: 301: 199: 160: 113: 69:colligative properties 13244:Equilibrium chemistry 13152:Bull. Amer. Math. Soc 12936:Wright, section 10.9. 12757:Wright, M.R. (2007). 12585: 12447: 12400: 12360: 12209:By the definition of 12202: 12089: 11965: 11730: 11532: 11503: 11501:{\displaystyle a_{i}} 11473: 11471:{\displaystyle a_{i}} 11446: 11422: 11420:{\displaystyle B_{i}} 11395: 11393:{\displaystyle B_{i}} 11368: 11219: 11117: 11115:{\displaystyle a_{i}} 11086: 11084:{\displaystyle a_{i}} 11056: 11028: 11003: 10978: 10958: 10736: 10714: 10484: 10461:spherical coordinates 10443: 10411: 10356: 10281: 10093: 10073: 10043: 9911: 9471: 9439: 9392:Thomas Hakon Gronwall 9379: 9182: 9026: 8864: 8840: 8813: 8751:by the corresponding 8746: 8714: 8712:{\displaystyle n_{i}} 8672: 8648: 8624: 8604: 8567: 8531: 8529:{\displaystyle z_{i}} 8498: 8496:{\displaystyle N_{i}} 8468: 8413: 8387: 8360: 8331: 8250: 8215: 8213:{\displaystyle x_{i}} 8182: 8149: 8116: 8114:{\displaystyle N_{i}} 8087: 8065: 8042: 7952: 7925: 7845: 7795: 7793:{\displaystyle P_{e}} 7762: 7675: 7648: 7628: 7535: 7509: 7467: 7368: 7299: 7246: 7224: 7201: 7152: 7093: 7032: 7006: 6983: 6910: 6908:{\displaystyle \Phi } 6885:Helmholtz free energy 6878: 6852: 6826: 6800: 6774: 6772:{\displaystyle \Phi } 6751: 6689: 6669: 6667:{\displaystyle \Phi } 6639: 6578: 6548: 6521: 6494: 6474: 6451: 6417: 6393: 6364: 6340: 6338:{\displaystyle x_{i}} 6311: 6285: 6259: 6236: 6179: 6172:in classical form is 6117: 6021: 5987: 5897: 5878:Electrolytes mixtures 5861: 5843: 5823: 5791: 5771: 5751: 5727: 5707: 5537:dissociation constant 5502: 5458: 5441: 5405: 5403:{\displaystyle Z_{0}} 5378: 5340: 5202: 5168: 5086: 5084:{\displaystyle Z_{0}} 5059: 5057:{\displaystyle z_{0}} 5032: 4996: 4976: 4956: 4936: 4892: 4872: 4843: 4805: 4776: 4754: 4616: 4472: 4470:{\displaystyle z_{0}} 4445: 4443:{\displaystyle Z_{0}} 4418: 4377: 4357: 4355:{\displaystyle \Phi } 4338: 4022: 3827:Nondimensionalization 3819: 3717: 3654: 3433: 3122: 3095: 3056: 3036: 2997: 2977: 2949: 2922: 2855: 2829: 2763: 2743: 2723: 2675: 2569: 2475: 2370: 2341: 2326: 2080: 2054: 2052:{\displaystyle u_{j}} 2020: 1863: 1838: 1714: 1681: 1679:{\displaystyle A''=0} 1655:must not diverge, so 1650: 1648:{\displaystyle \psi } 1630: 1604: 1579: 1551: 1359: 1150: 915: 793:spherical coordinates 767: 593: 490:relative permittivity 447: 433:Further information: 421: 302: 200: 161: 114: 45:activity coefficients 39:. It is a linearized 13234:Thermodynamic models 12973:Wright, section 12.3 12871:Wright, section 10.7 12817:Wright, Section 10.3 12808:Wright, Section 10.3 12790:Wright, Section 8.23 12781:Wright, section 1.10 12614:about not yet added. 12374: 12217: 12097: 11973: 11738: 11541: 11512: 11485: 11455: 11435: 11404: 11377: 11230: 11129: 11099: 11068: 11040: 11012: 10987: 10967: 10745: 10725: 10493: 10467: 10422: 10371: 10289: 10106: 10082: 10055: 9922: 9480: 9448: 9409: 9388:linear superposition 9201: 9044: 8873: 8853: 8822: 8758: 8723: 8696: 8683:Boltzmann statistics 8661: 8637: 8613: 8580: 8556: 8513: 8480: 8410: 8376: 8342: 8259: 8232: 8197: 8164: 8131: 8098: 8076: 8054: 7936: 7854: 7805: 7777: 7684: 7673:{\displaystyle \Xi } 7664: 7637: 7544: 7533:{\displaystyle \Xi } 7524: 7478: 7379: 7310: 7259: 7235: 7213: 7164: 7101: 7049: 7021: 6995: 6919: 6899: 6867: 6841: 6815: 6789: 6763: 6702: 6687:{\displaystyle \Xi } 6678: 6658: 6557: 6530: 6510: 6501:activity coefficient 6483: 6460: 6440: 6430:equilibrium constant 6406: 6375: 6353: 6322: 6300: 6274: 6248: 6176: 6041: 5998: 5938: 5832: 5812: 5780: 5760: 5740: 5716: 5604: 5478: 5414: 5387: 5351: 5210: 5176: 5095: 5068: 5041: 5005: 4985: 4965: 4945: 4901: 4881: 4870:{\displaystyle R(r)} 4852: 4814: 4785: 4781:groups to eliminate 4774:{\displaystyle \pi } 4765: 4624: 4483: 4454: 4427: 4391: 4366: 4346: 4037: 4031:Buckingham π theorem 3839: 3730: 3663: 3452: 3142: 3104: 3065: 3045: 3006: 2986: 2958: 2931: 2868: 2838: 2775: 2752: 2732: 2712: 2706:= 3.29 nmmoldm 2579: 2487: 2383: 2346: 2092: 2063: 2036: 1875: 1847: 1723: 1690: 1659: 1639: 1613: 1588: 1563: 1379: 1184: 1164:electrically neutral 957: 809: 676: 503: 325: 212: 182: 126: 103: 97:activity coefficient 67:is a solution whose 12799:Wright, chapter 10. 12495: 12279: 11940: 11922: 11847: 11054:{\displaystyle A''} 11026:{\displaystyle A''} 10482:{\displaystyle r=0} 10347: 10332: 10174: 10149: 9993: 9968: 9856: 9831: 9775: 9645: 9526: 9298: 9126: 8970: 8740: 8679:thermal equilibrium 6257:{\textstyle \prod } 6221: 6101: 6071: 6015: 5968: 5474:This equation for 3724:osmotic coefficient 3501: 3417: 3304: 3271: 3190: 2975:{\displaystyle I/2} 2432: 2251: 2078:{\displaystyle r=0} 1868:is found, yielding 1829: 1602:{\displaystyle A''} 1399: 1370:analytical solution 1353: 824: 633:dielectric constant 620:by the surrounding 21:Debye–Hückel theory 13129:2007-08-28 at the 12580: 12481: 12355: 12265: 12197: 12084: 11960: 11928: 11908: 11833: 11725: 11527: 11498: 11468: 11441: 11417: 11390: 11363: 11214: 11112: 11081: 11051: 11023: 11001:{\displaystyle A'} 10998: 10973: 10953: 10731: 10709: 10479: 10438: 10406: 10365:Helmholtz equation 10351: 10333: 10318: 10317: 10276: 10160: 10135: 10131: 10088: 10068: 10038: 9979: 9954: 9950: 9906: 9842: 9817: 9813: 9761: 9744: 9631: 9614: 9512: 9495: 9466: 9434: 9374: 9284: 9267: 9177: 9112: 9098: 9062: 9021: 8956: 8859: 8847:Boltzmann constant 8835: 8808: 8741: 8726: 8709: 8667: 8655:electric potential 8643: 8619: 8599: 8574:interaction energy 8562: 8526: 8493: 8463: 8382: 8355: 8326: 8245: 8210: 8189:Boltzmann constant 8177: 8144: 8111: 8082: 8060: 8037: 7920: 7840: 7790: 7771:potassium chloride 7757: 7670: 7643: 7623: 7530: 7519:Gibbs free entropy 7504: 7462: 7363: 7294: 7241: 7219: 7196: 7147: 7088: 7027: 7001: 6978: 6905: 6893:total differential 6873: 6847: 6821: 6795: 6769: 6746: 6684: 6664: 6634: 6543: 6516: 6489: 6469: 6446: 6412: 6388: 6359: 6335: 6306: 6280: 6264:is a notation for 6254: 6231: 6200: 6112: 6087: 6057: 6016: 6001: 5982: 5954: 5900: 5864: 5838: 5818: 5798:effective diameter 5786: 5766: 5746: 5722: 5702: 5497: 5472: 5436: 5400: 5373: 5335: 5197: 5163: 5081: 5054: 5027: 4991: 4971: 4951: 4931: 4887: 4867: 4838: 4800: 4771: 4749: 4611: 4467: 4440: 4413: 4372: 4352: 4333: 4331: 4017: 3814: 3712: 3701: 3649: 3558: 3487: 3428: 3403: 3282: 3257: 3176: 3117: 3090: 3051: 3031: 2992: 2972: 2944: 2917: 2850: 2824: 2758: 2738: 2718: 2670: 2564: 2470: 2418: 2377: 2365: 2321: 2237: 2075: 2049: 2015: 1861:{\displaystyle A'} 1858: 1833: 1815: 1709: 1676: 1645: 1625: 1599: 1577:{\displaystyle A'} 1574: 1546: 1385: 1354: 1339: 1328: 1176:Helmholtz equation 1145: 1033: 932:Boltzmann constant 910: 812: 762: 668:Poisson's equation 611:strong electrolyte 588: 450: 416: 297: 195: 170:each other due to 156: 109: 12768:978-0-470-84293-5 12658:Chemical activity 12635: 12634: 12575: 12547: 12500: 12398: 12350: 12322: 12192: 12164: 12079: 12034: 11935: 11923: 11889: 11848: 11703: 11678: 11649: 11611: 11444:{\displaystyle r} 11345: 11323: 11262: 11240: 11209: 11187: 11139: 10976:{\displaystyle A} 10948: 10919: 10888: 10859: 10856: 10830: 10798: 10788: 10679: 10650: 10637: 10586: 10542: 10527: 10308: 10306: 10271: 10264: 10211: 10204: 10122: 10102:of the solution: 10091:{\displaystyle I} 10033: 10026: 9941: 9896: 9889: 9804: 9799: 9735: 9717: 9710: 9669: 9605: 9595: 9588: 9550: 9486: 9367: 9360: 9322: 9258: 9250: 9170: 9163: 9089: 9053: 9014: 9007: 8949: 8942: 8904: 8862:{\displaystyle T} 8832: 8804: 8797: 8670:{\displaystyle P} 8631:elementary charge 8622:{\displaystyle q} 8576:in the amount of 8565:{\displaystyle P} 8385:{\displaystyle U} 8321: 8174: 8085:{\displaystyle s} 8063:{\displaystyle i} 8006: 7908: 7745: 7707: 7655:Gibbs free energy 7646:{\displaystyle G} 7618: 7602: 7578: 7450: 7413: 7351: 7244:{\displaystyle e} 7222:{\displaystyle k} 7142: 7119: 7083: 7060: 7030:{\displaystyle V} 7004:{\displaystyle P} 6966: 6939: 6891:D&H give the 6876:{\displaystyle A} 6850:{\displaystyle T} 6824:{\displaystyle U} 6798:{\displaystyle S} 6741: 6725: 6449:{\displaystyle K} 6415:{\displaystyle i} 6362:{\displaystyle i} 6309:{\displaystyle s} 6283:{\displaystyle i} 6110: 5980: 5841:{\displaystyle B} 5821:{\displaystyle A} 5789:{\displaystyle a} 5769:{\displaystyle I} 5749:{\displaystyle z} 5700: 5697: 5676: 5296: 5267: 5254: 5125: 5037:, then eliminate 4994:{\displaystyle I} 4974:{\displaystyle R} 4954:{\displaystyle r} 4941:, then eliminate 4890:{\displaystyle r} 4848:, then eliminate 4734: 4727: 4661: 4587: 4580: 4520: 4375:{\displaystyle R} 4309: 4213: 4186: 4095: 4088: 3987: 3980: 3925: 3896: 3883: 3809: 3806: 3796: 3776: 3773: 3762: 3749: 3726:is then given by 3692: 3686: 3673: 3644: 3615: 3608: 3549: 3529: 3511: 3494: 3481: 3462: 3423: 3392: 3390: 3368: 3343: 3289: 3246: 3239: 3100:, the multiplier 3082: 3071: 3054:{\displaystyle I} 3023: 3012: 2995:{\displaystyle I} 2927:. The multiplier 2901: 2878: 2808: 2785: 2761:{\displaystyle A} 2741:{\displaystyle I} 2721:{\displaystyle A} 2650: 2562: 2468: 2465: 2439: 2319: 2291: 2219: 2202: 2013: 1990: 1945: 1559:The coefficients 1544: 1513: 1459: 1411: 1319: 1317: 1252: 1134: 1024: 1022: 904: 741: 586: 509: 13256: 13239:Electrochemistry 13218: 13217: 13210: 13204: 13203: 13196: 13190: 13184: 13178: 13177: 13167: 13140: 13134: 13121: 13115: 13114: 13112: 13111: 13091: 13085: 13084: 13066: 13060: 13059: 13052: 12983: 12980: 12974: 12971: 12965: 12964:, 30 (1929), 412 12958: 12952: 12951: 12943: 12937: 12934: 12928: 12922: 12916: 12915: 12909: 12901: 12888:Electrochemistry 12883: 12872: 12869: 12863: 12860: 12854: 12851: 12845: 12842: 12836: 12833: 12827: 12824: 12818: 12815: 12809: 12806: 12800: 12797: 12791: 12788: 12782: 12779: 12773: 12772: 12754: 12748: 12747: 12729: 12688:Ionic atmosphere 