2843:
from that area until the chemical potential went back to zero; likewise, if the chemical potential somewhere was less than zero, photons would spontaneously appear until the chemical potential went back to zero. Since this process occurs extremely rapidly - at least, it occurs rapidly in the presence of dense charged matter or also in the walls of the textbook example for a photon gas of blackbody radiation - it is safe to assume that the photon chemical potential here is never different from zero. A physical situation where the chemical potential for photons can differ from zero are material-filled optical microcavities, with spacings between cavity mirrors in the wavelength regime. In such two-dimensional cases, photon gases with tuneable chemical potential, much reminiscent to gases of material particles, can be observed.
2037:, Gibbs introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying the interactions of homogeneous substances in contact, i.e. bodies, being in composition part solid, part liquid, and part vapor, and by using a three-dimensional
2891:
105:. When a ball rolls down a hill, it is moving from a higher gravitational potential (higher internal energy thus higher potential for work) to a lower gravitational potential (lower internal energy). In the same way, as molecules move, react, dissolve, melt, etc., they will always tend naturally to go from a higher chemical potential to a lower one, changing the
3911:
by
Ashcroft and Mermin, page 257 note 36. Page 593 of the same book uses, instead, an unusual "flipped" definition, where "chemical potential" is the total chemical potential, which is constant in equilibrium, and "electrochemical potential" is the internal chemical potential; presumably this unusual
128:
and the random motion of molecules. However, it is simpler to describe the process in terms of chemical potentials: For a given temperature, a molecule has a higher chemical potential in a higher-concentration area and a lower chemical potential in a low concentration area. Movement of molecules from
2577:
Electrons in solids have a chemical potential, defined the same way as the chemical potential of a chemical species: The change in free energy when electrons are added or removed from the system. In the case of electrons, the chemical potential is usually expressed in energy per particle rather than
2049:
graph, Gibbs was able to determine three states of equilibrium, i.e. "necessarily stable", "neutral", and "unstable", and whether or not changes will ensue. In 1876, Gibbs built on this framework by introducing the concept of chemical potential so to take into account chemical reactions and states
201:
contains acetic acid. When acid molecules dissociate, the concentration of the undissociated acid molecules (HA) decreases and the concentrations of the product ions (H and A) increase. Thus the chemical potential of HA decreases and the sum of the chemical potentials of H and A increases. When the
2842:
and can very easily and rapidly appear or disappear. Therefore, at thermodynamic equilibrium, the chemical potential of photons is in most physical situations always and everywhere zero. The reason is, if the chemical potential somewhere was higher than zero, photons would spontaneously disappear
2628:
As described above, when describing chemical potential, one has to say "relative to what". In the case of electrons in semiconductors, internal chemical potential is often specified relative to some convenient point in the band structure, e.g., to the bottom of the conduction band. It may also be
2846:
Electric charge is different because it is intrinsically conserved, i.e. it can be neither created nor destroyed. It can, however, diffuse. The "chemical potential of electric charge" controls this diffusion: Electric charge, like anything else, will tend to diffuse from areas of higher chemical
2009:
or combination of elements in given proportions may be considered a substance, whether capable or not of existing by itself as a homogeneous body. This freedom to choose the boundary of the system allows the chemical potential to be applied to a huge range of systems. The term can be used in
2127:
refers to the variation produced by any variations in the state of the parts of the body, and (when different parts of the body are in different states) in the proportion in which the body is divided between the different states. The condition of stable equilibrium is that the value of the
202:
sums of chemical potential of reactants and products are equal the system is at equilibrium and there is no tendency for the reaction to proceed in either the forward or backward direction. This explains why vinegar is acidic, because acetic acid dissociates to some extent, releasing
55:
of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. When both temperature and pressure are held constant, and the number of particles is expressed in moles, the chemical potential is the
2797:
3925:
3474:
930:
740:
1144:
2858:
present, electrons cannot be created or destroyed. Therefore, there is an electron chemical potential that might vary in space, causing diffusion. At very high temperatures, however, electrons and positrons can spontaneously appear out of the vacuum
431:
3899:. In this text, total chemical potential is usually called "electrochemical potential", but sometimes just "chemical potential". The internal chemical potential is referred to by the unwieldy phrase "chemical potential in the absence of the field".
1777:
1676:
1277:
623:
1870:
1483:
2347:). The internal chemical potential includes everything else besides the external potentials, such as density, temperature, and enthalpy. This formalism can be understood by assuming that the total energy of a system,
1366:
In thermodynamic equilibrium, when the system concerned is at constant temperature and pressure but can exchange particles with its external environment, the Gibbs free energy is at its minimum for the system, that is
2969:
1361:
3097:
2241:
3922:
2118:
2311:
2462:
1794:
is useful because it relates individual chemical potentials. For example, in a binary mixture, at constant temperature and pressure, the chemical potentials of the two participants A and B are related by
2660:
into the
Mulliken electronegativity, it is seen that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons, i.e.,
1929:
is the number of moles of B. Every instance of phase or chemical equilibrium is characterized by a constant. For instance, the melting of ice is characterized by a temperature, known as the
2666:
2502:
78:
is zero, as the free energy is at a minimum. In a system in diffusion equilibrium, the chemical potential of any chemical species is uniformly the same everywhere throughout the system.
781:
systems, such as chemical solutions, as it is hard to control the volume and entropy to be constant while particles are added. A more convenient expression may be obtained by making a
3284:
839:
646:
140:
O molecule that is in the solid phase (ice) has a higher chemical potential than a water molecule that is in the liquid phase (water) above 0 °C. When some of the ice melts, H
1038:
2863:), so the chemical potential of electrons by themselves becomes a less useful quantity than the chemical potential of the conserved quantities like (electrons minus positrons).
2633:, however, work function varies from surface to surface even on a completely homogeneous material. Total chemical potential, on the other hand, is usually specified relative to
328:
2529:
2054:
If we wish to express in a single equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when surrounded by a medium of constant pressure
97:
Particles tend to move from higher chemical potential to lower chemical potential because this reduces the free energy. In this way, chemical potential is a generalization of
1396:
2392:
1030:
1005:
1927:
1900:
1782:
These different forms for the chemical potential are all equivalent, meaning that they have the same physical content, and may be useful in different physical situations.
834:
3235:
1174:
3201:
3168:
1573:
1535:
3262:
3131:
1682:
1581:
1182:
528:
976:
953:
2365:
3566:
1801:
3634:
2168:. If two locations have different total chemical potentials for a species, some of it may be due to potentials associated with "external" force fields (
1404:
270:. The electrochemical potential completely characterizes all of the influences on an ion's motion, while the chemical potential includes everything
2851:
are the same. In fact, each conserved quantity is associated with a chemical potential and a corresponding tendency to diffuse to equalize it out.
2176:, etc.), while the rest would be due to "internal" factors (density, temperature, etc.) Therefore, the total chemical potential can be split into
2908:
144:
O molecules convert from solid to the warmer liquid where their chemical potential is lower, so the ice cube shrinks. At the temperature of the
1288:
2164:
The abstract definition of chemical potential given above—total change in free energy per extra mole of substance—is more specifically called
3020:
2190:
4245:
1933:
at which solid and liquid phases are in equilibrium with each other. Chemical potentials can be used to explain the slopes of lines on a
2534:
The phrase "chemical potential" sometimes means "total chemical potential", but that is not universal. In some fields, in particular
2068:
2252:
2397:
124:
to low concentration, until eventually, the concentration is the same everywhere. The microscopic explanation for this is based on
148:, 0 °C, the chemical potentials in water and ice are the same; the ice cube neither grows nor shrinks, and the system is in
4112:
4084:
4065:
4016:
3942:
J. Klaers; J. Schmitt; F. Vewinger & M. Weitz (2010). "Bose–Einstein condensation of photons in an optical microcavity".
51:
with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the
3741:
110:
2640:
In atomic physics, the chemical potential of the electrons in an atom is sometimes said to be the negative of the atom's
319:
315:
2050:
of bodies that are chemically different from each other. In his own words from the aforementioned paper, Gibbs states:
2792:{\displaystyle \mu _{\text{Mulliken}}=-\chi _{\text{Mulliken}}=-{\frac {IP+EA}{2}}=\left}{\delta N}}\right]_{N=N_{0}}.}
311:
304:
4240:
129:
higher chemical potential to lower chemical potential is accompanied by a release of free energy. Therefore, it is a
3592:
4235:
2467:
3268:
with itself and other species . This can be corrected for by factoring in the coefficient of activity of species
245:. In each case the chemical potential of a given species at equilibrium is the same in all phases of the system.
3562:
3469:{\displaystyle \mu _{i}=\mu _{i0}(T,P)+RT\ln(x_{i})+RT\ln(\gamma _{i})=\mu _{i0}(T,P)+RT\ln(x_{i}\gamma _{i}).}
2645:
2173:
925:{\displaystyle \mathrm {d} G=\mathrm {d} U+P\,\mathrm {d} V+V\,\mathrm {d} P-T\,\mathrm {d} S-S\,\mathrm {d} T}
735:{\displaystyle \mathrm {d} U=\sum _{i=1}^{N}\left({\frac {\partial U}{\partial x_{i}}}\right)\mathrm {d} x_{i}}
3683:
519:
is volume. Other work terms, such as those involving electric, magnetic or gravitational fields may be added.
1282:
and the change in Gibbs free energy of a system that is held at constant temperature and pressure is simply
1139:{\displaystyle \mathrm {d} G=-S\,\mathrm {d} T+V\,\mathrm {d} P+\sum _{i=1}^{n}\mu _{i}\,\mathrm {d} N_{i}.}
2875:
426:{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V+\sum _{i=1}^{n}\mu _{i}\,\mathrm {d} N_{i},}
1985:
we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its
4225:
2854:
In the case of electrons, the behaviour depends on temperature and context. At low temperatures, with no
2035:
A Method of
Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces
136:
Another example, not based on concentration but on phase, is an ice cube on a plate above 0 °C. An H
2902:
Generally the chemical potential is given as a sum of an ideal contribution and an excess contribution:
2879:
2649:
2644:. Likewise, the process of chemical potential equalization is sometimes referred to as the process of
2507:
75:
85:
physics, the chemical potential of a system of electrons at zero absolute temperature is known as the
3523:
3503:
3493:
2551:
2169:
1370:
266:
149:
44:
3662:
2370:
2316:
i.e., the external potential is the sum of electric potential, gravitational potential, etc. (where
1965:
Chemical potential was first described by the
American engineer, chemist and mathematical physicist
2618:
1946:
1791:
1010:
985:
778:
238:
160:
1905:
1878:
786:
782:
463:
125:
102:
3657:
3518:
3513:
2816:
2606:
2602:
2023:
1942:
795:
308:
4102:
3923:
Introduction to
Density Functional Theory of Chemical Reactivity: The so-called Conceptual DFT
3896:
3880:
3823:
3213:
1152:
120:
in a homogeneous environment. In this system, the molecules tend to move from areas with high
4039:
3498:
3173:
3140:
2539:
1772:{\displaystyle \mu _{i}=\left({\frac {\partial F}{\partial N_{i}}}\right)_{T,V,N_{j\neq i}}.}
1671:{\displaystyle \mu _{i}=\left({\frac {\partial H}{\partial N_{i}}}\right)_{S,P,N_{j\neq i}},}
1538:
1272:{\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial N_{i}}}\right)_{T,P,N_{j\neq i}},}
618:{\displaystyle \mu _{i}=\left({\frac {\partial U}{\partial N_{i}}}\right)_{S,V,N_{j\neq i}}.}
234:
210:
3864:
1543:
1505:
4176:
4139:
3961:
3826:. This text uses "internal", "external", and "total chemical potential" as in this article.
3649:
3607:
3488:
3240:
3109:
2653:
1489:
500:
470:
67:
48:
8:
4230:
2586:
2543:
2394:, and an external energy due to the interaction of each particle with an external field,
1978:
130:
117:
59:
4180:
4143:
3965:
3653:
3611:
1865:{\displaystyle d\mu _{\text{B}}=-{\frac {n_{\text{A}}}{n_{\text{B}}}}\,d\mu _{\text{A}}}
958:
935:
4204:
3985:
3951:
3675:
2350:
2336:
2026:). Chemical potential is measured in units of energy/particle or, equivalently, energy/
1982:
1941:, which in turn can be derived from the Gibbs–Duhem equation. They are used to explain
1938:
637:
52:
43:. The chemical potential of a species in a mixture is defined as the rate of change of
3264:) contained in the solution. This neglects intermolecular interaction between species
4192:
4155:
4108:
4080:
4061:
4022:
4012:
3977:
3737:
3671:
3558:
2657:
2641:
2634:
790:
156:
71:
62:
3679:
2159:
4184:
4147:
4005:
3989:
3969:
3667:
3615:
2812:
2610:
2535:
2344:
2006:
249:
98:
40:
1974:
4130:
Cook, G.; Dickerson, R. H. (1995-08-01). "Understanding the chemical potential".