12630: 12627: 12621: 12605: 12597: 12589: 12587: 12586: 12581: 12576: 12574: 12573: 12572: 12550: 12548: 12546: 12545: 12544: 12535: 12534: 12518: 12514: 12513: 12503: 12501: 12496: 12494: 12489: 12480: 12479: 12469: 12466: 12461: 12440: 12439: 12430: 12429: 12419: 12414: 12399: 12391: 12386: 12385: 12364: 12362: 12361: 12356: 12351: 12349: 12348: 12347: 12325: 12323: 12321: 12320: 12319: 12310: 12309: 12293: 12289: 12288: 12278: 12273: 12263: 12255: 12254: 12242: 12241: 12229: 12228: 12206: 12204: 12203: 12198: 12193: 12191: 12190: 12189: 12167: 12165: 12163: 12162: 12161: 12152: 12151: 12135: 12128: 12127: 12117: 12109: 12108: 12093: 12091: 12090: 12085: 12080: 12078: 12077: 12076: 12057: 12056: 12055: 12054: 12037: 12035: 12033: 12032: 12031: 12022: 12021: 12005: 12001: 12000: 11990: 11985: 11984: 11969: 11967: 11966: 11961: 11953: 11952: 11936: 11933: 11924: 11921: 11916: 11907: 11903: 11902: 11892: 11890: 11888: 11887: 11886: 11877: 11876: 11857: 11849: 11846: 11841: 11832: 11828: 11827: 11806: 11805: 11804: 11803: 11783: 11782: 11772: 11761: 11760: 11748: 11734: 11732: 11731: 11726: 11718: 11717: 11705: 11704: 11701: 11692: 11691: 11679: 11677: 11676: 11667: 11663: 11662: 11652: 11650: 11648: 11647: 11646: 11637: 11636: 11617: 11612: 11610: 11609: 11600: 11599: 11598: 11597: 11577: 11575: 11574: 11559: 11558: 11536: 11534: 11533: 11528: 11507: 11505: 11504: 11499: 11497: 11496: 11477: 11475: 11474: 11469: 11467: 11466: 11450: 11448: 11447: 11442: 11426: 11424: 11423: 11418: 11416: 11415: 11399: 11397: 11396: 11391: 11389: 11388: 11372: 11370: 11369: 11364: 11359: 11358: 11346: 11341: 11337: 11336: 11326: 11324: 11322: 11321: 11320: 11311: 11310: 11291: 11286: 11285: 11264: 11263: 11260: 11242: 11241: 11238: 11223: 11221: 11220: 11215: 11210: 11205: 11201: 11200: 11190: 11188: 11186: 11185: 11184: 11175: 11174: 11155: 11141: 11140: 11137: 11121: 11119: 11118: 11113: 11111: 11110: 11090: 11088: 11087: 11082: 11080: 11079: 11060: 11058: 11057: 11052: 11050: 11032: 11030: 11029: 11024: 11022: 11007: 11005: 11004: 10999: 10997: 10982: 10980: 10979: 10974: 10962: 10960: 10959: 10954: 10949: 10944: 10943: 10928: 10920: 10915: 10914: 10902: 10900: 10889: 10884: 10883: 10868: 10860: 10858: 10857: 10855: 10854: 10845: 10836: 10835: 10831: 10829: 10828: 10819: 10812: 10810: 10799: 10794: 10793: 10789: 10787: 10786: 10777: 10767: 10740: 10738: 10737: 10732: 10718: 10716: 10715: 10710: 10693: 10692: 10680: 10678: 10670: 10653: 10651: 10643: 10638: 10636: 10635: 10634: 10621: 10608: 10607: 10597: 10592: 10588: 10587: 10585: 10577: 10560: 10558: 10557: 10543: 10541: 10530: 10528: 10526: 10525: 10513: 10505: 10504: 10488: 10486: 10485: 10480: 10447: 10445: 10444: 10439: 10437: 10436: 10415: 10413: 10412: 10407: 10399: 10398: 10383: 10382: 10360: 10358: 10357: 10352: 10346: 10341: 10331: 10326: 10316: 10307: 10299: 10285: 10283: 10282: 10277: 10272: 10270: 10266: 10265: 10262: 10256: 10255: 10246: 10245: 10235: 10234: 10233: 10217: 10212: 10210: 10206: 10205: 10202: 10196: 10195: 10186: 10185: 10175: 10173: 10168: 10159: 10158: 10148: 10143: 10133: 10130: 10118: 10117: 10097: 10095: 10094: 10089: 10077: 10075: 10074: 10069: 10067: 10066: 10047: 10045: 10044: 10039: 10034: 10032: 10028: 10027: 10024: 10018: 10017: 10008: 10007: 9997: 9992: 9987: 9978: 9977: 9967: 9962: 9952: 9949: 9934: 9933: 9915: 9913: 9912: 9907: 9902: 9898: 9897: 9895: 9891: 9890: 9887: 9881: 9880: 9871: 9870: 9860: 9855: 9850: 9841: 9840: 9830: 9825: 9815: 9812: 9800: 9798: 9797: 9796: 9787: 9786: 9776: 9774: 9769: 9757: 9756: 9746: 9743: 9723: 9719: 9718: 9716: 9712: 9711: 9708: 9701: 9694: 9693: 9683: 9670: 9668: 9667: 9666: 9657: 9656: 9646: 9644: 9639: 9627: 9626: 9616: 9613: 9598: 9597: 9596: 9594: 9590: 9589: 9586: 9579: 9572: 9571: 9561: 9551: 9549: 9548: 9547: 9538: 9537: 9527: 9525: 9520: 9508: 9507: 9497: 9494: 9475: 9473: 9472: 9467: 9443: 9441: 9440: 9435: 9421: 9420: 9383: 9381: 9380: 9375: 9370: 9369: 9368: 9366: 9362: 9361: 9358: 9351: 9344: 9343: 9333: 9323: 9321: 9320: 9319: 9310: 9309: 9299: 9297: 9292: 9280: 9279: 9269: 9266: 9251: 9249: 9248: 9247: 9238: 9237: 9224: 9213: 9212: 9191:Poisson equation 9186: 9184: 9183: 9178: 9173: 9172: 9171: 9169: 9165: 9164: 9161: 9154: 9147: 9146: 9136: 9125: 9120: 9108: 9107: 9097: 9085: 9084: 9072: 9071: 9061: 9030: 9028: 9027: 9022: 9017: 9016: 9015: 9013: 9009: 9008: 9005: 8998: 8991: 8990: 8980: 8969: 8964: 8952: 8951: 8950: 8948: 8944: 8943: 8940: 8933: 8926: 8925: 8915: 8905: 8900: 8899: 8890: 8885: 8884: 8868: 8866: 8865: 8860: 8844: 8842: 8841: 8836: 8834: 8833: 8830: 8817: 8815: 8814: 8809: 8807: 8806: 8805: 8803: 8799: 8798: 8795: 8788: 8781: 8780: 8770: 8753:Boltzmann factor 8750: 8748: 8747: 8742: 8739: 8734: 8718: 8716: 8715: 8710: 8708: 8707: 8691:number densities 8676: 8674: 8673: 8668: 8652: 8650: 8649: 8644: 8628: 8626: 8625: 8620: 8608: 8606: 8605: 8600: 8592: 8591: 8571: 8569: 8568: 8563: 8535: 8533: 8532: 8527: 8525: 8524: 8507:in the solution, 8502: 8500: 8499: 8494: 8492: 8491: 8472: 8470: 8469: 8464: 8453: 8452: 8443: 8442: 8432: 8427: 8395:ionic atmosphere 8391: 8389: 8388: 8383: 8364: 8362: 8361: 8356: 8354: 8353: 8335: 8333: 8332: 8327: 8322: 8317: 8316: 8315: 8300: 8299: 8289: 8284: 8283: 8271: 8270: 8254: 8252: 8251: 8246: 8244: 8243: 8219: 8217: 8216: 8211: 8209: 8208: 8186: 8184: 8183: 8178: 8176: 8175: 8172: 8153: 8151: 8150: 8145: 8143: 8142: 8120: 8118: 8117: 8112: 8110: 8109: 8091: 8089: 8088: 8083: 8069: 8067: 8066: 8061: 8046: 8044: 8043: 8038: 8027: 8026: 8008: 8007: 8004: 7995: 7994: 7982: 7981: 7971: 7966: 7948: 7947: 7929: 7927: 7926: 7921: 7909: 7907: 7906: 7897: 7896: 7887: 7879: 7878: 7866: 7865: 7849: 7847: 7846: 7841: 7836: 7835: 7823: 7822: 7799: 7797: 7796: 7791: 7789: 7788: 7766: 7764: 7763: 7758: 7746: 7744: 7743: 7734: 7720: 7708: 7700: 7679: 7677: 7676: 7671: 7652: 7650: 7649: 7644: 7632: 7630: 7629: 7624: 7619: 7611: 7603: 7598: 7590: 7579: 7574: 7560: 7539: 7537: 7536: 7531: 7513: 7511: 7510: 7505: 7503: 7502: 7490: 7489: 7471: 7469: 7468: 7463: 7451: 7449: 7448: 7439: 7438: 7429: 7414: 7409: 7408: 7399: 7391: 7390: 7372: 7370: 7369: 7364: 7352: 7350: 7349: 7340: 7339: 7330: 7322: 7321: 7303: 7301: 7300: 7295: 7290: 7289: 7277: 7276: 7250: 7248: 7247: 7242: 7228: 7226: 7225: 7220: 7205: 7203: 7202: 7197: 7195: 7194: 7182: 7181: 7156: 7154: 7153: 7148: 7143: 7141: 7133: 7125: 7120: 7118: 7117: 7105: 7097: 7095: 7094: 7089: 7084: 7082: 7074: 7066: 7061: 7053: 7036: 7034: 7033: 7028: 7010: 7008: 7007: 7002: 6987: 6985: 6984: 6979: 6967: 6965: 6964: 6952: 6940: 6932: 6914: 6912: 6911: 6906: 6882: 6880: 6879: 6874: 6856: 6854: 6853: 6848: 6830: 6828: 6827: 6822: 6804: 6802: 6801: 6796: 6778: 6776: 6775: 6770: 6755: 6753: 6752: 6747: 6742: 6734: 6726: 6718: 6693: 6691: 6690: 6685: 6673: 6671: 6670: 6665: 6651:D&H use the 6643: 6641: 6640: 6635: 6627: 6626: 6608: 6607: 6597: 6592: 6552: 6550: 6549: 6544: 6542: 6541: 6525: 6523: 6522: 6517: 6498: 6496: 6495: 6490: 6478: 6476: 6475: 6470: 6455: 6453: 6452: 6447: 6421: 6419: 6418: 6413: 6397: 6395: 6394: 6389: 6387: 6386: 6368: 6366: 6365: 6360: 6344: 6342: 6341: 6336: 6334: 6333: 6315: 6313: 6312: 6307: 6289: 6287: 6286: 6281: 6263: 6261: 6260: 6255: 6240: 6238: 6237: 6232: 6220: 6219: 6218: 6208: 6198: 6193: 6147: 6121: 6119: 6118: 6113: 6111: 6106: 6100: 6095: 6070: 6065: 6053: 6052: 6025: 6023: 6022: 6017: 6014: 6009: 5991: 5989: 5988: 5983: 5981: 5976: 5967: 5962: 5950: 5949: 5925:Kohlrausch's Law 5853:Pitzer equations 5847: 5845: 5844: 5839: 5827: 5825: 5824: 5819: 5795: 5793: 5792: 5787: 5775: 5773: 5772: 5767: 5755: 5753: 5752: 5747: 5734:common logarithm 5731: 5729: 5728: 5723: 5711: 5709: 5708: 5703: 5701: 5699: 5698: 5693: 5678: 5677: 5672: 5670: 5665: 5664: 5655: 5654: 5645: 5636: 5619: 5618: 5591:Pitzer equations 5583:ionic atmosphere 5506: 5504: 5503: 5498: 5496: 5495: 5445: 5443: 5442: 5437: 5435: 5434: 5409: 5407: 5406: 5401: 5399: 5398: 5382: 5380: 5379: 5374: 5372: 5371: 5344: 5342: 5341: 5336: 5319: 5318: 5297: 5295: 5287: 5270: 5268: 5260: 5255: 5253: 5252: 5251: 5238: 5225: 5224: 5214: 5206: 5204: 5203: 5198: 5172: 5170: 5169: 5164: 5162: 5161: 5146: 5145: 5134: 5133: 5126: 5124: 5116: 5099: 5090: 5088: 5087: 5082: 5080: 5079: 5063: 5061: 5060: 5055: 5053: 5052: 5036: 5034: 5033: 5028: 5026: 5025: 5000: 4998: 4997: 4992: 4980: 4978: 4977: 4972: 4960: 4958: 4957: 4952: 4940: 4938: 4937: 4932: 4915: 4914: 4913: 4896: 4894: 4893: 4888: 4876: 4874: 4873: 4868: 4847: 4845: 4844: 4839: 4809: 4807: 4806: 4801: 4780: 4778: 4777: 4772: 4758: 4756: 4755: 4750: 4748: 4747: 4735: 4733: 4729: 4728: 4725: 4719: 4718: 4709: 4708: 4689: 4688: 4687: 4678: 4677: 4667: 4662: 4660: 4659: 4658: 4649: 4648: 4638: 4637: 4628: 4620: 4618: 4617: 4612: 4610: 4609: 4588: 4586: 4582: 4581: 4578: 4572: 4571: 4562: 4561: 4551: 4547: 4546: 4537: 4536: 4526: 4521: 4519: 4518: 4517: 4508: 4507: 4497: 4496: 4487: 4476: 4474: 4473: 4468: 4466: 4465: 4449: 4447: 4446: 4441: 4439: 4438: 4422: 4420: 4419: 4414: 4412: 4411: 4385:screening length 4381: 4379: 4378: 4373: 4361: 4359: 4358: 4353: 4342: 4340: 4339: 4334: 4332: 4310: 4302: 4293: 4292: 4279: 4278: 4262: 4261: 4245: 4244: 4228: 4227: 4214: 4212: 4211: 4202: 4201: 4200: 4188: 4187: 4184: 4174: 4165: 4164: 4151: 4150: 4134: 4133: 4096: 4094: 4090: 4089: 4086: 4079: 4062: 4053: 4052: 4026: 4024: 4023: 4018: 4001: 4000: 3988: 3986: 3982: 3981: 3978: 3972: 3971: 3962: 3961: 3951: 3931: 3926: 3924: 3916: 3899: 3897: 3889: 3884: 3882: 3881: 3880: 3867: 3854: 3853: 3843: 3823: 3821: 3820: 3815: 3810: 3808: 3807: 3804: 3798: 3797: 3794: 3788: 3777: 3775: 3774: 3771: 3765: 3764: 3763: 3760: 3751: 3750: 3747: 3740: 3721: 3719: 3718: 3713: 3711: 3710: 3700: 3688: 3687: 3684: 3675: 