4098:
3929:
3137:
contained in the solution. The chemical potential becomes negative infinity when
2860:
2832:
2808:
2578:
energy per mole, and the energy per particle is conventionally given in units of
2137:
2046:
1994:
1954:
1478:{\displaystyle \mu _{1}\,\mathrm {d} N_{1}+\mu _{2}\,\mathrm {d} N_{2}+\dots =0.}
441:
106:
36:
28:
3479:
The plots above give a very rough picture of the ideal and non-ideal situation.
2847:
potential to areas of lower chemical potential. Other conserved quantities like
4094:
2975:
2011:
1950:
633:
275:
242:
20:
4188:
777:
with respect to the corresponding species particle number is inconvenient for
4219:
4196:
4159:
2848:
2828:
2630:
2590:
1966:
1934:
1930:
218:
214:
145:
121:
82:
4026:
2018:
for any system undergoing change. Chemical potential is also referred to as
1575:, expressions for the chemical potential may be obtained in terms of these:
16:
Change in energies of a thermodynamic system with respect to particle number
3981:
2594:
2579:
979:
203:
2964:{\displaystyle \mu _{i}=\mu _{i}^{\text{ideal}}+\mu _{i}^{\text{excess}},}
2622:
2617:
chemical potential) varies from the p-type to the n-type side, while the
2598:
2572:
2027:
1356:{\displaystyle \mathrm {d} G=\sum _{i=1}^{n}\mu _{i}\,\mathrm {d} N_{i}.}
222:
172:
86:
3973:
2820:
226:
3941:
3619:
3092:{\displaystyle \mu _{i}^{\text{ideal}}\approx \mu _{i0}+RT\ln(x_{i}),}
2236:{\displaystyle \mu _{\text{tot}}=\mu _{\text{int}}+\mu _{\text{ext}},}
2005:
Gibbs later noted also that for the purposes of this definition, any
773:
This expression of the chemical potential as a partial derivative of
279:
164:
4151:
260:
always tend to go from higher to lower chemical potential, but they
3508:
2855:
2605:
silicon has a higher internal chemical potential of electrons than
1500:
3956:
1997:
of the mass divided by the quantity of the substance added is the
2871:
2160:
Electrochemical, internal, external, and total chemical potential
2145:
2042:
2015:
1986:
508:
452:
230:
209:
Chemical potentials are important in many aspects of multi-phase
198:
2890:
2827:, a chemical potential for electrons, a chemical potential for
2824:
2811:
has applied the definition of chemical potential to systems in
2153:
2113:{\displaystyle \displaystyle \delta (\epsilon -T\eta +P\nu )=0}
2038:
1990:
32:
2306:{\displaystyle \mu _{\text{ext}}=qV_{\text{ele}}+mgh+\cdots ,}
2874:
is related to the number of particles and the temperature by
2867:
2839:
2629:
specified "relative to vacuum", to yield a quantity known as
2457:{\displaystyle U_{\text{ext}}=N(qV_{\text{ele}}+mgh+\cdots )}
636:, so if its differential exists, then the differential is an
2823:, at every point in space there is a chemical potential for
522:
From the above equation, the chemical potential is given by
3010:. Given this definition, the chemical potential of species
74:, the total sum of the product of chemical potentials and
3851:
by Bard and
Faulkner, 2nd edition, Section 2.2.4(a), 4–5.
2898:
in solution for (left) ideal and (right) real solutions
2585:
Chemical potential plays an especially important role in
1488:
Use of this equality provides the means to establish the
253:
3635:"Chemical potential–a quantity in search of recognition"
3170:, but this does not lead to nonphysical results because
35:
that can be absorbed or released due to a change of the
3006:) is defined as the chemical potential of pure species
3861:
2652:
scale. By inserting the energetic definitions of the
4205:"Values of the chemical potential of 1300 substances"
4093:
3287:
3243:
3216:
3176:
3143:
3112:
3023:
2911:
2669:
2510:
2470:
2400:
2373:
2353:
2255:
2193:
2072:
2071:
1908:
1881:
1804:
1685:
1584:
1546:
1508:
1407:
1373:
1291:
1185:
1155:
1041:
1013:
988:
961:
938:
842:
798:
649:
531:
331:
39:
of the given species, e.g. in a chemical reaction or
4209:
3771:
2464:. The definition of chemical potential applied to
4004:
3468:
3256:
3229:
3195:
3162:
3125:
3091:
2963:
2791:
2523:
2496:
2456:
2386:
2359:
2305:
2235:
2112:
1921:
1894:
1864:
1771:
1670:
1567:
1529:
1477:
1390:
1355:
1271:
1168:
1138:
1024:
999:
982:is applied to) and using the above expression for
970:
947:
924:
828:
734:
617:
425:
303:(atomic, molecular or nuclear) is defined, as all
4079:(7th ed.). Oxford: Oxford University Press.
4042:, by Jean Letessier, Johann Rafelski, p. 91.
2128:expression in the parenthesis shall be a minimum.
116:A simple example is a system of dilute molecules
4217:
4167:Kaplan, T. A. (2006). "The Chemical Potential".
4002:
2885:
2815:and its associated processes. For example, in a
1495:By making further Legendre transformations from
4075:Atkins, Peter William; Paula, Julio De (2002).
3705:
3703:
3632:
3557:Opacity, Walter F. Huebner, W. David Barfield,
2367:, is the sum of two parts: an internal energy,
2339:and height of the container, respectively, and
482:is the infinitesimal change of particle number
4055:
3820:Thermodynamics in Earth and Planetary Sciences
4129:
3829:
2613:diode at equilibrium the chemical potential (
2497:{\displaystyle U_{\text{int}}+U_{\text{ext}}}
3893:The Physics of Solids: Essentials and Beyond
3700:
2988:) is dependent on temperature and pressure.
4074:
3935:
1957:for the solvent using chemical potentials.
1499:to other thermodynamic potentials like the
285:
2589:and is closely related to the concepts of
2001:for that substance in the mass considered.
4060:(8th ed.). Oxford University Press.
3955:
3912:terminology was an unintentional mistake.
3803:
3801:
3799:
3781:, p. 968, University Science Books, 1997.
3779:Physical Chemistry – A Molecular Approach
3715:
3661:
3590:
1993:remaining unchanged, the increase of the
1848:
1447:
1418:
1334:
1149:As a consequence, another expression for
1117:
1074:
1059:
913:
898:
883:
868:
404:
361:
346:
3839:, 3rd ed., p. 352, Academic Press, 2008.
2889:
2802:
2324:are the charge and mass of the species,
4056:Atkins, Peter; de Paula, Julio (2006).
3633:Job, G.; Herrmann, F. (February 2006).
2561:
2132:In this description, as used by Gibbs,
4218:
4166:
3807:
3796:
3790:
3765:
3753:
3721:
3709:
3578:
3546:
2566:
2546:, the term "chemical potential" means
495:as particles are added or subtracted.
312:fundamental equation of thermodynamics
155:A third example is illustrated by the
2978:, the chemical potential of species
2894:The chemical potential of component
2838:In the case of photons, photons are
2625:) is constant throughout the diode.
628:This is because the internal energy
4246:Chemical engineering thermodynamics
3237:only depends on the mole fraction (
2550:chemical potential, while the term
1953:for the solute can be derived from
13:
3877:Physics Of Transition Metal Oxides
3736:, F Mandl, (Wiley, London, 11971)
1715:
1707:
1614:
1606:
1449:
1420:
1375:
1336:
1293:
1215:
1207:
1119:
1076:
1061:
1043:
1015:
990:
915:
900:
885:
870:
855:
844:
718:
697:
689:
651:
561:
553:
406:
363:
348:
333:
14:
4257:
4123:
2648:. This connection comes from the
2524:{\displaystyle \mu _{\text{tot}}}
3593:"The elusive chemical potential"
3133:is the mole fraction of species
2621:(electrochemical potential, or,
2504:yields the above expression for
2062:, this equation may be written:
1949:by the application of pressure.
1902:is the number of moles of A and
4107:(2nd ed.). W. H. Freeman.
4033:
3996:
3915:
3902:
3886:
3870:
3854:
3842:
3813:
3793:, pp. 143–145, section 5.3
3784:
3777:McQuarrie, D. A.; Simon, J. D.
3768:, pp. 150–155, section 5.5
3759:
3591:Baierlein, Ralph (April 2001).
1785:
1391:{\displaystyle \mathrm {d} G=0}
462:is the infinitesimal change of
440:is the infinitesimal change of
282:for more on this terminology.)
264:always go from higher to lower
4169:Journal of Statistical Physics
4040:Hadrons and Quark-Gluon Plasma
4011:. Cambridge University Press.
3895:, by Eleftherios N. Economou,
3747:
3727:
3626:
3584:
3572:
3551:
3540:
3460:
3437:
3419:
3407:
3388:
3375:
3357:
3344:
3326:
3314:
3207:is not present in the system.
3083:
3070:
2749:
2743:
2646:electronegativity equalization
2451:
2417:
2387:{\displaystyle U_{\text{int}}}
2174:gravitational potential energy
2100:
2076:
1007:, a differential relation for
1:
4048:
2886:Ideal vs. non-ideal solutions
1025:{\displaystyle \mathrm {d} G}
1000:{\displaystyle \mathrm {d} U}
473:in thermal equilibrium, and d
3534:
1969:. He defined it as follows:
1922:{\displaystyle n_{\text{B}}}
1895:{\displaystyle n_{\text{A}}}
451:the infinitesimal change of
7:
4132:American Journal of Physics
3756:, pp. 126, section 4.1
3712:, pp. 141, section 6.4
3642:European Journal of Physics
3600:American Journal of Physics
3581:, pp. 227, section 9.2
3482:
3210:This equation assumes that
2866:The chemical potentials of
2345:acceleration due to gravity
2182:external chemical potential
2178:internal chemical potential
92:
76:stoichiometric coefficients
10:
4264:
4077:Atkins' Physical Chemistry
4058:Atkins' Physical Chemistry
3672:10.1088/0143-0807/27/2/018
3529:
2650:Mulliken electronegativity
2570:
2020:partial molar Gibbs energy
1960:
745:for independent variables
4189:10.1007/s10955-005-8067-x
4003:Baierlein, Ralph (2003).
3524:Thermodynamic equilibrium
3504:Excess chemical potential
3494:Electrochemical potential
3278:. This correction yields
3106:is the gas constant, and
2552:electrochemical potential
2170:electric potential energy
1492:for a chemical reaction.
829:{\displaystyle G=U+PV-TS}
322:infinitesimal processes:
267:electrochemical potential
4241:Thermodynamic properties
3879:, by Sadamichi Maekawa,
3230:{\displaystyle \mu _{i}}
3014:in an ideal solution is
2876:Bose–Einstein statistics
2619:total chemical potential
2166:total chemical potential
1947:melting-point depression
1169:{\displaystyle \mu _{i}}
836:. From the differential
286:Thermodynamic definition
239:liquid-liquid extraction
4236:Chemical thermodynamics
3849:Electrochemical Methods
3196:{\displaystyle x_{i}=0}
3163:{\displaystyle x_{i}=0}
787:thermodynamic potential
783:Legendre transformation
307:quantities are, by the
290:The chemical potential
113:to chemical potential.
103:gravitational potential
99:"potentials" in physics
3822:by Jibamitra Ganguly,
3519:Thermodynamic activity
3514:Partial molar property
3470:
3258:
3231:
3197:
3164:
3127:
3093:
2965:
2899:
2880:Fermi–Dirac statistics
2793:
2525:
2498:
2458:
2388:
2361:
2307:
2237:
2130:
2114:
2024:partial molar property
2003:
1943:colligative properties
1923:
1896:
1866:
1773:
1672:
1569:
1568:{\displaystyle F=U-TS}
1531:
1530:{\displaystyle H=U+PV}
1479:
1392:
1357:
1323:
1273:
1170:
1140:
1106:
1026:
1001:
972:
949:
926:
830:
736:
681:
619:
427:
393:
314:. This holds for both
3932:, retrieved May 2016.