3674: 3671: 3658: 3656: 3655: 3650: 3645: 3643: 3635: 3634: 3633: 3629: 3620: 3616: 3614: 3610: 3609: 3606: 3600: 3599: 3590: 3589: 3579: 3578: 3577: 3568: 3567: 3557: 3541: 3531: 3530: 3527: 3520: 3512: 3510: 3502: 3500: 3495: 3492: 3483: 3482: 3479: 3472: 3464: 3463: 3460: 3442:osmotic pressure 3437: 3435: 3434: 3429: 3424: 3419: 3416: 3411: 3393: 3391: 3383: 3381: 3380: 3371: 3369: 3367: 3366: 3365: 3361: 3345: 3344: 3341: 3335: 3334: 3325: 3324: 3305: 3303: 3299: 3290: 3287: 3281: 3280: 3270: 3265: 3255: 3247: 3245: 3241: 3240: 3237: 3231: 3230: 3221: 3220: 3204: 3200: 3199: 3189: 3184: 3174: 3163: 3162: 3126: 3124: 3123: 3118: 3116: 3115: 3099: 3097: 3096: 3091: 3089: 3088: 3083: 3080: 3077: 3072: 3069: 3060: 3058: 3057: 3052: 3040: 3038: 3037: 3032: 3030: 3029: 3024: 3021: 3018: 3013: 3010: 3001: 2999: 2998: 2993: 2981: 2979: 2978: 2973: 2968: 2953: 2951: 2950: 2945: 2943: 2942: 2926: 2924: 2923: 2918: 2916: 2915: 2911: 2902: 2899: 2896: 2895: 2891: 2879: 2876: 2859: 2857: 2856: 2851: 2833: 2831: 2830: 2825: 2823: 2822: 2818: 2809: 2806: 2803: 2802: 2798: 2786: 2783: 2767: 2765: 2764: 2759: 2747: 2745: 2744: 2739: 2727: 2725: 2724: 2719: 2679: 2677: 2676: 2671: 2669: 2668: 2664: 2655: 2651: 2649: 2645: 2644: 2643: 2633: 2632: 2623: 2622: 2612: 2608: 2607: 2594: 2573: 2571: 2570: 2565: 2563: 2561: 2557: 2556: 2555: 2545: 2544: 2535: 2534: 2512: 2508: 2507: 2497: 2479: 2477: 2476: 2471: 2469: 2467: 2466: 2461: 2459: 2458: 2435: 2434: 2431: 2426: 2408: 2407: 2395: 2394: 2374: 2372: 2371: 2366: 2364: 2363: 2330: 2328: 2327: 2322: 2320: 2318: 2317: 2316: 2294: 2292: 2290: 2289: 2288: 2279: 2278: 2262: 2261: 2260: 2250: 2245: 2235: 2227: 2226: 2220: 2218: 2217: 2205: 2203: 2201: 2200: 2199: 2190: 2189: 2173: 2169: 2168: 2158: 2150: 2149: 2137: 2136: 2127: 2126: 2117: 2116: 2104: 2103: 2084: 2082: 2081: 2076: 2059:, of the ion at 2058: 2056: 2055: 2050: 2048: 2047: 2024: 2022: 2021: 2016: 2014: 2009: 2008: 1993: 1991: 1989: 1988: 1987: 1968: 1967: 1966: 1965: 1948: 1946: 1944: 1943: 1942: 1933: 1932: 1916: 1912: 1911: 1901: 1887: 1886: 1867: 1865: 1864: 1859: 1857: 1842: 1840: 1839: 1834: 1828: 1823: 1814: 1813: 1804: 1803: 1785: 1777: 1776: 1758: 1757: 1745: 1744: 1735: 1734: 1718: 1716: 1715: 1710: 1708: 1707: 1685: 1683: 1682: 1677: 1669: 1654: 1652: 1651: 1646: 1634: 1632: 1631: 1626: 1608: 1606: 1605: 1600: 1598: 1583: 1581: 1580: 1575: 1573: 1555: 1553: 1552: 1547: 1545: 1540: 1539: 1527: 1525: 1514: 1509: 1508: 1493: 1491: 1471: 1470: 1460: 1457: 1448: 1447: 1435: 1434: 1422: 1421: 1412: 1404: 1398: 1393: 1363: 1361: 1360: 1355: 1352: 1347: 1338: 1337: 1327: 1318: 1316: 1312: 1311: 1310: 1300: 1299: 1290: 1289: 1279: 1278: 1269: 1264: 1263: 1253: 1250: 1238: 1237: 1228: 1227: 1206: 1205: 1196: 1195: 1154: 1152: 1151: 1146: 1144: 1140: 1139: 1135: 1133: 1129: 1128: 1127: 1116: 1106: 1105: 1093: 1092: 1079: 1061: 1060: 1048: 1047: 1032: 1023: 1021: 1020: 1019: 1010: 1009: 996: 979: 978: 969: 968: 919: 917: 916: 911: 909: 905: 903: 899: 898: 897: 886: 876: 875: 863: 862: 849: 837: 836: 820: 771: 769: 768: 763: 752: 751: 742: 740: 739: 738: 729: 728: 715: 698: 697: 688: 687: 640:electrostriction 597: 595: 594: 589: 587: 585: 584: 583: 574: 573: 564: 563: 547: 546: 545: 536: 535: 526: 525: 515: 510: 507: 425: 423: 422: 417: 415: 414: 398: 389: 385: 384: 383: 378: 377: 376: 365: 364: 359: 358: 357: 337: 336: 306: 304: 303: 298: 296: 295: 291: 282: 278: 277: 276: 275: 274: 273: 255: 254: 253: 252: 251: 224: 223: 204: 202: 201: 196: 194: 193: 165: 163: 162: 157: 155: 154: 145: 118: 116: 115: 110: 23:was proposed by 13264: 13263: 13259: 13258: 13257: 13255: 13254: 13253: 13224: 13223: 13222: 13221: 13212: 13211: 13207: 13198: 13197: 13193: 13185: 13181: 13158:(11): 775–786. 13141: 13137: 13131:Wayback Machine 13122: 13118: 13109: 13107: 13092: 13088: 13067: 13063: 13054: 13053: 12986: 12981: 12977: 12972: 12968: 12959: 12955: 12948:Ion Association 12944: 12940: 12935: 12931: 12923: 12919: 12903: 12902: 12898: 12884: 12875: 12870: 12866: 12861: 12857: 12852: 12848: 12843: 12839: 12834: 12830: 12825: 12821: 12816: 12812: 12807: 12803: 12798: 12794: 12789: 12785: 12780: 12776: 12769: 12755: 12751: 12730: 12726: 12721: 12708:Pitzer equation 12703:Davies equation 12698:Ion association 12649: 12640: 12631: 12625: 12622: 12615: 12606: 12595: 12568: 12564: 12554: 12549: 12540: 12536: 12530: 12526: 12519: 12509: 12505: 12504: 12502: 12490: 12485: 12475: 12471: 12470: 12468: 12462: 12451: 12435: 12431: 12425: 12421: 12415: 12404: 12390: 12381: 12377: 12375: 12372: 12371: 12343: 12339: 12329: 12324: 12315: 12311: 12305: 12301: 12294: 12284: 12280: 12274: 12269: 12264: 12262: 12250: 12246: 12237: 12233: 12224: 12220: 12218: 12215: 12214: 12185: 12181: 12171: 12166: 12157: 12153: 12147: 12143: 12136: 12123: 12119: 12118: 12116: 12104: 12100: 12098: 12095: 12094: 12072: 12068: 12058: 12050: 12046: 12042: 12038: 12036: 12027: 12023: 12017: 12013: 12006: 11996: 11992: 11991: 11989: 11980: 11976: 11974: 11971: 11970: 11948: 11944: 11932: 11917: 11912: 11898: 11894: 11893: 11891: 11882: 11878: 11872: 11868: 11861: 11856: 11842: 11837: 11823: 11819: 11799: 11795: 11788: 11784: 11778: 11774: 11773: 11771: 11756: 11752: 11741: 11739: 11736: 11735: 11713: 11709: 11700: 11696: 11687: 11683: 11672: 11668: 11658: 11654: 11653: 11651: 11642: 11638: 11632: 11628: 11621: 11616: 11605: 11601: 11593: 11589: 11582: 11578: 11576: 11570: 11566: 11554: 11550: 11542: 11539: 11538: 11513: 11510: 11509: 11492: 11488: 11486: 11483: 11482: 11462: 11458: 11456: 11453: 11452: 11436: 11433: 11432: 11411: 11407: 11405: 11402: 11401: 11384: 11380: 11378: 11375: 11374: 11354: 11350: 11332: 11328: 11327: 11325: 11316: 11312: 11306: 11302: 11295: 11290: 11281: 11277: 11259: 11255: 11237: 11233: 11231: 11228: 11227: 11196: 11192: 11191: 11189: 11180: 11176: 11170: 11166: 11159: 11154: 11136: 11132: 11130: 11127: 11126: 11106: 11102: 11100: 11097: 11096: 11075: 11071: 11069: 11066: 11065: 11043: 11041: 11038: 11037: 11015: 11013: 11010: 11009: 10990: 10988: 10985: 10984: 10968: 10965: 10964: 10933: 10929: 10927: 10907: 10903: 10901: 10893: 10873: 10869: 10867: 10850: 10846: 10844: 10837: 10824: 10820: 10818: 10817: 10813: 10811: 10803: 10782: 10778: 10776: 10772: 10768: 10766: 10746: 10743: 10742: 10726: 10723: 10722: 10688: 10684: 10671: 10654: 10652: 10642: 10630: 10626: 10622: 10603: 10599: 10598: 10596: 10578: 10561: 10559: 10553: 10549: 10548: 10544: 10534: 10529: 10521: 10517: 10512: 10500: 10496: 10494: 10491: 10490: 10468: 10465: 10464: 10429: 10425: 10423: 10420: 10419: 10394: 10390: 10378: 10374: 10372: 10369: 10368: 10342: 10337: 10327: 10322: 10312: 10298: 10290: 10287: 10286: 10261: 10257: 10251: 10247: 10241: 10237: 10236: 10229: 10225: 10218: 10216: 10201: 10197: 10191: 10187: 10181: 10177: 10176: 10169: 10164: 10154: 10150: 10144: 10139: 10134: 10132: 10126: 10113: 10109: 10107: 10104: 10103: 10083: 10080: 10079: 10062: 10058: 10056: 10053: 10052: 10023: 10019: 10013: 10009: 10003: 9999: 9998: 9988: 9983: 9973: 9969: 9963: 9958: 9953: 9951: 9945: 9929: 9925: 9923: 9920: 9919: 9886: 9882: 9876: 9872: 9866: 9862: 9861: 9851: 9846: 9836: 9832: 9826: 9821: 9816: 9814: 9808: 9792: 9788: 9782: 9778: 9777: 9770: 9765: 9752: 9748: 9747: 9745: 9739: 9734: 9730: 9707: 9703: 9702: 9689: 9685: 9684: 9682: 9675: 9671: 9662: 9658: 9652: 9648: 9647: 9640: 9635: 9622: 9618: 9617: 9615: 9609: 9585: 9581: 9580: 9567: 9563: 9562: 9560: 9556: 9552: 9543: 9539: 9533: 9529: 9528: 9521: 9516: 9503: 9499: 9498: 9496: 9490: 9481: 9478: 9477: 9449: 9446: 9445: 9416: 9412: 9410: 9407: 9406: 9357: 9353: 9352: 9339: 9335: 9334: 9332: 9328: 9324: 9315: 9311: 9305: 9301: 9300: 9293: 9288: 9275: 9271: 9270: 9268: 9262: 9243: 9239: 9233: 9229: 9228: 9223: 9208: 9204: 9202: 9199: 9198: 9160: 9156: 9155: 9142: 9138: 9137: 9135: 9131: 9127: 9121: 9116: 9103: 9099: 9093: 9080: 9076: 9067: 9063: 9057: 9045: 9042: 9041: 9004: 9000: 8999: 8986: 8982: 8981: 8979: 8975: 8971: 8965: 8960: 8939: 8935: 8934: 8921: 8917: 8916: 8914: 8910: 8906: 8895: 8891: 8889: 8880: 8876: 8874: 8871: 8870: 8854: 8851: 8850: 8829: 8825: 8823: 8820: 8819: 8794: 8790: 8789: 8776: 8772: 8771: 8769: 8765: 8761: 8759: 8756: 8755: 8735: 8730: 8724: 8721: 8720: 8703: 8699: 8697: 8694: 8693: 8689:. All species' 8662: 8659: 8658: 8638: 8635: 8634: 8614: 8611: 8610: 8587: 8583: 8581: 8578: 8577: 8557: 8554: 8553: 8520: 8516: 8514: 8511: 8510: 8487: 8483: 8481: 8478: 8477: 8448: 8444: 8438: 8434: 8428: 8417: 8411: 8408: 8407: 8404: 8377: 8374: 8373: 8349: 8345: 8343: 8340: 8339: 8311: 8307: 8295: 8291: 8290: 8288: 8279: 8275: 8266: 8262: 8260: 8257: 8256: 8239: 8235: 8233: 8230: 8229: 8204: 8200: 8198: 8195: 8194: 8171: 8167: 8165: 8162: 8161: 8138: 8134: 8132: 8129: 8128: 8105: 8101: 8099: 8096: 8095: 8077: 8074: 8073: 8055: 8052: 8051: 8022: 8018: 8003: 7999: 7990: 7986: 7977: 7973: 7967: 7956: 7943: 7939: 7937: 7934: 7933: 7902: 7898: 7892: 7888: 7886: 7874: 7870: 7861: 7857: 7855: 7852: 7851: 7831: 7827: 7818: 7814: 7806: 7803: 7802: 7784: 7780: 7778: 7775: 7774: 7739: 7735: 7721: 7719: 7699: 7685: 7682: 7681: 7665: 7662: 7661: 7638: 7635: 7634: 7610: 7591: 7589: 7561: 7559: 7545: 7542: 7541: 7525: 7522: 7521: 7498: 7494: 7485: 7481: 7479: 7476: 7475: 7444: 7440: 7434: 7430: 7428: 7404: 7400: 7398: 7386: 7382: 7380: 7377: 7376: 7345: 7341: 7335: 7331: 7329: 7317: 7313: 7311: 7308: 7307: 7285: 7281: 7272: 7268: 7260: 7257: 7256: 7236: 7233: 7232: 7214: 7211: 7210: 7190: 7186: 7177: 7173: 7165: 7162: 7161: 7134: 7126: 7124: 7113: 7109: 7104: 7102: 7099: 7098: 7075: 7067: 7065: 7052: 7050: 7047: 7046: 7022: 7019: 7018: 6996: 6993: 6992: 6960: 6956: 6951: 6931: 6920: 6917: 6916: 6900: 6897: 6896: 6868: 6865: 6864: 6842: 6839: 6838: 6833:internal energy 6816: 6813: 6812: 6790: 6787: 6786: 6764: 6761: 6760: 6733: 6717: 6703: 6700: 6699: 6679: 6676: 6675: 6659: 6656: 6655: 6649: 6622: 6618: 6603: 6599: 6593: 6582: 6558: 6555: 6554: 6537: 6533: 6531: 6528: 6527: 6511: 6508: 6507: 6484: 6481: 6480: 6461: 6458: 6457: 6441: 6438: 6437: 6407: 6404: 6403: 6382: 6378: 6376: 6373: 6372: 6354: 6351: 6350: 6329: 6325: 6323: 6320: 6319: 6301: 6298: 6297: 6275: 6272: 6271: 6249: 6246: 6245: 6214: 6210: 6209: 6204: 6194: 6183: 6177: 6174: 6173: 6166: 6140: 6105: 6096: 6091: 6066: 6061: 6048: 6044: 6042: 6039: 6038: 6010: 6005: 5999: 5996: 5995: 5975: 5963: 5958: 5945: 5941: 5939: 5936: 5935: 5930: 5921:electrophoretic 5892: 5880: 5833: 5830: 5829: 5813: 5810: 5809: 5796:is the size or 5781: 5778: 5777: 5761: 5758: 5757: 5741: 5738: 5737: 5717: 5714: 5713: 5692: 5679: 5671: 5666: 5660: 5656: 5650: 5646: 5641: 5637: 5635: 5614: 5610: 5605: 5602: 5601: 5587:Davies equation 5567:solvation shell 5557: 5518:Ion association 5491: 5487: 5479: 5476: 5475: 5453: 5447:work, however. 5430: 5426: 5415: 5412: 5411: 5394: 5390: 5388: 5385: 5384: 5367: 5363: 5352: 5349: 5348: 5314: 5310: 5288: 5271: 5269: 5259: 5247: 5243: 5239: 5220: 5216: 5215: 5213: 5211: 5208: 5207: 5177: 5174: 5173: 5157: 5153: 5135: 5129: 5128: 5127: 5117: 5100: 5098: 5096: 5093: 5092: 5075: 5071: 5069: 5066: 5065: 5048: 5044: 5042: 5039: 5038: 5021: 5017: 5006: 5003: 5002: 4986: 4983: 4982: 4966: 4963: 4962: 4946: 4943: 4942: 4909: 4905: 4904: 4902: 4899: 4898: 4882: 4879: 4878: 4853: 4850: 4849: 4815: 4812: 4811: 4786: 4783: 4782: 4766: 4763: 4762: 4743: 4739: 4724: 4720: 4714: 4710: 4704: 4700: 4690: 4683: 4679: 4673: 4669: 4668: 4666: 4654: 4650: 4644: 4640: 4639: 4633: 4629: 4627: 4625: 4622: 4621: 4605: 4601: 4577: 4573: 4567: 4563: 4557: 4553: 4552: 4542: 4538: 4532: 4528: 4527: 4525: 4513: 4509: 4503: 4499: 4498: 4492: 4488: 4486: 4484: 4481: 4480: 4461: 4457: 4455: 4452: 4451: 4434: 4430: 4428: 4425: 4424: 4407: 4403: 4392: 4389: 4388: 4367: 4364: 4363: 4347: 4344: 4343: 4330: 4329: 4301: 4294: 4288: 4284: 4281: 4280: 4274: 4270: 4263: 4257: 4253: 4250: 4249: 4240: 4236: 4229: 4223: 4219: 4216: 4215: 4207: 4203: 4196: 4192: 4183: 4179: 4175: 4173: 4166: 4160: 4156: 4153: 4152: 4146: 4142: 4135: 4129: 4125: 4122: 4121: 4085: 4081: 4080: 4063: 4061: 4054: 4048: 4044: 4040: 4038: 4035: 4034: 3996: 3992: 3977: 3973: 3967: 3963: 3957: 3953: 3952: 3932: 3930: 3917: 3900: 3898: 3888: 3876: 3872: 3868: 3849: 3845: 3844: 3842: 3840: 3837: 3836: 3829: 3803: 3799: 3793: 3789: 3787: 3770: 3766: 3759: 3755: 3746: 3742: 3741: 3739: 3731: 3728: 3727: 3706: 3702: 3696: 3683: 3679: 3670: 3666: 3664: 3661: 3660: 3636: 3625: 3621: 3605: 3601: 3595: 3591: 3585: 3581: 3580: 3573: 3569: 3563: 3559: 3553: 3542: 3540: 3536: 3535: 3526: 3522: 3521: 3519: 3503: 3496: 3491: 3478: 3474: 3473: 3471: 3459: 3455: 3453: 3450: 3449: 3418: 3412: 3407: 3382: 3376: 3372: 3370: 3357: 3353: 3349: 3340: 3336: 3330: 3326: 3320: 3316: 3306: 3295: 3291: 3286: 3276: 3272: 3266: 3261: 3256: 3254: 3236: 3232: 3226: 3222: 3216: 3212: 3205: 3195: 3191: 3185: 3180: 3175: 3173: 3158: 3154: 3143: 3140: 3139: 3130: 3111: 3107: 3105: 3102: 3101: 3084: 3079: 3078: 3073: 3068: 3066: 3063: 3062: 3046: 3043: 3042: 3025: 3020: 3019: 3014: 3009: 3007: 3004: 3003: 2987: 2984: 2983: 2964: 2959: 2956: 2955: 2938: 2934: 2932: 2929: 2928: 2907: 2903: 2898: 2897: 2887: 2880: 2875: 2874: 2869: 2866: 2865: 2839: 2836: 2835: 2814: 2810: 2805: 2804: 2794: 2787: 2782: 2781: 2776: 2773: 2772: 2753: 2750: 2749: 2733: 2730: 2729: 2713: 2710: 2709: 2697: 2660: 2656: 2639: 2638: 2634: 2628: 2624: 2618: 2614: 2613: 2603: 2599: 2595: 2593: 2589: 2588: 2580: 2577: 2576: 2551: 2550: 2546: 2540: 2536: 2530: 2526: 2513: 2503: 2499: 2498: 2496: 2488: 2485: 2484: 2460: 2454: 2450: 2440: 2433: 2427: 2422: 2403: 2399: 2390: 2386: 2384: 2381: 2380: 2359: 2355: 2347: 2344: 2343: 2312: 2308: 2298: 2293: 2284: 2280: 2274: 2270: 2263: 2256: 2252: 2246: 2241: 2236: 2234: 2222: 2221: 2213: 2209: 2204: 2195: 2191: 2185: 2181: 2174: 2164: 2160: 2159: 2157: 2145: 2141: 2132: 2128: 2122: 2121: 2112: 2108: 2099: 2095: 2093: 2090: 2089: 2064: 2061: 2060: 2043: 2039: 2037: 2034: 2033: 1998: 1994: 1992: 1983: 1979: 1969: 1961: 1957: 1953: 1949: 1947: 1938: 1934: 1928: 1924: 1917: 1907: 1903: 1902: 1900: 1882: 1878: 1876: 1873: 1872: 1850: 1848: 1845: 1844: 1824: 1819: 1809: 1805: 1799: 1795: 1781: 1772: 1768: 1753: 1749: 1740: 1736: 1730: 1726: 1724: 1721: 1720: 1703: 1699: 1691: 1688: 1687: 1662: 1660: 1657: 1656: 1640: 1637: 1636: 1614: 1611: 1610: 1591: 1589: 1586: 1585: 1566: 1564: 1561: 1560: 1532: 1528: 1526: 1518: 1498: 1494: 1492: 1484: 1466: 1462: 1456: 1443: 1439: 1430: 1426: 1417: 1413: 1403: 1394: 1389: 1380: 1377: 1376: 1348: 1343: 1333: 1329: 1323: 1306: 1305: 1301: 1295: 1291: 1285: 1281: 1280: 1274: 1270: 1268: 1259: 1255: 1249: 1233: 1229: 1223: 1219: 1201: 1197: 1191: 1187: 1185: 1182: 1181: 1173: 1169: 1123: 1122: 1118: 1117: 1101: 1097: 1088: 1084: 1080: 1078: 1074: 1056: 1052: 1043: 1039: 1038: 1034: 1028: 1015: 1011: 1005: 1001: 1000: 995: 974: 970: 964: 960: 958: 955: 954: 929: 893: 892: 888: 887: 871: 867: 858: 854: 850: 848: 844: 832: 828: 816: 810: 807: 806: 747: 743: 734: 730: 724: 720: 719: 714: 693: 689: 683: 679: 677: 674: 673: 656: 579: 575: 569: 565: 559: 555: 548: 541: 537: 531: 527: 521: 517: 516: 514: 506: 504: 501: 500: 495: 488:in a medium of 472: 462: 442: 437: 394: 390: 379: 372: 368: 367: 366: 360: 353: 349: 348: 347: 346: 342: 341: 332: 328: 326: 323: 322: 317: 313: 287: 283: 269: 265: 261: 260: 256: 247: 243: 239: 238: 234: 233: 229: 228: 219: 215: 213: 210: 209: 189: 185: 183: 180: 179: 150: 146: 141: 127: 124: 123: 104: 101: 100: 53: 17: 12: 11: 5: 13262: 13252: 13251: 13246: 13241: 13236: 13220: 13219: 13205: 13191: 13179: 13135: 13133:, section 19.3 13116: 13086: 13061: 12984: 12975: 12966: 12953: 12938: 12929: 12917: 12896: 12873: 12864: 12855: 12846: 12837: 12828: 12819: 12810: 12801: 12792: 12783: 12774: 12767: 12749: 12723: 12722: 12720: 12717: 12716: 12715: 12710: 12705: 12700: 12695: 12690: 12685: 12680: 12678:Bjerrum length 12675: 12670: 12665: 12663:Ionic strength 12660: 12655: 12648: 12645: 12639: 12636: 12633: 12632: 12609: 12607: 12600: 12594: 12591: 12579: 12571: 12567: 12563: 12560: 12557: 12553: 12543: 12539: 12533: 12529: 12525: 12522: 12517: 12512: 12508: 12499: 12493: 12488: 12484: 12478: 12474: 12465: 12460: 12457: 12454: 12450: 12446: 12443: 12438: 12434: 12428: 12424: 12418: 12413: 12410: 12407: 12403: 12397: 12394: 12389: 12384: 12380: 12354: 12346: 12342: 12338: 12335: 12332: 12328: 12318: 12314: 12308: 12304: 12300: 12297: 12292: 12287: 12283: 12277: 12272: 12268: 12261: 12258: 12253: 12249: 12245: 12240: 12236: 12232: 12227: 12223: 12196: 12188: 12184: 12180: 12177: 12174: 12170: 12160: 12156: 12150: 12146: 12142: 12139: 12134: 12131: 12126: 12122: 12115: 12112: 12107: 12103: 12083: 12075: 12071: 12067: 12064: 12061: 12053: 12049: 12045: 12041: 12030: 12026: 12020: 12016: 12012: 12009: 12004: 11999: 11995: 11988: 11983: 11979: 11959: 11956: 11951: 11947: 11943: 11939: 11931: 11927: 11920: 11915: 11911: 11906: 11901: 11897: 11885: 11881: 11875: 11871: 11867: 11864: 11860: 11855: 11852: 11845: 11840: 11836: 11831: 11826: 11822: 11818: 11815: 11812: 11809: 11802: 11798: 11794: 11791: 11787: 11781: 11777: 11770: 11767: 11764: 11759: 11755: 11751: 11747: 11744: 11724: 11721: 11716: 11712: 11708: 11699: 11695: 11690: 11686: 11682: 11675: 11671: 11666: 11661: 11657: 11645: 11641: 11635: 11631: 11627: 11624: 11620: 11615: 11608: 11604: 11596: 11592: 11588: 11585: 11581: 11573: 11569: 11565: 11562: 11557: 11553: 11549: 11546: 11526: 11523: 11520: 11517: 11495: 11491: 11465: 11461: 11440: 11414: 11410: 11387: 11383: 11362: 11357: 11353: 11349: 11344: 11340: 11335: 11331: 11319: 11315: 11309: 11305: 11301: 11298: 11294: 11289: 11284: 11280: 11276: 11273: 11270: 11267: 11258: 11254: 11251: 11248: 11245: 11236: 11213: 11208: 11204: 11199: 11195: 11183: 11179: 11173: 11169: 11165: 11162: 11158: 11153: 11150: 11147: 11144: 11135: 11109: 11105: 11078: 11074: 11061:must be zero. 11049: 11046: 11021: 11018: 10996: 10993: 10972: 10952: 10947: 10942: 10939: 10936: 10932: 10926: 10923: 10918: 10913: 10910: 10906: 10899: 10896: 10892: 10887: 10882: 10879: 10876: 10872: 10866: 10863: 10853: 10849: 10843: 10840: 10834: 10827: 10823: 10816: 10809: 10806: 10802: 10797: 10792: 10785: 10781: 10775: 10771: 10765: 10762: 10759: 10756: 10753: 10750: 10730: 10708: 10705: 10702: 10699: 10696: 10691: 10687: 10683: 10677: 10674: 10669: 10666: 10663: 10660: 10657: 10649: 10646: 10641: 10633: 10629: 10625: 10620: 10617: 10614: 10611: 10606: 10602: 10595: 10591: 10584: 10581: 10576: 10573: 10570: 10567: 10564: 10556: 10552: 10547: 10540: 10537: 10533: 10524: 10520: 10516: 10511: 10508: 10503: 10499: 10478: 10475: 10472: 10448:is called the 10435: 10432: 10428: 10405: 10402: 10397: 10393: 10389: 10386: 10381: 10377: 10350: 10345: 10340: 10336: 10330: 10325: 10321: 10315: 10311: 10305: 10302: 10297: 10294: 10275: 10269: 10260: 10254: 10250: 10244: 10240: 10232: 10228: 10224: 10221: 10215: 10209: 10200: 10194: 10190: 10184: 10180: 10172: 10167: 10163: 10157: 10153: 10147: 10142: 10138: 10129: 10125: 10121: 10116: 10112: 10100:ionic strength 10087: 10065: 10061: 10037: 10031: 10022: 10016: 10012: 10006: 10002: 9996: 9991: 9986: 9982: 9976: 9972: 9966: 9961: 9957: 9948: 9944: 9940: 9937: 9932: 9928: 9905: 9901: 9894: 9885: 9879: 9875: 9869: 9865: 9859: 9854: 9849: 9845: 9839: 9835: 9829: 9824: 9820: 9811: 9807: 9803: 9795: 9791: 9785: 9781: 9773: 9768: 9764: 9760: 9755: 9751: 9742: 9738: 9733: 9729: 9726: 9722: 9715: 9706: 9700: 9697: 9692: 9688: 9681: 