3499:Equilibrium chemistry
3471:
3259:
3257:{\displaystyle x_{i}}
3232:
3198:
3165:
3128:
3126:{\displaystyle x_{i}}
3094:
2966:
2893:
2803:Sub-nuclear particles
2794:
2540:semiconductor physics
2526:
2499:
2459:
2389:
2362:
2308:
2238:
2115:
2052:
1971:
1924:
1897:
1867:
1774:
1673:
1570:
1539:Helmholtz free energy
1532:
1480:
1393:
1358:
1303:
1274:
1171:
1141:
1086:
1027:
1002:
973:
950:
927:
831:
737:
661:
620:
428:
373:
235:partition coefficient
211:equilibrium chemistry
3921:Morell, Christophe,
3489:Chemical equilibrium
3285:
3241:
3214:
3174:
3141:
3110:
3021:
2909:
2667:
2654:ionization potential
2562:Systems of particles
2558:chemical potential.
2508:
2468:
2398:
2371:
2351:
2253:
2191:
2069:
1967:Josiah Willard Gibbs
1906:
1879:
1802:
1792:Gibbs–Duhem equation
1683:
1582:
1544:
1506:
1490:equilibrium constant
1405:
1371:
1289:
1183:
1153:
1039:
1011:
986:
959:
936:
840:
796:
647:
529:
501:absolute temperature
471:thermodynamic system
329:
68:chemical equilibrium
49:thermodynamic system
4181:2006JSP...122.1237K
4144:1995AmJPh..63..737C
3974:10.1038/nature09567
3966:2010Natur.468..545K
3909:Solid State Physics
3734:Statistical Physics
3654:2006EJPh...27..353J
3612:2001AmJPh..69..423B
3203:means that species
3038:
2957:
2939:
2587:solid-state physics
2567:Electrons in solids
2544:solid-state physics
206:into the solution.
131:spontaneous process
4226:Physical chemistry
3928:2017-08-28 at the
3837:Physical Chemistry
3466:
3254:
3227:
3193:
3160:
3123:
3089:
3024:
2961:
2943:
2925:
2900:
2817:quark–gluon plasma
2789:
2521:
2494:
2454:
2384:
2357:
2337:electric potential
2303:
2233:
2110:
2109:
2033:In his 1873 paper
1983:hydrostatic stress
1939:Clapeyron equation
1919:
1892:
1862:
1769:
1668:
1565:
1527:
1475:
1398:. It follows that
1388:
1353:
1269:
1166:
1136:
1022:
997:
971:{\displaystyle TS}
968:
948:{\displaystyle PV}
945:
922:
826:
732:
638:exact differential
615:
423:
111:conjugate variable
53:partial derivative
25:chemical potential
4114:978-0-7167-1088-2
4086:978-0-19-879285-7
4067:978-0-19-870072-2
4018:978-0-521-65838-6
3950:(7323): 545–548.
3863:, by Norio Sato,
3620:10.1119/1.1336839
3036:
2955:
2937:
2807:In recent years,
2761:
2724:
2693:
2677:
2658:electron affinity
2642:electronegativity
2635:electrical ground
2518:
2491:
2478:
2430:
2408:
2381:
2360:{\displaystyle U}
2279:
2263:
2227:
2214:
2201:
2148:of the body, and
1916:
1889:
1859:
1846:
1843:
1833:
1815:
1729:
1628:
1229:
791:Gibbs free energy
711:
575:
515:is pressure, and
157:chemical reaction
72:phase equilibrium
63:Gibbs free energy
4253:
4212:
4200:
4175:(6): 1237–1260.
4163:
4118:
4090:
4071:
4043:
4037:
4031:
4030:
4010:
4000:
3994:
3993:
3959:
3939:
3933:
3919:
3913:
3906:
3900:
3890:
3884:
3874:
3868:
3858:
3852:
3846:
3840:
3835:Mortimer, R. G.
3833:
3827:
3817:
3811:
3805:
3794:
3788:
3782:
3775:
3769:
3763:
3757:
3751:
3745:
3731:
3725:
3719:
3713:
3707:
3698:
3697:
3695:
3694:
3688:
3682:. Archived from
3665:
3639:
3630:
3624:
3623:
3597:
3588:
3582:
3576:
3570:
3555:
3549:
3544:
3475:
3473:
3472:
3467:
3459:
3458:
3449:
3448:
3406:
3405:
3387:
3386:
3356:
3355:
3313:
3312:
3297:
3296:
3263:
3261:
3260:
3255:
3253:
3252:
3236:
3234:
3233:
3228:
3226:
3225:
3202:
3200:
3199:
3194:
3186:
3185:
3169:
3167:
3166:
3161:
3153:
3152:
3132:
3130:
3129:
3124:
3122:
3121:
3098:
3096:
3095:
3090:
3082:
3081:
3054:
3053:
3037:
3034:
3032:
2970:
2968:
2967:
2962:
2956:
2953:
2951:
2938:
2935:
2933:
2921:
2920:
2835:, and so forth.