9678: 9674: 9665: 9661: 9655: 9651: 9643: 9638: 9634: 9630: 9625: 9621: 9612: 9608: 9604: 9601: 9593: 9584: 9578: 9575: 9570: 9566: 9559: 9555: 9546: 9542: 9536: 9532: 9524: 9519: 9515: 9511: 9506: 9502: 9493: 9489: 9485: 9465: 9462: 9459: 9456: 9453: 9433: 9430: 9427: 9424: 9419: 9415: 9373: 9365: 9356: 9350: 9347: 9342: 9338: 9331: 9327: 9318: 9314: 9308: 9304: 9296: 9291: 9287: 9283: 9278: 9274: 9265: 9261: 9257: 9254: 9246: 9242: 9236: 9232: 9227: 9222: 9219: 9216: 9211: 9207: 9176: 9168: 9159: 9153: 9150: 9145: 9141: 9134: 9130: 9124: 9119: 9115: 9111: 9106: 9102: 9096: 9092: 9088: 9083: 9079: 9075: 9070: 9066: 9060: 9056: 9052: 9049: 9038:charge density 9020: 9012: 9003: 8997: 8994: 8989: 8985: 8978: 8974: 8968: 8963: 8959: 8955: 8947: 8938: 8932: 8929: 8924: 8920: 8913: 8909: 8903: 8898: 8894: 8888: 8883: 8879: 8858: 8828: 8802: 8793: 8787: 8784: 8779: 8775: 8768: 8764: 8738: 8733: 8729: 8706: 8702: 8666: 8642: 8618: 8598: 8595: 8590: 8586: 8561: 8546: 8545: 8523: 8519: 8508: 8490: 8486: 8462: 8459: 8456: 8451: 8447: 8441: 8437: 8431: 8426: 8423: 8420: 8416: 8403: 8400: 8381: 8352: 8348: 8325: 8320: 8314: 8310: 8306: 8303: 8298: 8294: 8287: 8282: 8278: 8274: 8269: 8265: 8242: 8238: 8226: 8225: 8207: 8203: 8192: 8170: 8159: 8141: 8137: 8126: 8108: 8104: 8093: 8081: 8071: 8059: 8036: 8033: 8030: 8025: 8021: 8017: 8014: 8011: 8002: 7998: 7993: 7989: 7985: 7980: 7976: 7970: 7965: 7962: 7959: 7955: 7951: 7946: 7942: 7919: 7916: 7913: 7905: 7901: 7895: 7891: 7885: 7882: 7877: 7873: 7869: 7864: 7860: 7839: 7834: 7830: 7826: 7821: 7817: 7813: 7810: 7787: 7783: 7756: 7753: 7750: 7742: 7738: 7733: 7730: 7727: 7724: 7718: 7715: 7712: 7706: 7703: 7698: 7695: 7692: 7689: 7669: 7642: 7622: 7617: 7614: 7609: 7606: 7601: 7597: 7594: 7588: 7585: 7582: 7577: 7573: 7570: 7567: 7564: 7558: 7555: 7552: 7549: 7529: 7501: 7497: 7493: 7488: 7484: 7461: 7458: 7455: 7447: 7443: 7437: 7433: 7427: 7424: 7421: 7418: 7412: 7407: 7403: 7397: 7394: 7389: 7385: 7362: 7359: 7356: 7348: 7344: 7338: 7334: 7328: 7325: 7320: 7316: 7293: 7288: 7284: 7280: 7275: 7271: 7267: 7264: 7253: 7252: 7240: 7230: 7218: 7193: 7189: 7185: 7180: 7176: 7172: 7169: 7146: 7140: 7137: 7132: 7129: 7123: 7116: 7112: 7108: 7087: 7081: 7078: 7073: 7070: 7064: 7059: 7056: 7043: 7042: 7026: 7016: 7000: 6977: 6974: 6971: 6963: 6959: 6955: 6950: 6947: 6944: 6938: 6935: 6930: 6927: 6924: 6904: 6889: 6888: 6872: 6862: 6846: 6836: 6820: 6810: 6794: 6784: 6768: 6745: 6740: 6737: 6732: 6729: 6724: 6721: 6716: 6713: 6710: 6707: 6683: 6663: 6648: 6645: 6633: 6630: 6625: 6621: 6617: 6614: 6611: 6606: 6602: 6596: 6591: 6588: 6585: 6581: 6577: 6574: 6571: 6568: 6565: 6562: 6540: 6536: 6515: 6488: 6468: 6465: 6445: 6434: 6433: 6423: 6411: 6385: 6381: 6370: 6358: 6332: 6328: 6317: 6305: 6295: 6292:dummy variable 6279: 6269: 6266:multiplication 6253: 6230: 6227: 6224: 6217: 6213: 6207: 6203: 6197: 6192: 6189: 6186: 6182: 6165: 6162: 6150:electrostatics 6139: 6136: 6123: 6122: 6109: 6104: 6099: 6094: 6090: 6086: 6083: 6080: 6077: 6074: 6069: 6064: 6060: 6056: 6051: 6047: 6013: 6008: 6004: 5993: 5992: 5979: 5974: 5971: 5966: 5961: 5957: 5953: 5948: 5944: 5928: 5891: 5888: 5879: 5876: 5837: 5817: 5800:of the ion in 5785: 5765: 5745: 5721: 5696: 5691: 5688: 5685: 5682: 5675: 5669: 5663: 5659: 5653: 5649: 5644: 5640: 5634: 5631: 5628: 5625: 5622: 5617: 5613: 5609: 5575: 5574: 5559: 5555: 5540: 5533: 5526:Bjerrum length 5510:oversimplified 5494: 5490: 5486: 5483: 5452: 5449: 5433: 5429: 5425: 5422: 5419: 5397: 5393: 5370: 5366: 5362: 5359: 5356: 5334: 5331: 5328: 5325: 5322: 5317: 5313: 5309: 5306: 5303: 5300: 5294: 5291: 5286: 5283: 5280: 5277: 5274: 5266: 5263: 5258: 5250: 5246: 5242: 5237: 5234: 5231: 5228: 5223: 5219: 5196: 5193: 5190: 5187: 5184: 5181: 5160: 5156: 5152: 5149: 5144: 5141: 5138: 5132: 5123: 5120: 5115: 5112: 5109: 5106: 5103: 5078: 5074: 5051: 5047: 5024: 5020: 5016: 5013: 5010: 4990: 4970: 4950: 4930: 4927: 4924: 4921: 4918: 4912: 4908: 4886: 4866: 4863: 4860: 4857: 4837: 4834: 4831: 4828: 4825: 4822: 4819: 4799: 4796: 4793: 4790: 4770: 4746: 4742: 4738: 4732: 4723: 4717: 4713: 4707: 4703: 4699: 4696: 4693: 4686: 4682: 4676: 4672: 4665: 4657: 4653: 4647: 4643: 4636: 4632: 4608: 4604: 4600: 4597: 4594: 4591: 4585: 4576: 4570: 4566: 4560: 4556: 4550: 4545: 4541: 4535: 4531: 4524: 4516: 4512: 4506: 4502: 4495: 4491: 4464: 4460: 4437: 4433: 4410: 4406: 4402: 4399: 4396: 4371: 4351: 4328: 4325: 4322: 4319: 4316: 4313: 4308: 4305: 4300: 4297: 4295: 4291: 4287: 4283: 4282: 4277: 4273: 4269: 4266: 4264: 4260: 4256: 4252: 4251: 4248: 4243: 4239: 4235: 4232: 4230: 4226: 4222: 4218: 4217: 4210: 4206: 4199: 4195: 4191: 4182: 4178: 4172: 4169: 4167: 4163: 4159: 4155: 4154: 4149: 4145: 4141: 4138: 4136: 4132: 4128: 4124: 4123: 4120: 4117: 4114: 4111: 4108: 4105: 4102: 4099: 4093: 4084: 4078: 4075: 4072: 4069: 4066: 4060: 4057: 4055: 4051: 4047: 4043: 4042: 4016: 4013: 4010: 4007: 4004: 3999: 3995: 3991: 3985: 3976: 3970: 3966: 3960: 3956: 3950: 3947: 3944: 3941: 3938: 3935: 3929: 3923: 3920: 3915: 3912: 3909: 3906: 3903: 3895: 3892: 3887: 3879: 3875: 3871: 3866: 3863: 3860: 3857: 3852: 3848: 3828: 3825: 3813: 3802: 3792: 3786: 3783: 3780: 3769: 3758: 3754: 3745: 3738: 3735: 3709: 3705: 3699: 3695: 3691: 3682: 3678: 3669: 3648: 3642: 3639: 3632: 3628: 3624: 3619: 3613: 3604: 3598: 3594: 3588: 3584: 3576: 3572: 3566: 3562: 3556: 3552: 3548: 3545: 3539: 3534: 3525: 3518: 3515: 3509: 3506: 3499: 3490: 3486: 3477: 3470: 3467: 3458: 3427: 3422: 3415: 3410: 3406: 3402: 3399: 3396: 3389: 3386: 3379: 3375: 3364: 3360: 3356: 3352: 3348: 3339: 3333: 3329: 3323: 3319: 3315: 3312: 3309: 3302: 3298: 3294: 3285: 3279: 3275: 3269: 3264: 3260: 3253: 3250: 3244: 3235: 3229: 3225: 3219: 3215: 3211: 3208: 3203: 3198: 3194: 3188: 3183: 3179: 3172: 3169: 3166: 3161: 3157: 3153: 3150: 3147: 3133:ionic strength 3114: 3110: 3087: 3076: 3050: 3028: 3017: 2991: 2971: 2967: 2963: 2941: 2937: 2914: 2910: 2906: 2894: 2890: 2886: 2883: 2873: 2849: 2846: 2843: 2821: 2817: 2813: 2801: 2797: 2793: 2790: 2780: 2757: 2737: 2717: 2695: 2689:ionic strength 2681: 2680: 2667: 2663: 2659: 2654: 2648: 2642: 2637: 2631: 2627: 2621: 2617: 2611: 2606: 2602: 2598: 2592: 2587: 2584: 2574: 2560: 2554: 2549: 2543: 2539: 2533: 2529: 2525: 2522: 2519: 2516: 2511: 2506: 2502: 2495: 2492: 2481: 2480: 2464: 2457: 2453: 2449: 2446: 2443: 2438: 2430: 2425: 2421: 2417: 2414: 2411: 2406: 2402: 2398: 2393: 2389: 2362: 2358: 2354: 2351: 2332: 2331: 2315: 2311: 2307: 2304: 2301: 2297: 2287: 2283: 2277: 2273: 2269: 2266: 2259: 2255: 2249: 2244: 2240: 2233: 2230: 2225: 2216: 2212: 2208: 2198: 2194: 2188: 2184: 2180: 2177: 2172: 2167: 2163: 2156: 2153: 2148: 2144: 2140: 2135: 2131: 2125: 2120: 2115: 2111: 2107: 2102: 2098: 2074: 2071: 2068: 2046: 2042: 2026: 2025: 2012: 2007: 2004: 2001: 1997: 1986: 1982: 1978: 1975: 1972: 1964: 1960: 1956: 1952: 1941: 1937: 1931: 1927: 1923: 1920: 1915: 1910: 1906: 1899: 1896: 1893: 1890: 1885: 1881: 1856: 1853: 1832: 1827: 1822: 1818: 1812: 1808: 1802: 1798: 1794: 1791: 1788: 1784: 1780: 1775: 1771: 1767: 1764: 1761: 1756: 1752: 1748: 1743: 1739: 1733: 1729: 1706: 1702: 1698: 1695: 1675: 1672: 1668: 1665: 1644: 1624: 1621: 1618: 1597: 1594: 1572: 1569: 1557: 1556: 1543: 1538: 1535: 1531: 1524: 1521: 1517: 1512: 1507: 1504: 1501: 1497: 1490: 1487: 1483: 1480: 1477: 1474: 1469: 1465: 1458:with solutions 1454: 1451: 1446: 1442: 1438: 1433: 1429: 1425: 1420: 1416: 1410: 1407: 1402: 1397: 1392: 1388: 1384: 1366: 1365: 1351: 1346: 1342: 1336: 1332: 1326: 1322: 1315: 1309: 1304: 1298: 1294: 1288: 1284: 1277: 1273: 1267: 1262: 1258: 1247: 1244: 1241: 1236: 1232: 1226: 1222: 1218: 1215: 1212: 1209: 1204: 1200: 1194: 1190: 1171: 1167: 1156: 1155: 1143: 1138: 1132: 1126: 1121: 1115: 1112: 1109: 1104: 1100: 1096: 1091: 1087: 1083: 1077: 1073: 1070: 1067: 1064: 1059: 1055: 1051: 1046: 1042: 1037: 1031: 1027: 1018: 1014: 1008: 1004: 999: 994: 991: 988: 985: 982: 977: 973: 967: 963: 927: 921: 920: 908: 902: 896: 891: 885: 882: 879: 874: 870: 866: 861: 857: 853: 847: 843: 840: 835: 831: 827: 823: 819: 815: 789:charge density 773: 772: 761: 758: 755: 750: 746: 737: 733: 727: 723: 718: 713: 710: 707: 704: 701: 696: 692: 686: 682: 655: 652: 647: 646: 643: 636: 629: 622:electric field 614: 599: 598: 582: 578: 572: 568: 562: 558: 554: 551: 544: 540: 534: 530: 524: 520: 513: 493: 470: 460: 441: 438: 427: 426: 413: 410: 407: 404: 401: 397: 393: 388: 382: 375: 371: 363: 356: 352: 345: 340: 335: 331: 315: 311: 308: 307: 294: 290: 286: 281: 272: 268: 264: 259: 250: 246: 242: 237: 232: 227: 222: 218: 192: 188: 167: 166: 153: 149: 144: 140: 137: 134: 131: 108: 81:thermodynamics 65:ideal solution 63:solutions, an 52: 49: 15: 9: 6: 4: 3: 2: 13261: 13250: 13247: 13245: 13242: 13240: 13237: 13235: 13232: 13231: 13229: 13215: 13209: 13201: 13195: 13188: 13183: 13175: 13171: 13166: 13161: 13157: 13153: 13149: 13145: 13139: 13132: 13128: 13125: 13120: 13106:on 2007-06-07 13105: 13101: 13097: 13090: 13082: 13078: 13077: 13072: 13065: 13057: 13051: 13049: 13047: 13045: 13043: 13041: 13039: 13037: 13035: 13033: 13031: 13029: 13027: 13025: 13023: 13021: 13019: 13017: 13015: 13013: 13011: 13009: 13007: 13005: 13003: 13001: 12999: 12997: 12995: 12993: 12991: 12989: 12979: 12970: 12963: 12957: 12949: 12942: 12933: 12926: 12921: 12913: 12907: 12899: 12897:3-527-29096-6 12893: 12889: 12882: 12880: 12878: 12868: 12859: 12850: 12841: 12832: 12823: 12814: 12805: 12796: 12787: 12778: 12770: 12764: 12760: 12753: 12745: 12741: 12740: 12735: 12728: 12724: 12714: 12711: 12709: 12706: 12704: 12701: 12699: 12696: 12694: 12691: 12689: 12686: 12684: 12681: 12679: 12676: 12674: 12671: 12669: 12666: 12664: 12661: 12659: 12656: 12654: 12651: 12650: 12644: 12629: 12619: 12613: 12610:This article 12608: 12604: 12599: 12598: 12590: 12577: 12569: 12565: 12561: 12558: 12555: 12551: 12541: 12537: 12531: 12527: 12523: 12520: 12515: 12510: 12506: 12497: 12491: 12486: 12482: 12476: 12472: 12463: 12458: 12455: 12452: 12448: 12444: 12441: 12436: 12432: 12426: 12422: 12416: 12411: 12408: 12405: 12401: 12395: 12392: 12387: 12382: 12378: 12368: 12365: 12352: 12344: 12340: 12336: 12333: 12330: 12326: 12316: 12312: 12306: 12302: 12298: 12295: 12290: 12285: 12281: 12275: 12270: 12266: 12259: 12256: 12251: 12247: 12243: 12238: 12234: 12230: 12225: 12221: 12212: 12207: 12194: 12186: 12182: 12178: 12175: 12172: 12168: 12158: 12154: 12148: 12144: 12140: 12137: 12132: 12129: 12124: 12120: 12113: 12110: 12105: 12101: 12081: 12073: 12069: 12065: 12062: 12059: 12051: 12047: 12043: 12039: 12028: 12024: 12018: 12014: 12010: 12007: 12002: 11997: 11993: 11986: 11981: 11977: 11957: 11949: 11945: 11937: 11929: 11925: 11918: 11913: 11909: 11904: 11899: 11895: 11883: 11879: 11873: 11869: 11865: 11862: 11858: 11853: 11850: 11843: 11838: 11834: 11824: 11820: 11816: 11813: 11810: 11800: 11796: 11792: 11789: 11785: 11779: 11775: 11768: 11765: 11757: 11753: 11745: 11742: 11722: 11714: 11710: 11697: 11693: 11688: 11684: 11680: 11673: 11669: 11664: 11659: 11655: 11643: 11639: 11633: 11629: 11625: 11622: 11618: 11613: 11606: 11602: 11594: 11590: 11586: 11583: 11579: 11571: 11567: 11563: 11555: 11551: 11544: 11521: 11515: 11493: 11489: 11479: 11463: 11459: 11438: 11430: 11429:shell theorem 11412: 11408: 11385: 11381: 11360: 11355: 11351: 11347: 11342: 11338: 11333: 11329: 11317: 11313: 11307: 11303: 11299: 11296: 11292: 11287: 11282: 11278: 11274: 11268: 11256: 11252: 11246: 11234: 11224: 11211: 11206: 11202: 11197: 11193: 11181: 11177: 11171: 11167: 11163: 11160: 11156: 11151: 11145: 11133: 11123: 11107: 11103: 11094: 11076: 11072: 11062: 11047: 11044: 11034: 11019: 11016: 10994: 10991: 10970: 10950: 10945: 10940: 10937: 10934: 10930: 10924: 10921: 10916: 10911: 10908: 10904: 10897: 10894: 10890: 10885: 10880: 10877: 10874: 10870: 10864: 10861: 10851: 10847: 10841: 10838: 10832: 10825: 10821: 10814: 10807: 10804: 10800: 10795: 10790: 10783: 10779: 10773: 10769: 10763: 10760: 10754: 10748: 10728: 10719: 10706: 10700: 10694: 10689: 10685: 10681: 10675: 10664: 10658: 10647: 10644: 10639: 10631: 10627: 10615: 10609: 10604: 10593: 10589: 10582: 10571: 10565: 10554: 10550: 10545: 10538: 10522: 10518: 10514: 10509: 10506: 10501: 10476: 10473: 10470: 10462: 10457: 10455: 10451: 10433: 10430: 10426: 10416: 10403: 10400: 10395: 10391: 10387: 10384: 10379: 10366: 10361: 10348: 10343: 10338: 10334: 10328: 10323: 10319: 10313: 10309: 10303: 10300: 10295: 10292: 10273: 10267: 10258: 10252: 10248: 10242: 10238: 10230: 10226: 10222: 10219: 10213: 10207: 10198: 10192: 10188: 10182: 10178: 10170: 10165: 10161: 10155: 10151: 10145: 10140: 10136: 10127: 10123: 10119: 10114: 10110: 10101: 10085: 10063: 10059: 10049: 10035: 10029: 10020: 10014: 10010: 10004: 10000: 9994: 9989: 9984: 9980: 9974: 9970: 9964: 9959: 9955: 9946: 9942: 9938: 9935: 9930: 9916: 9903: 9899: 9892: 9883: 9877: 9873: 9867: 9863: 9857: 9852: 9847: 9843: 9837: 9833: 9827: 9822: 9818: 9809: 9805: 9801: 9793: 9789: 9783: 9779: 9771: 9766: 9762: 9758: 9753: 9749: 9740: 9736: 9731: 9727: 9724: 9720: 9713: 9704: 9698: 9695: 9690: 9686: 9679: 9676: 9672: 9663: 9659: 9653: 9649: 9641: 9636: 9632: 9628: 9623: 9619: 9610: 9606: 9602: 9599: 9591: 9582: 9576: 9573: 9568: 9564: 9557: 9553: 9544: 9540: 9534: 9530: 9522: 9517: 9513: 9509: 9504: 9500: 9491: 9487: 9483: 9463: 9460: 9457: 9454: 9451: 9431: 9428: 9425: 9422: 9417: 9413: 9404: 9399: 9397: 9393: 9389: 9384: 9371: 9363: 9354: 9348: 9345: 9340: 9336: 9329: 9325: 9316: 9312: 9306: 9302: 9294: 9289: 9285: 9281: 9276: 9272: 9263: 9259: 9255: 9252: 9244: 9240: 9234: 9230: 9225: 9220: 9217: 9214: 9209: 9196: 9192: 9187: 9174: 9166: 9157: 9151: 9148: 9143: 9139: 9132: 9128: 9122: 9117: 9113: 9109: 9104: 9100: 9094: 9090: 9086: 9081: 9077: 9073: 9068: 9064: 9058: 9054: 9050: 9047: 9039: 9034: 9031: 9018: 9010: 9001: 8995: 8992: 8987: 8983: 8976: 8972: 8966: 8961: 8957: 8953: 8945: 8936: 8930: 8927: 8922: 8918: 8911: 8907: 8901: 8896: 8892: 8886: 8881: 8877: 8856: 8848: 8826: 8800: 8791: 8785: 8782: 8777: 8773: 8766: 8762: 8754: 8736: 8731: 8727: 8704: 8700: 8692: 8688: 8684: 8680: 8664: 8656: 8640: 8632: 8616: 8596: 8593: 8588: 8584: 8575: 8559: 8551: 8543: 8539: 8538:charge number 8521: 8517: 8509: 8506: 8488: 8484: 8476: 8475: 8474: 8460: 8457: 8454: 8449: 8445: 8439: 8435: 8429: 8424: 8421: 8418: 8414: 8399: 8396: 8379: 8370: 8368: 8367:ideal mixture 8350: 8336: 8323: 8318: 8312: 8308: 8304: 8301: 8296: 8292: 8285: 8280: 8276: 8272: 8267: 8263: 8240: 8236: 8223: 8205: 8201: 8193: 8190: 8168: 8160: 8157: 8139: 8135: 8127: 8124: 8106: 8102: 8094: 8079: 8072: 8070:is a species, 8057: 8050: 8049: 8048: 8034: 8023: 8019: 8012: 8009: 8000: 7996: 7991: 7987: 7978: 7974: 7968: 7963: 7960: 7957: 7953: 7949: 7944: 7930: 7917: 7914: 7911: 7903: 7899: 7893: 7889: 7883: 7880: 7875: 7867: 7862: 7837: 7832: 7824: 7819: 7811: 7785: 7781: 7772: 7767: 7754: 7751: 7748: 7740: 7736: 7731: 7728: 7725: 7722: 7716: 7713: 7710: 7704: 7701: 7696: 7693: 7687: 7658: 7656: 7640: 7620: 7615: 7612: 7607: 7604: 7599: 7595: 7592: 7586: 7580: 7575: 7571: 7568: 7565: 7562: 7556: 7553: 7550: 7520: 7515: 7499: 7491: 7486: 7472: 7459: 7456: 7453: 7445: 7441: 7435: 7431: 7425: 7422: 7419: 7416: 7410: 7405: 7401: 7395: 7392: 7387: 7373: 7360: 7357: 7354: 7346: 7342: 7336: 7332: 7326: 7323: 7318: 7304: 7291: 7286: 7278: 7273: 7265: 7238: 7231: 7216: 7209: 7208: 7207: 7191: 7187: 7183: 7178: 7174: 7170: 7167: 7158: 7144: 7138: 7121: 7114: 7110: 7106: 7085: 7079: 7062: 7057: 7054: 7040: 7024: 7017: 7014: 6998: 6991: 6990: 6989: 6975: 6972: 6969: 6961: 6957: 6953: 6948: 6945: 6942: 6936: 6933: 6928: 6922: 6894: 6886: 6870: 6863: 6860: 6844: 6837: 6834: 6818: 6811: 6808: 6792: 6785: 6782: 6759: 6758: 6757: 6743: 6738: 6735: 6730: 6727: 6722: 6719: 6714: 6711: 6708: 6697: 6654: 6644: 6631: 6623: 6619: 6612: 6609: 6604: 6600: 6594: 6589: 6586: 6583: 6579: 6575: 6569: 6563: 6560: 6538: 6534: 6513: 6504: 6502: 6486: 6466: 6463: 6443: 6431: 6427: 6424: 6409: 6401: 6383: 6379: 6371: 6356: 6348: 6347:mole fraction 6330: 6326: 6318: 6303: 6296: 6293: 6277: 6270: 6267: 6251: 6244: 6243: 6242: 6228: 6225: 6222: 6215: 6211: 6205: 6201: 6195: 6190: 6187: 6184: 6180: 6171: 6161: 6157: 6153: 6151: 6146: 6135: 6132: 6128: 6107: 6097: 6092: 6084: 6081: 6078: 6072: 6067: 6062: 6054: 6049: 6037: 6036: 6035: 6033: 6029: 6011: 6006: 5977: 5972: 5969: 5964: 5959: 5951: 5946: 5934: 5933: 5932: 5926: 5922: 5918: 5913: 5909: 5905: 5896: 5887: 5885: 5875: 5873: 5869: 5860: 5856: 5854: 5849: 5835: 5815: 5807: 5806:hydronium ion 5803: 5799: 5783: 5763: 5743: 5735: 5719: 5694: 5689: 5686: 5683: 5680: 5673: 5661: 5657: 5651: 5647: 5638: 5632: 5626: 5620: 5615: 5611: 5607: 5598: 5596: 5592: 5588: 5584: 5580: 5572: 5568: 5564: 5560: 5553: 5549: 5545: 5544:point charges 5541: 5538: 5534: 5531: 5527: 5523: 5522:Niels Bjerrum 5519: 5515: 5514: 5513: 5511: 5492: 5488: 5484: 5481: 5469: 5465: 5462: 5457: 5448: 5431: 5423: 5420: 5395: 5391: 5368: 5360: 5357: 5345: 5332: 5326: 5315: 5307: 5304: 5298: 5292: 5281: 5264: 5261: 5256: 5248: 5244: 5232: 5221: 5194: 5191: 5158: 5154: 5150: 5147: 5142: 5139: 5136: 5121: 5110: 5076: 5072: 5049: 5045: 5022: 5014: 5011: 4988: 4968: 4948: 4928: 4925: 4919: 4906: 4884: 4861: 4855: 4829: 4823: 4794: 4788: 4768: 4759: 4744: 4740: 4736: 4730: 4721: 4715: 4711: 4705: 4701: 4697: 4694: 4691: 4684: 4680: 4674: 4670: 4663: 4655: 4651: 4645: 4641: 4634: 4630: 4606: 4598: 4595: 4589: 4583: 4574: 4568: 4564: 4558: 4554: 4548: 4543: 4539: 4533: 4529: 4522: 4514: 4510: 4504: 4500: 4493: 4489: 4478: 4462: 4458: 4435: 4431: 4408: 4400: 4397: 4386: 4369: 4326: 4320: 4314: 4311: 4306: 4303: 4298: 4296: 4289: 4285: 4275: 4271: 4267: 4265: 4258: 4254: 4246: 4241: 4237: 4233: 4231: 4224: 4220: 4208: 4204: 4197: 4193: 4189: 4180: 4176: 4170: 4168: 4161: 4157: 4147: 4143: 4139: 4137: 4130: 4126: 4112: 4106: 4097: 4091: 4082: 4073: 4067: 4064: 4058: 4056: 4049: 4045: 4032: 4027: 4014: 4008: 4002: 3997: 3993: 3989: 3983: 3974: 3968: 3964: 3958: 3954: 3945: 3939: 3936: 3933: 3927: 3921: 3910: 3904: 3893: 3890: 3885: 3877: 3873: 3861: 3855: 3850: 3834: 3824: 3811: 3800: 3790: 3784: 3781: 3778: 3767: 3756: 3752: 3743: 3736: 3733: 3725: 3707: 3703: 3697: 3693: 3689: 3680: 3676: 3667: 3646: 3640: 3637: 3630: 3626: 3622: 3617: 3611: 3602: 3596: 3592: 3586: 3582: 3574: 3570: 3564: 3560: 3554: 3550: 3546: 3543: 3537: 3532: 3523: 3516: 3513: 3507: 3504: 3497: 3488: 3484: 3475: 3468: 3465: 3456: 3447: 3443: 3438: 3425: 3420: 3413: 3408: 3404: 3400: 3397: 3394: 3387: 3384: 3377: 3373: 3362: 3358: 3354: 3346: 3337: 3331: 3327: 3321: 3317: 3310: 3307: 3300: 3296: 3292: 3283: 3277: 3273: 3267: 3262: 3258: 3251: 3248: 3242: 3233: 3227: 3223: 3217: 3213: 3209: 3206: 3201: 3196: 3192: 3186: 3181: 3177: 3170: 3167: 3159: 3155: 3148: 3145: 3137: 3134: 3128: 3112: 3108: 3085: 3074: 3048: 3026: 3015: 2989: 2969: 2965: 2961: 2939: 2935: 2912: 2908: 2904: 2892: 2888: 2884: 2881: 2871: 2863: 2847: 2844: 2841: 2819: 2815: 2811: 2799: 2795: 2791: 2788: 2778: 2770: 2755: 2735: 2715: 2707: 2705: 2701: 2694: 2690: 2686: 2665: 2661: 2657: 2652: 2646: 2635: 2629: 2625: 2619: 2615: 2609: 2604: 2600: 2596: 2590: 2585: 2582: 2575: 2558: 2547: 2541: 2537: 2531: 2527: 2523: 2520: 2517: 2514: 2509: 2504: 2500: 2493: 2490: 2483: 2482: 2462: 2455: 2451: 2447: 2444: 2441: 2436: 2428: 2423: 2419: 2415: 2412: 2409: 2404: 2400: 2396: 2391: 2387: 2379: 2378: 2360: 2356: 2352: 2349: 2342:Experimental 2340: 2336: 2313: 2309: 2305: 2302: 2299: 