2813:particle physics
2798:
2796:
2795:
2790:
2785:
2784:
2783:
2782:
2766:
2762:
2760:
2752:
2735:
2725:
2720:
2703:
2695:
2694:
2691:
2679:
2678:
2675:
2554:is used to mean
2536:electrochemistry
2530:
2528:
2527:
2522:
2520:
2519:
2516:
2503:
2501:
2500:
2495:
2493:
2492:
2489:
2480:
2479:
2476:
2463:
2461:
2460:
2455:
2432:
2431:
2428:
2410:
2409:
2406:
2393:
2391:
2390:
2385:
2383:
2382:
2379:
2366:
2364:
2363:
2358:
2312:
2310:
2309:
2304:
2281:
2280:
2277:
2265:
2264:
2261:
2242:
2240:
2239:
2234:
2229:
2228:
2225:
2216:
2215:
2212:
2203:
2202:
2199:
2119:
2117:
2116:
2111:
2058:and temperature
2007:chemical element
1928:
1926:
1925:
1920:
1918:
1917:
1914:
1901:
1899:
1898:
1893:
1891:
1890:
1887:
1871:
1869:
1868:
1863:
1861:
1860:
1857:
1847:
1845:
1844:
1841:
1835:
1834:
1831:
1825:
1817:
1816:
1813:
1778:
1776:
1775:
1770:
1765:
1764:
1763:
1762:
1734:
1730:
1728:
1727:
1726:
1713:
1705:
1695:
1694:
1677:
1675:
1674:
1669:
1664:
1663:
1662:
1661:
1633:
1629:
1627:
1626:
1625:
1612:
1604:
1594:
1593:
1574:
1572:
1571:
1566:
1536:
1534:
1533:
1528:
1484:
1482:
1481:
1476:
1462:
1461:
1452:
1446:
1445:
1433:
1432:
1423:
1417:
1416:
1397:
1395:
1394:
1389:
1378:
1362:
1360:
1359:
1354:
1349:
1348:
1339:
1333:
1332:
1322:
1317:
1296:
1278:
1276:
1275:
1270:
1265:
1264:
1263:
1262:
1234:
1230:
1228:
1227:
1226:
1213:
1205:
1195:
1194:
1175:
1173:
1172:
1167:
1165:
1164:
1145:
1143:
1142:
1137:
1132:
1131:
1122:
1116:
1115:
1105:
1100:
1079:
1064:
1046:
1031:
1029:
1028:
1023:
1018:
1006:
1004:
1003:
998:
993:
977:
975:
974:
969:
954:
952:
951:
946:
931:
929:
928:
923:
918:
903:
888:
873:
858:
847:
835:
833:
832:
827:
779:condensed-matter
741:
739:
738:
733:
731:
730:
721:
716:
712:
710:
709:
708:
695:
687:
680:
675:
654:
624:
622:
621:
616:
611:
610:
609:
608:
580:
576:
574:
573:
572:
559:
551:
541:
540:
432:
430:
429:
424:
419:
418:
409:
403:
402:
392:
387:
366:
351:
336:
309:phenomenological
250:electrochemistry
41:phase transition
4263:
4262:
4256:
4255:
4254:
4252:
4251:
4250:
4216:
4215:
4203:
4152:10.1119/1.17844
4126:
4121:
4115:
4104:Thermal Physics
4099:Herbert Kroemer
4095:Kittel, Charles
4087:
4068:
4051:
4046:
4038:
4034:
4019:
4007:Thermal Physics
4001:
3997:
3940:
3936:
3930:Wayback Machine
3920:
3916:
3907:
3903:
3891:
3887:
3875:
3871:
3859:
3855:
3847:
3843:
3834:
3830:
3818:
3814:
3806:
3797:
3789:
3785:
3776:
3772:
3764:
3760:
3752:
3748:
3732:
3728:
3720:
3716:
3708:
3701:
3692:
3690:
3686:
3663:10.1.1.568.9205
3637:
3631:
3627:
3595:
3589:
3585:
3577:
3573:
3556:
3552:
3545:
3541:
3537:
3532:
3485:
3454:
3450:
3444:
3440:
3398:
3394:
3382:
3378:
3351:
3347:
3305:
3301:
3292:
3288:
3286:
3283:
3282:
3277:
3248:
3244:
3242:
3239:
3238:
3221:
3217:
3215:
3212:
3211:
3181:
3177:
3175:
3172:
3171:
3148:
3144:
3142:
3139:
3138:
3117:
3113:
3111:
3108:
3107:
3077:
3073:
3046:
3042:
3033:
3028:
3022:
3019:
3018:
2997:
2987:
2952:
2947:
2934:
2929:
2916:
2912:
2910:
2907:
2906:
2888:
2861:pair production
2833:electric charge
2809:thermal physics
2805:
2778:
2774:
2767:
2753:
2736:
2734:
2730:
2729:
2704:
2702:
2690:
2686:
2674:
2670:
2668:
2665:
2664:
2601:. For example,
2575:
2569:
2564:
2515:
2511:
2509:
2506:
2505:
2488:
2484:
2475:
2471:
2469:
2466:
2465:
2427:
2423:
2405:
2401:
2399:
2396:
2395:
2378:
2374:
2372:
2369:
2368:
2352:
2349:
2348:
2330:
2276:
2272:
2260:
2256:
2254:
2251:
2250:
2224:
2220:
2211:
2207:
2198:
2194:
2192:
2189:
2188:
2162:
2138:internal energy
2070:
2067:
2066:
2047:internal energy
1963:
1913:
1909:
1907:
1904:
1903:
1886:
1882:
1880:
1877:
1876:
1856:
1852:
1840:
1836:
1830:
1826:
1824:
1812:
1808:
1803:
1800:
1799:
1788:
1752:
1748:
1735:
1722:
1718:
1714:
1706:
1704:
1700:
1699:
1690:
1686:
1684:
1681:
1680:
1651:
1647:
1634:
1621:
1617:
1613:
1605:
1603:
1599:
1598:
1589:
1585:
1583:
1580:
1579:
1545:
1542:
1541:
1507:
1504:
1503:
1457:
1453:
1448:
1441:
1437:
1428:
1424:
1419:
1412:
1408:
1406:
1403:
1402:
1374:
1372:
1369:
1368:
1344:
1340:
1335:
1328:
1324:
1318:
1307:
1292:
1290:
1287:
1286:
1252:
1248:
1235:
1222:
1218:
1214:
1206:
1204:
1200:
1199:
1190:
1186:
1184:
1181:
1180:
1160:
1156:
1154:
1151:
1150:
1127:
1123:
1118:
1111:
1107:
1101:
1090:
1075:
1060:
1042:
1040:
1037:
1036:
1014:
1012:
1009:
1008:
989:
987:
984:
983:
960:
957:
956:
937:
934:
933:
914:
899:
884:
869:
854:
843:
841:
838:
837:
797:
794:
793:
764:
758:
751:
726:
722:
717:
704:
700:
696:
688:
686:
682:
676:
665:
650:
648:
645:
644:
598:
594:
581:
568:
564:
560:
552:
550:
546:
545:
536:
532:
530:
527:
526:
490:
481:
442:internal energy
414:
410:
405:
398:
394:
388:
377:
362:
347:
332:
330:
327:
326:
298:
288:
182:
143:
139:
109:, which is the
107:particle number
95:
37:particle number
17:
12:
11:
5:
4261:
4260:
4249:
4248:
4243:
4238:
4233:
4228:
4214:
4213:
4201:
4164:
4138:(8): 737–742.