2295: 2285: 2281: 2275: 2271: 2267: 2264: 2257: 2253: 2247: 2242: 2238: 2231: 2228: 2214: 2210: 2206: 2196: 2192: 2186: 2182: 2178: 2175: 2170: 2165: 2161: 2154: 2146: 2142: 2133: 2129: 2118: 2113: 2109: 2105: 2100: 2096: 2088: 2087: 2086: 2072: 2069: 2066: 2044: 2040: 2031: 2010: 2005: 2002: 1999: 1995: 1984: 1980: 1976: 1973: 1970: 1962: 1958: 1954: 1950: 1939: 1935: 1929: 1925: 1921: 1918: 1913: 1908: 1904: 1897: 1891: 1883: 1879: 1871: 1870: 1869: 1854: 1851: 1843:, from which 1825: 1820: 1816: 1810: 1806: 1800: 1796: 1792: 1789: 1782: 1778: 1773: 1769: 1765: 1762: 1754: 1750: 1741: 1737: 1731: 1704: 1700: 1696: 1693: 1673: 1670: 1666: 1663: 1642: 1616: 1595: 1592: 1570: 1567: 1541: 1536: 1533: 1529: 1522: 1519: 1515: 1510: 1505: 1502: 1499: 1495: 1488: 1485: 1481: 1475: 1467: 1463: 1452: 1449: 1444: 1440: 1431: 1427: 1423: 1418: 1408: 1405: 1400: 1395: 1390: 1375: 1374: 1373: 1371: 1368:which has an 1349: 1344: 1340: 1334: 1330: 1324: 1320: 1313: 1302: 1296: 1292: 1286: 1282: 1275: 1271: 1265: 1260: 1256: 1242: 1234: 1230: 1224: 1220: 1216: 1210: 1202: 1198: 1192: 1180: 1179: 1178: 1177: 1165: 1161: 1160:Taylor series 1141: 1136: 1130: 1119: 1110: 1102: 1098: 1094: 1089: 1085: 1081: 1075: 1071: 1068: 1062: 1057: 1053: 1044: 1040: 1035: 1029: 1025: 1016: 1012: 1006: 1002: 997: 992: 989: 983: 975: 971: 965: 953: 952: 951: 949: 945: 941: 937: 933: 926: 906: 900: 889: 880: 872: 868: 864: 859: 855: 851: 845: 841: 838: 833: 829: 825: 821: 817: 813: 805: 804: 803: 801: 796: 794: 790: 786: 782: 778: 756: 748: 744: 735: 731: 725: 721: 716: 711: 708: 702: 694: 690: 684: 672: 671: 670: 669: 665: 660: 651: 644: 641: 637: 634: 630: 627: 623: 619: 615: 612: 608: 604: 603: 602: 580: 576: 570: 566: 560: 556: 552: 549: 542: 538: 532: 528: 522: 518: 511: 499: 498: 497: 491: 487: 483: 482:Coulomb's law 479: 475: 469: 465: 459: 453: 446: 436: 431: 408: 405: 402: 395: 391: 386: 380: 373: 369: 361: 354: 350: 343: 338: 333: 329: 321: 320: 319: 292: 288: 284: 279: 270: 257: 248: 235: 230: 225: 220: 216: 208: 207: 206: 190: 186: 175: 173: 172:electrostatic 151: 147: 142: 138: 135: 132: 129: 122: 121: 120: 106: 98: 94: 90: 86: 82: 78: 74: 73:concentration 70: 66: 62: 58: 48: 46: 42: 38: 34: 30: 26: 22: 13208: 13194: 13189:section 3.1) 13182: 13155: 13151: 13144:Hille, Einar 13138: 13119: 13108:. Retrieved 13104:the original 13099: 13089: 13080: 13074: 13064: 13055: 12978: 12969: 12956: 12947: 12941: 12932: 12920: 12887: 12867: 12858: 12849: 12840: 12831: 12822: 12813: 12804: 12795: 12786: 12777: 12758: 12752: 12743: 12737: 12727: 12673:Debye length 12641: 12623: 12611: 12369: 12366: 12208: 11480: 11225: 11124: 11093:point charge 11063: 11035: 10720: 10458: 10417: 10362: 10078:. Also, let 10050: 9917: 9400: 9385: 9188: 9035: 9032: 8549: 8547: 8541: 8504: 8405: 8371: 8337: 8227: 8221: 8155: 8122: 7931: 7768: 7659: 7516: 7473: 7374: 7305: 7254: 7159: 7044: 6890: 6650: 6647:Fundamentals 6505: 6435: 6425: 6167: 6164:Introduction 6158: 6154: 6141: 6130: 6126: 6124: 6031: 6027: 5994: 5912:Lars Onsager 5904:conductivity 5901: 5890:Conductivity 5881: 5865: 5850: 5797: 5599: 5579:ionic radius 5576: 5546:and are not 5529: 5473: 5346: 5064:in favor of 5001:in favor of 4961:in favor of 4877:in favor of 4810:in favor of 4760: 4479: 4028: 3830: 3439: 3138: 3132: 3129: 2861: 2768: 2708: 2703: 2699: 2692: 2684: 2682: 2333: 2027: 1558: 1367: 1166:(so that Σ n 1157: 939: 935: 924: 922: 797: 784: 780: 776: 774: 666:by means of 663: 661: 657: 648: 638:There is no 600: 496:is given by 485: 477: 473: 467: 463: 457: 454: 451: 428: 318:is given by 309: 176: 168: 92: 88: 54: 33:electrolytes 29:Erich Hückel 20: 18: 13249:Peter Debye 12653:Electrolyte 8685:and form a 8540:of species 6859:temperature 6402:of species 6349:of species 5868:surfactants 3440:The excess 61:electrolyte 25:Peter Debye 13228:Categories 13110:2007-10-23 13083:: 185–206. 12960:R. Wasoe, 12746:: 185–206. 12719:References 10463:by taking 5468:octahedron 4029:Using the 12927:, page 9. 12906:cite book 12761:. Wiley. 12618:talk page 12562:κ 12538:ε 12528:ε 12524:π 12516:κ 12449:∑ 12445:− 12402:∑ 12337:κ 12313:ε 12303:ε 12299:π 12291:κ 12260:− 12179:κ 12155:ε 12145:ε 12141:π 12133:κ 12114:− 12066:κ 12044:κ 12025:ε 12015:ε 12011:π 11930:φ 11880:ε 11870:ε 11866:π 11854:− 11817:κ 11793:κ 11790:− 11769:− 11743:φ 11698:φ 11640:ε 11630:ε 11626:π 11587:κ 11584:− 11545:φ 11516:φ 11314:ε 11304:ε 11300:π 11257:φ 11235:φ 11178:ε 11168:ε 11164:π 11134:φ 10938:κ 10935:− 10909:κ 10878:κ 10875:− 10848:κ 10822:κ 10780:κ 10774:− 10749:φ 10729:κ 10695:φ 10686:κ 10673:∂ 10659:φ 10656:∂ 10624:∂ 10610:φ 10601:∂ 10580:∂ 10566:φ 10563:∂ 10536:∂ 10532:∂ 10507:φ 10498:∇ 10431:− 10427:κ 10401:φ 10392:κ 10385:φ 10376:∇ 10310:∑ 10249:ε 10239:ε 10189:ε 10179:ε 10124:∑ 10111:κ 10060:κ 10011:ε 10001:ε 9995:φ 9943:∑ 9936:φ 9927:∇ 9874:ε 9864:ε 9858:φ 9806:∑ 9802:− 9790:ε 9780:ε 9737:∑ 9728:− 9699:φ 9680:− 9660:ε 9650:ε 9607:∑ 9603:− 9600:≈ 9577:φ 9558:− 9541:ε 9531:ε 9488:∑ 9484:− 9461:≪ 9423:≈ 9349:φ 9330:− 9313:ε 9303:ε 9260:∑ 9256:− 9241:ε 9231:ε 9226:ρ 9221:− 9215:φ 9206:∇ 9197:results: 9152:φ 9133:− 9091:∑ 9055:∑ 9048:ρ 8996:φ 8977:− 8931:φ 8912:− 8786:φ 8767:− 8657:field at 8641:φ 8597:φ 8415:∑ 8347:Ξ 8286:− 8264:ξ 8237:ξ 8136:ξ 7997:− 7988:ξ 7954:∑ 7941:Ξ 7884:∫ 7872:Φ 7859:Ξ 7829:Ξ 7816:Ξ 7809:Ξ 7697:− 7691:Ξ 7668:Ξ 7608:− 7587:− 7584:Φ 7557:− 7548:Ξ 7528:Ξ 7496:Ξ 7483:Φ 7426:∫ 7396:∫ 7384:Φ 7327:∫ 7315:Φ 7283:Φ 7270:Φ 7263:Φ 7136:∂ 7131:Φ 7128:∂ 7077:∂ 7072:Φ 7069:∂ 6926:Φ 6903:Φ 6767:Φ 6731:− 6715:− 6706:Φ 6682:Ξ 6662:Φ 6620:γ 6613:⁡ 6601:ν 6580:∑ 6570:γ 6564:⁡ 6535:γ 6514:γ 6487:γ 6464:γ 6380:ν 6252:∏ 6212:ν 6181:∏ 6089:Λ 6073:− 6059:Λ 6046:Λ 6003:Λ 5970:− 5956:Λ 5943:Λ 5917:viscosity 5720:γ 5662:− 5627:γ 5621:⁡ 5608:− 5548:polarized 5493:± 5489:γ 5485:⁡ 5421:κ 5358:κ 5321:Φ 5305:κ 5290:∂ 5276:Φ 5273:∂ 5241:∂ 5227:Φ 5218:∂ 5186:∞ 5180:Φ 5151:− 5119:∂ 5105:Φ 5102:∂ 5012:κ 4911:′ 4818:Φ 4789:φ 4769:π 4712:ε 4702:ε 4695:π 4652:π 4642:π 4631:π 4596:κ 4565:ε 4555:ε 4511:π 4501:π 4490:π 4398:κ 4350:Φ 4286:π 4255:π 4221:π 4194:ε 4158:π 4144:ε 4127:π 4101:Φ 4068:φ 4046:π 4003:φ 3994:κ 3965:ε 3955:ε 3940:φ 3919:∂ 3905:φ 3902:∂ 3870:∂ 3856:φ 3847:∂ 3734:ϕ 3694:∑ 3641:π 3593:ε 3583:ε 3551:∑ 3547:π 3517:− 3508:π 3489:κ 3469:− 3446:cgs units 3398:− 3328:ε 3318:ε 3311:π 3252:− 3224:ε 3214:ε 3210:π 3202:κ 3171:− 3156:γ 3149:⁡ 2882:− 2877: mol 2845:⁡ 2789:− 2784: mol 2626:ϵ 2616:ϵ 2538:ϵ 2528:ϵ 2524:π 2518:× 2413:− 2405:± 2401:γ 2397:⁡ 2361:± 2357:γ 2353:⁡ 2306:κ 2296:κ 2282:ε 2272:ε 2268:π 2232:− 2193:ε 2183:ε 2179:π 2155:− 2130:ψ 2003:κ 2000:− 1977:κ 1955:κ 1936:ε 1926:ε 1922:π 1880:ψ 1807:ϵ 1797:ϵ 1793:π 1766:− 1738:ψ 1728:∂ 1643:ψ 1623:∞ 1620:→ 1534:κ 1503:κ 1500:− 1464:ψ 1441:ψ 1428:κ 1424:− 1415:∂ 1387:∂ 1321:∑ 1293:ϵ 1283:ϵ 1257:κ 1231:ψ 1221:κ 1199:ψ 1189:∇ 1099:ψ 1082:− 1072:⁡ 1026:∑ 1013:ϵ 1003:ϵ 993:− 972:ψ 962:∇ 869:ψ 852:− 842:⁡ 745:ρ 732:ϵ 722:ϵ 712:− 691:ψ 681:∇ 626:Solvation 618:polarized 567:ϵ 557:ϵ 553:π 440:The model 370:γ 351:γ 334:± 330:γ 271:− 258:γ 236:γ 221:± 217:γ 191:± 187:γ 136:γ 107:γ 57:chemistry 13146:(1932). 13127:Archived 12962:Phys. Z. 12647:See also 12626:May 2018 11938:′ 11746:′ 11048:″ 11020:″ 10995:′ 10898:″ 10808:′ 8818:, where 8609:, where 7850:and put 7013:pressure 6479:, where 5872:micelles 5802:angstrom 2769:of water 1855:′ 1667:″ 1596:″ 1571:′ 1523:″ 1489:′ 822:′ 476:, where 85:activity 51:Overview 13174:1562506 10418:Today, 10098:be the 8845:is the 8629:is the 8536:is the 8187:is the 6807:entropy 6428:is the 6398:is the 6345:is the 5808:is 9Å. 5732:as its 5563:dipolar 5554:ion, NO 5552:nitrate 3833:earlier 2954:before 2687:is the 75:of the 55:In the 37:plasmas 13172: 12894: 12765: 11373:where 11008:, and 10963:where 8849:, and 8633:, and 8473:where 8047:where 7633:where 7206:where 7039:volume 6988:where 6756:where 6241:where 6125:where 5712:where 5524:. The 5461:sodium 3722:. The 2683:where 1686:. At 923:where 607:solute 77:solute 6456:with 6290:is a 2872:0.509 2860:, so 2779:1.172 2515:2.303 508:force 12912:link 12892:ISBN 12763:ISBN 9455:< 9444:for 9036:The 6674:and 6129:and 5919:and 5828:and 5593:and 3070:mole 3061:are 3002:are 2691:and 2028:The 1584:and 1251:with 934:and 605:The 466:and 35:and 27:and 19:The 13160:doi 7680:as 7653:is 7540:is 7037:is 7011:is 6915:as 6895:of 6883:is 6857:is 6831:is 6805:is 6779:is 6610:log 6561:log 6553:is 5612:log 5482:log 5464:ion 3493:cgs 3011:mol 2864:is 2771:is 2388:log 2350:log 2085:is 1069:exp 930:is 839:exp 59:of 13230:: 13170:MR 13168:. 13156:38 13154:. 13150:. 13098:. 13081:24 13079:. 13073:. 12987:^ 12908:}} 12904:{{ 12876:^ 12744:24 12742:. 12736:. 11934:sp 11702:sp 11261:pc 11239:sp 11138:pc 11122:. 10983:, 10367:: 7657:. 6152:. 5931:. 5908:AC 5616:10 5597:. 5589:, 5530:kT 4387:, 3805:id 3795:ex 3772:id 3761:ex 3748:id 3672:id 3638:24 3505:24 3461:ex 3448:: 3374:10 3146:ln 3109:10 3022:dm 2936:10 2900:kg 2848:10 2842:ln 2807:kg 2392:10 2032:, 1635:, 950:. 802:. 775:ψ( 635:). 624:. 119:. 99:, 13216:. 13202:. 13176:. 13162:: 13113:. 12914:) 12900:. 12771:. 12628:) 12624:( 12620:. 12578:. 12570:i 12566:a 12559:+ 12556:1 12552:1 12542:0 12532:r 12521:4 12511:2 12507:q 12498:2 12492:2 12487:i 12483:z 12477:i 12473:N 12464:s 12459:1 12456:= 12453:i 12442:= 12437:i 12433:u 12427:i 12423:N 12417:s 12412:1 12409:= 12406:i 12396:2 12393:1 12388:= 12383:e 12379:U 12353:. 12345:i 12341:a 12334:+ 12331:1 12327:1 12317:0 12307:r 12296:4 12286:2 12282:q 12276:2 12271:i 12267:z 12257:= 12252:i 12248:B 12244:q 12239:i 12235:z 12231:= 12226:i 12222:u 12195:. 12187:i 12183:a 12176:+ 12173:1 12169:1 12159:0 12149:r 12138:4 12130:q 12125:i 12121:z 12111:= 12106:i 12102:B 12082:, 12074:i 12070:a 12063:+ 12060:1 12052:i 12048:a 12040:e 12029:0 12019:r 12008:4 12003:q 11998:i 11994:z 11987:= 11982:i 11978:A 11958:, 11955:) 11950:i 11946:a 11942:( 11926:= 11919:2 11914:i 11910:a 11905:q 11900:i 11896:z 11884:0 11874:r 11863:4 11859:1 11851:= 11844:2 11839:i 11835:a 11830:) 11825:i 11821:a 11814:+ 11811:1 11808:( 11801:i 11797:a 11786:e 11780:i 11776:A 11766:= 11763:) 11758:i 11754:a 11750:( 11723:, 11720:) 11715:i 11711:a 11707:( 11694:= 11689:i 11685:B 11681:+ 11674:i 11670:a 11665:q 11660:i 11656:z 11644:0 11634:r 11623:4 11619:1 11614:= 11607:i 11603:a 11595:i 11591:a 11580:e 11572:i 11568:A 11564:= 11561:) 11556:i 11552:a 11548:( 11525:) 11522:r 11519:( 11494:i 11490:a 11464:i 11460:a 11439:r 11413:i 11409:B 11386:i 11382:B 11361:, 11356:i 11352:B 11348:+ 11343:r 11339:q 11334:i 11330:z 11318:0 11308:r 11297:4 11293:1 11288:= 11283:i 11279:B 11275:+ 11272:) 11269:r 11266:( 11253:= 11250:) 11247:r 11244:( 11212:. 11207:r 11203:q 11198:i 11194:z 11182:0 11172:r 11161:4 11157:1 11152:= 11149:) 11146:r 11143:( 11108:i 11104:a 11077:i 11073:a 11045:A 11017:A 10992:A 10971:A 10951:, 10946:r 10941:r 10931:e 10925:A 10922:= 10917:r 10912:r 10905:e 10895:A 10891:+ 10886:r 10881:r 10871:e 10865:A 10862:= 10852:2 10842:r 10839:2 10833:r 10826:2 10815:e 10805:A 10801:+ 10796:r 10791:r 10784:2 10770:e 10764:A 10761:= 10758:) 10755:r 10752:( 10707:. 10704:) 10701:r 10698:( 10690:2 10682:= 10676:r 10668:) 10665:r 10662:( 10648:r 10645:2 10640:+ 10632:2 10628:r 10619:) 10616:r 10613:( 10605:2 10594:= 10590:) 10583:r 10575:) 10572:r 10569:( 10555:2 10551:r 10546:( 10539:r 10523:2 10519:r 10515:1 10510:= 10502:2 10477:0 10474:= 10471:r 10434:1 10404:. 10396:2 10388:= 10380:2 10349:. 10344:0 10339:i 10335:n 10329:2 10324:i 10320:z 10314:i 10304:2 10301:1 10296:= 10293:I 10274:, 10268:T 10263:B 10259:k 10253:0 10243:r 10231:2 10227:q 10223:I 10220:2 10214:= 10208:T 10203:B 10199:k 10193:0 10183:r 10171:0 10166:i 10162:n 10156:2 10152:q 10146:2 10141:i 10137:z 10128:i 10120:= 10115:2 10086:I 10064:2 10036:, 10030:T 10025:B 10021:k 10015:0 10005:r 9990:0 9985:i 9981:n 9975:2 9971:q 9965:2 9960:i 9956:z 9947:i 9939:= 9931:2 9904:. 9900:) 9893:T 9888:B 9884:k 9878:0 9868:r 9853:0 9848:i 9844:n 9838:2 9834:q 9828:2 9823:i 9819:z 9810:i 9794:0 9784:r 9772:0 9767:i 9763:n 9759:q 9754:i 9750:z 9741:i 9732:( 9725:= 9721:) 9714:T 9709:B 9705:k 9696:q 9691:i 9687:z 9677:1 9673:( 9664:0 9654:r 9642:0 9637:i 9633:n 9629:q 9624:i 9620:z 9611:i 9592:T 9587:B 9583:k 9574:q 9569:i 9565:z 9554:e 9545:0 9535:r 9523:0 9518:i 9514:n 9510:q 9505:i 9501:z 9492:i 9464:1 9458:x 9452:0 9432:x 9429:+ 9426:1 9418:x 9414:e 9372:. 9364:T 9359:B 9355:k 9346:q 9341:i 9337:z 9326:e 9317:0 9307:r 9295:0 9290:i 9286:n 9282:q 9277:i 9273:z 9264:i 9253:= 9245:0 9235:r 9218:= 9210:2 9175:. 9167:T 9162:B 9158:k 9149:q 9144:i 9140:z 9129:e 9123:0 9118:i 9114:n 9110:q 9105:i 9101:z 9095:i 9087:= 9082:i 9078:n 9074:q 9069:i 9065:z 9059:i 9051:= 9019:. 9011:T 9006:B 9002:k 8993:q 8988:i 8984:z 8973:e 8967:0 8962:i 8958:n 8954:= 8946:T 8941:B 8937:k 8928:q 8923:i 8919:z 8908:e 8902:V 8897:i 8893:N 8887:= 8882:i 8878:n 8857:T 8831:B 8827:k 8801:T 8796:B 8792:k 8783:q 8778:i 8774:z 8763:e 8737:0 8732:i 8728:n 8705:i 8701:n 8665:P 8617:q 8594:q 8589:i 8585:z 8560:P 8550:i 8544:. 8542:i 8522:i 8518:z 8505:i 8489:i 8485:N 8461:, 8458:0 8455:= 8450:i 8446:z 8440:i 8436:N 8430:s 8425:1 8422:= 8419:i 8380:U 8351:k 8324:. 8319:T 8313:i 8309:v 8305:P 8302:+ 8297:i 8293:u 8281:i 8277:s 8273:= 8268:i 8241:i 8224:. 8222:i 8206:i 8202:x 8191:, 8173:B 8169:k 8158:, 8156:i 8140:i 8125:, 8123:i 8107:i 8103:N 8080:s 8058:i 8035:, 8032:) 8029:) 8024:i 8020:x 8016:( 8013:n 8010:l 8005:B 8001:k 7992:i 7984:( 7979:i 7975:N 7969:s 7964:0 7961:= 7958:i 7950:= 7945:k 7918:. 7915:T 7912:d 7904:2 7900:T 7894:e 7890:U 7881:= 7876:e 7868:= 7863:e 7838:, 7833:e 7825:+ 7820:k 7812:= 7786:e 7782:P 7755:. 7752:T 7749:d 7741:2 7737:T 7732:V 7729:P 7726:+ 7723:U 7717:+ 7714:P 7711:d 7705:T 7702:V 7694:= 7688:d 7641:G 7621:, 7616:T 7613:G 7605:= 7600:T 7596:V 7593:P 7581:= 7576:T 7572:V 7569:P 7566:+ 7563:U 7554:S 7551:= 7500:e 7492:= 7487:e 7460:. 7457:T 7454:d 7446:2 7442:T 7436:e 7432:U 7423:+ 7420:V 7417:d 7411:T 7406:e 7402:P 7393:= 7388:e 7361:. 7358:T 7355:d 7347:2 7343:T 7337:e 7333:U 7324:= 7319:e 7292:. 7287:e 7279:+ 7274:k 7266:= 7239:e 7217:k 7192:e 7188:U 7184:+ 7179:k 7175:U 7171:= 7168:U 7145:, 7139:T 7122:= 7115:2 7111:T 7107:U 7086:, 7080:V 7063:= 7058:T 7055:P 7041:. 7025:V 7015:, 6999:P 6976:, 6973:T 6970:d 6962:2 6958:T 6954:U 6949:+ 6946:V 6943:d 6937:T 6934:P 6929:= 6923:d 6887:. 6871:A 6861:, 6845:T 6835:, 6819:U 6809:, 6793:S 6783:, 6744:, 6739:T 6736:A 6728:= 6723:T 6720:U 6712:S 6709:= 6632:. 6629:) 6624:i 6616:( 6605:i 6595:s 6590:1 6587:= 6584:i 6576:= 6573:) 6567:( 6539:i 6467:K 6444:K 6432:. 6426:K 6422:, 6410:i 6384:i 6369:, 6357:i 6331:i 6327:x 6304:s 6278:i 6268:, 6229:, 6226:K 6223:= 6216:i 6206:i 6202:x 6196:s 6191:1 6188:= 6185:i 6131:B 6127:A 6108:c 6103:) 6098:0 6093:m 6085:B 6082:+ 6079:A 6076:( 6068:0 6063:m 6055:= 6050:m 6032:c 6028:K 6012:0 6007:m 5978:c 5973:K 5965:0 5960:m 5952:= 5947:m 5929:m 5836:B 5816:A 5784:a 5764:I 5744:z 5695:I 5690:a 5687:B 5684:+ 5681:1 5674:I 5668:| 5658:z 5652:+ 5648:z 5643:| 5639:A 5633:= 5630:) 5624:( 5556:3 5532:. 5432:2 5428:) 5424:a 5418:( 5396:0 5392:Z 5369:2 5365:) 5361:a 5355:( 5333:. 5330:) 5327:R 5324:( 5316:2 5312:) 5308:a 5302:( 5299:= 5293:R 5285:) 5282:R 5279:( 5265:R 5262:2 5257:+ 5249:2 5245:R 5236:) 5233:R 5230:( 5222:2 5195:0 5192:= 5189:) 5183:( 5159:0 5155:Z 5148:= 5143:1 5140:= 5137:R 5131:| 5122:R 5114:) 5111:R 5108:( 5077:0 5073:Z 5050:0 5046:z 5023:2 5019:) 5015:a 5009:( 4989:I 4969:R 4949:r 4929:a 4926:= 4923:) 4920:r 4917:( 4907:R 4885:r 4865:) 4862:r 4859:( 4856:R 4836:) 4833:) 4830:r 4827:( 4824:R 4821:( 4798:) 4795:r 4792:( 4745:0 4741:Z 4737:= 4731:T 4726:B 4722:k 4716:0 4706:r 4698:a 4692:4 4685:2 4681:q 4675:0 4671:z 4664:= 4656:3 4646:2 4635:5 4607:2 4603:) 4599:a 4593:( 4590:= 4584:T 4579:B 4575:k 4569:0 4559:r 4549:I 4544:2 4540:q 4534:2 4530:a 4523:= 4515:3 4505:2 4494:4 4463:0 4459:z 4436:0 4432:Z 4409:2 4405:) 4401:a 4395:( 4370:R 4327:. 4324:) 4321:r 4318:( 4315:R 4312:= 4307:a 4304:r 4299:= 4290:6 4276:0 4272:z 4268:= 4259:5 4247:I 4242:3 4238:a 4234:= 4225:4 4209:2 4205:q 4198:0 4190:T 4185:B 4181:k 4177:a 4171:= 4162:3 4148:r 4140:= 4131:2 4119:) 4116:) 4113:r 4110:( 4107:R 4104:( 4098:= 4092:T 4087:B 4083:k 4077:) 4074:r 4071:( 4065:q 4059:= 4050:1 4015:. 4012:) 4009:r 4006:( 3998:2 3990:= 3984:T 3979:B 3975:k 3969:0 3959:r 3949:) 3946:r 3943:( 3937:q 3934:I 3928:= 3922:r 3914:) 3911:r 3908:( 3894:r 3891:2 3886:+ 3878:2 3874:r 3865:) 3862:r 3859:( 3851:2 3812:. 3801:P 3791:P 3785:+ 3782:1 3779:= 3768:P 3757:P 3753:+ 3744:P 3737:= 3708:i 3704:c 3698:i 3690:T 3685:B 3681:k 3677:= 3668:P 3647:. 3631:2 3627:/ 3623:3 3618:) 3612:T 3607:B 3603:k 3597:r 3587:0 3575:j 3571:q 3565:j 3561:c 3555:j 3544:4 3538:( 3533:T 3528:B 3524:k 3514:= 3498:3 3485:T 3480:B 3476:k 3466:= 3457:P 3426:, 3421:I 3414:2 3409:i 3405:z 3401:A 3395:= 3388:2 3385:I 3378:3 3363:2 3359:/ 3355:3 3351:) 3347:T 3342:B 3338:k 3332:0 3322:r 3314:( 3308:4 3301:2 3297:/ 3293:1 3288:A 3284:N 3278:3 3274:q 3268:2 3263:i 3259:z 3249:= 3243:T 3238:B 3234:k 3228:0 3218:r 3207:8 3197:2 3193:q 3187:2 3182:i 3178:z 3168:= 3165:) 3160:i 3152:( 3113:3 3086:3 3081:m 3075:/ 3049:I 3027:3 3016:/ 2990:I 2970:2 2966:/ 2962:I 2940:3 2913:2 2909:/ 2905:1 2893:2 2889:/ 2885:1 2862:A 2820:2 2816:/ 2812:1 2800:2 2796:/ 2792:1 2756:A 2736:I 2716:A 2704:B 2700:A 2696:0 2693:a 2685:I 2666:2 2662:/ 2658:1 2653:) 2647:T 2641:B 2636:k 2630:r 2620:0 2610:N 2605:2 2601:e 2597:2 2591:( 2586:= 2583:B 2559:T 2553:B 2548:k 2542:r 2532:0 2521:8 2510:B 2505:2 2501:e 2494:= 2491:A 2463:I 2456:0 2452:a 2448:B 2445:+ 2442:1 2437:I 2429:2 2424:j 2420:z 2416:A 2410:= 2314:0 2310:a 2303:+ 2300:1 2286:r 2276:0 2265:4 2258:2 2254:e 2248:2 2243:j 2239:z 2229:= 2224:) 2215:0 2211:a 2207:1 2197:r 2187:0 2176:4 2171:e 2166:j 2162:z 2152:) 2147:0 2143:a 2139:( 2134:j 2124:( 2119:e 2114:j 2110:z 2106:= 2101:j 2097:u 2073:0 2070:= 2067:r 2045:j 2041:u 2011:r 2006:r 1996:e 1985:0 1981:a 1974:+ 1971:1 1963:0 1959:a 1951:e 1940:r 1930:0 1919:4 1914:e 1909:j 1905:z 1898:= 1895:) 1892:r 1889:( 1884:j 1852:A 1831:) 1826:2 1821:0 1817:a 1811:r 1801:0 1790:4 1787:( 1783:/ 1779:e 1774:j 1770:z 1763:= 1760:) 1755:0 1751:a 1747:( 1742:j 1732:r 1705:0 1701:a 1697:= 1694:r 1674:0 1671:= 1664:A 1617:r 1593:A 1568:A 1542:r 1537:r 1530:e 1520:A 1516:+ 1511:r 1506:r 1496:e 1486:A 1482:= 1479:) 1476:r 1473:( 1468:j 1453:0 1450:= 1445:j 1437:) 1432:2 1419:r 1409:r 1406:2 1401:+ 1396:2 1391:r 1383:( 1364:, 1350:2 1345:i 1341:z 1335:i 1331:n 1325:i 1314:T 1308:B 1303:k 1297:r 1287:0 1276:2 1272:e 1266:= 1261:2 1246:) 1243:r 1240:( 1235:j 1225:2 1217:= 1214:) 1211:r 1208:( 1203:j 1193:2 1172:i 1170:z 1168:i 1142:} 1137:) 1131:T 1125:B 1120:k 1114:) 1111:r 1108:( 1103:j 1095:e 1090:i 1086:z 1076:( 1066:) 1063:e 1058:i 1054:z 1050:( 1045:i 1041:n 1036:{ 1030:i 1017:r 1007:0 998:1 990:= 987:) 984:r 981:( 976:j 966:2 940:r 936:T 928:B 925:k 907:) 901:T 895:B 890:k 884:) 881:r 878:( 873:j 865:e 860:i 856:z 846:( 834:i 830:n 826:= 818:i 814:n 785:r 781:r 777:r 760:) 757:r 754:( 749:j 736:r 726:0 717:1 709:= 706:) 703:r 700:( 695:j 685:2 664:j 642:. 613:. 581:2 577:r 571:r 561:0 550:4 543:2 539:e 533:2 529:z 523:1 519:z 512:= 494:r 492:ε 486:r 478:z 474:e 471:2 468:z 464:e 461:1 458:z 412:) 409:m 406:+ 403:n 400:( 396:/ 392:1 387:) 381:m 374:B 362:n 355:A 344:( 339:= 316:m 314:B 312:n 293:2 289:/ 285:1 280:) 267:l 263:C 249:+ 245:a 241:N 231:( 226:= 152:0 148:c 143:/ 139:c 133:= 130:a 93:c 89:a

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Knowledge

Debye–Hückel theory

Index