4125:
4124:External links
4122:
4120:
4119:
4113:
4101:(1980-01-15).
4091:
4085:
4072:
4066:
4052:
4050:
4047:
4045:
4044:
4032:
4017:
3995:
3934:
3914:
3901:
3885:
3869:
3853:
3841:
3828:
3812:
3810:, pp. 124
3795:
3783:
3770:
3758:
3746:
3726:
3724:, pp. 357
3714:
3699:
3648:(2): 353–371.
3625:
3606:(4): 423–434.
3583:
3571:
3550:
3538:
3536:
3533:
3531:
3528:
3527:
3526:
3521:
3516:
3511:
3506:
3501:
3496:
3491:
3484:
3481:
3477:
3476:
3465:
3462:
3457:
3453:
3447:
3443:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3404:
3401:
3397:
3393:
3390:
3385:
3381:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3354:
3350:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3311:
3308:
3304:
3300:
3295:
3291:
3273:
3272:, defined as Îł
3251:
3247:
3224:
3220:
3192:
3189:
3184:
3180:
3159:
3156:
3151:
3147:
3120:
3116:
3100:
3099:
3088:
3085:
3080:
3076:
3072:
3069:
3066:
3063:
3060:
3057:
3052:
3049:
3045:
3041:
3031:
3027:
2992:
2983:
2976:ideal solution
2972:
2971:
2960:
2950:
2946:
2942:
2932:
2928:
2924:
2919:
2915:
2887:
2884:
2882:respectively.
2804:
2801:
2800:
2799:
2788:
2781:
2777:
2773:
2770:
2765:
2759:
2756:
2751:
2748:
2745:
2742:
2739:
2733:
2728:
2723:
2719:
2716:
2713:
2710:
2707:
2701:
2698:
2689:
2685:
2682:
2673:
2609:silicon. In a
2571:Main article:
2568:
2565:
2563:
2560:
2514:
2487:
2483:
2474:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2426:
2422:
2419:
2416:
2413:
2404:
2377:
2356:
2328:
2314:
2313:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2275:
2271:
2268:
2259:
2244:
2243:
2232:
2223:
2219:
2210:
2206:
2197:
2161:
2158:
2144:refers to the
2136:refers to the
2121:
2120:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2012:thermodynamics
1962:
1959:
1912:
1885:
1873:
1872:
1855:
1851:
1839:
1829:
1823:
1820:
1811:
1807:
1787:
1784:
1780:
1779:
1768:
1761:
1758:
1755:
1751:
1747:
1744:
1741:
1738:
1733:
1725:
1721:
1717:
1712:
1709:
1703:
1698:
1693:
1689:
1678:
1667:
1660:
1657:
1654:
1650:
1646:
1643:
1640:
1637:
1632:
1624:
1620:
1616:
1611:
1608:
1602:
1597:
1592:
1588:
1564:
1561:
1558:
1555:
1552:
1549:
1526:
1523:
1520:
1517:
1514:
1511:
1486:
1485:
1474:
1471:
1468:
1465:
1460:
1456:
1451:
1444:
1440:
1436:
1431:
1427:
1422:
1415:
1411:
1387:
1384:
1381:
1377:
1364:
1363:
1352:
1347:
1343:
1338:
1331:
1327:
1321:
1316:
1313:
1310:
1306:
1302:
1299:
1295:
1280:
1279:
1268:
1261:
1258:
1255:
1251:
1247:
1244:
1241:
1238:
1233:
1225:
1221:
1217:
1212:
1209:
1203:
1198:
1193:
1189:
1163:
1159:
1147:
1146:
1135:
1130:
1126:
1121:
1114:
1110:
1104:
1099:
1096:
1093:
1089:
1085:
1082:
1078:
1073:
1070:
1067:
1063:
1058:
1055:
1052:
1049:
1045:
1021:
1017:
996:
992:
967:
964:
944:
941:
921:
917:
912:
909:
906:
902:
897:
894:
891:
887:
882:
879:
876:
872:
867:
864:
861:
857:
853:
850:
846:
825:
822:
819:
816:
813:
810:
807:
804:
801:
762:
756:
749:
743:
742:
729:
725:
720:
715:
707:
703:
699:
694:
691:
685:
679:
674:
671:
668:
664:
660:
657:
653:
634:state function
626:
625:
614:
607:
604:
601:
597:
593:
590:
587:
584:
579:
571:
567:
563:
558:
555:
549:
544:
539:
535:
486:
477:
434:
433:
422:
417:
413:
408:
401:
397:
391:
386:
383:
380:
376:
372:
369:
365:
360:
357:
354:
350:
345:
342:
339:
335:
294:
287:
284:
276:electric force
243:chromatography
196:
195:
180:
141:
137:
126:kinetic theory
94:
91:
21:thermodynamics
15:
9:
6:
4:
3:
2:
4259:
4258:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4229:
4227:
4224:
4223:
4221:
4210:
4206:
4202:
4198:
4194:
4190:
4186:
4182:
4178:
4174:
4170:
4165:
4161:
4157:
4153:
4149:
4145:
4141:
4137:
4133:
4128:
4127:
4116:
4110:
4106:
4105:
4100:
4096:
4092:
4088:
4082:
4078:
4073:
4069:
4063:
4059:
4054:
4053:
4041:
4036:
4028:
4024:
4020:
4014:
4009:
4008:
3999:
3991:
3987:
3983:
3979:
3975:
3971:
3967:
3963:
3958:
3953:
3949:
3945:
3938:
3931:
3927:
3924:
3918:
3910:
3905:
3898:
3894:
3889:
3882:
3878:
3873:
3866:
3862:
3857:
3850:
3845:
3838:
3832:
3825:
3821:
3816:
3809:
3804:
3802:
3800:
3792:
3787:
3780:
3774:
3767:
3762:
3755:
3750:
3743:
3742:0 471 56658 6
3739:
3735:
3730:
3723:
3718:
3711:
3706:
3704:
3689:on 2015-09-24
3685:
3681:
3677:
3673:
3669:
3664:
3659:
3655:
3651:
3647:
3643:
3636:
3629:
3621:
3617:
3613:
3609:
3605:
3601:
3594:
3587:
3580:
3575:
3568:
3564:
3560:
3554:
3548:
3543:
3539:
3525:
3522:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3486:
3480:
3463:
3455:
3451:
3445:
3441:
3434:
3431:
3428:
3425:
3422:
3416:
3413:
3410:
3402:
3399:
3395:
3391:
3383:
3379:
3372:
3369:
3366:
3363:
3360:
3352:
3348:
3341:
3338:
3335:
3332:
3329:
3323:
3320:
3317:
3309:
3306:
3302:
3298:
3293:
3289:
3281:
3280:
3279:
3276:
3271:
3267:
3249:
3245:
3222:
3218:
3208:
3206:
3190:
3187:
3182:
3178:
3157:
3154:
3149:
3145:
3136:
3118:
3114:
3105:
3086:
3078:
3074:
3067:
3064:
3061:
3058:
3055:
3050:
3047:
3043:
3039:
3029:
3025:
3017:
3016:
3015:
3013:
3009:
3005:
3001:
2995:
2989:
2986:
2981:
2977:
2958:
2948:
2944:
2940:
2930:
2926:
2922:
2917:
2913:
2905:
2904:
2903:
2897:
2892:
2883:
2881:
2877:
2873:
2869:
2864:
2862:
2857:
2852:
2850:
2849:baryon number
2844:
2841:
2836:
2834:
2830:
2829:baryon number
2826:
2822:
2818:
2814:
2810:
2786:
2779:
2775:
2771:
2768:
2763:
2757:
2754:
2746:
2740:
2737:
2731:
2726:
2721:
2717:
2714:
2711:
2708:
2705:
2699:
2696:
2687:
2683:
2680:
2671:
2663:
2662:
2661:
2659:
2655:
2651:
2647:
2643:
2638:
2636:
2632:
2631:work function
2626:
2624:
2620:
2616:
2612:
2608:
2604:
2600:
2596:
2592:
2591:work function
2588:
2583:
2581:
2574:
2559:
2557:
2553:
2549:
2545:
2541:
2537:
2532:
2512:
2485:
2481:
2472:
2448:
2445:
2442:
2439:
2436:
2433:
2424:
2420:
2414:
2411:
2402:
2375:
2354:
2346:
2342:
2338:
2334:
2327:
2323:
2319:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2273:
2269:
2266:
2257:
2249:
2248:
2247:
2230:
2221:
2217:
2208:
2204:
2195:
2187:
2186:
2185:
2183:
2179:
2175:
2171:
2167:
2157:
2156:of the body.
2155:
2151:
2147:
2143:
2140:of the body,
2139:
2135:
2129:
2126:
2106:
2103:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2073:
2065:
2064:
2063:
2061:
2057:
2051:
2048:
2044:
2040:
2036:
2031:
2029:
2025:
2021:
2017:
2013:
2008:
2002:
2000:
1996:
1992:
1988:
1984:
1980:
1976:
1970:
1968:
1958:
1956:
1952:
1948:
1944:
1940:
1937:by using the
1936:
1935:phase diagram
1932:
1931:melting point
1910:
1883:
1853:
1849:
1837:
1827:
1821:
1818:
1809:
1805:
1798:
1797:
1796:
1793:
1783:
1766:
1759:
1756:
1753:
1749:
1745:
1742:
1739:
1736:
1731:
1723:
1719:
1710:
1701:
1696:
1691:
1687:
1679:
1665:
1658:
1655:
1652:
1648:
1644:
1641:
1638:
1635:
1630:
1622:
1618:
1609:
1600:
1595:
1590:
1586:
1578:
1577:
1576:
1562:
1559:
1556:
1553:
1550:
1547:
1540:
1524:
1521:
1518:
1515:
1512:
1509:
1502:
1498:
1493:
1491:
1472:
1469:
1466:
1463:
1458:
1454:
1442:
1438:
1434:
1429:
1425:
1413:
1409:
1401:
1400:
1399:
1385:
1382:
1379:
1350:
1345:
1341:
1329:
1325:
1319:
1314:
1311:
1308:
1304:
1300:
1297:
1285:
1284:
1283:
1266:
1259:
1256:
1253:
1249:
1245:
1242:
1239:
1236:
1231:
1223:
1219:
1210:
1201:
1196:
1191:
1187:
1179:
1178:
1177:
1161:
1157:
1133:
1128:
1124:
1112:
1108:
1102:
1097:
1094:
1091:
1087:
1083:
1080:
1071:
1068:
1065:
1056:
1053:
1050:
1047:
1035:
1034:
1033:
1032:is obtained:
1019:
994:
981:
965:
962:
942:
939:
919:
910:
907:
904:
895:
892:
889:
880:
877:
874:
865:
862:
859:
851:
848:
823:
820:
817:
814:
811:
808:
805:
802:
799:
792:
788:
784:
780:
776:
771:
769:
765:
755:
748:
727:
723:
713:
705:
701:
692:
683:
677:
672:
669:
666:
662:
658:
655:
643:
642:
641:
639:
635:
631:
612:
605:
602:
599:
595:
591:
588:
585:
582:
577:
569:
565:
556:
547:
542:
537:
533:
525:
524:
523:
520:
518:
514:
510:
506:
502:
498:
494:
489:
485:
480:
476:
472:
468:
465:
461:
457:
454:
450:
446:
443:
439:
420:
415:
411:
399:
395:
389:
384:
381:
378:
374:
370:
367:
358:
355:
352:
343:
340:
337:
325:
324:
323:
321:
317:
313:
310:
306:
302:
297:
293:
283:
281:
277:
273:
269:
268:
263:
259:
255:
251:
246:
244:
240:
236:
232:
228:
224:
220:
216:
212:
207:
205:
204:hydrogen ions
200:
194:
190:
186:
185:
184:
178:
174:
170:
166:
162:
158:
153:
151:
147:
146:melting point
134:
132:
127:
123:
122:concentration
119:
114:
112:
108:
104:
100:
90:
88:
84:
83:semiconductor
79:
77:
73:
69:
65:
64:
61:
54:
50:
46:
42:
38:
34:
30:
26:
22:
4208:
4172:
4168:
4135:
4131:
4103:
4076:
4057:
4035:
4006:
3998:
3947:
3943:
3937:
3917:
3908:
3904:
3892:
3888:
3876:
3872:
3860:
3856:
3848:
3844:
3836:
3831:
3819:
3815:
3786:
3778:
3773:
3761:
3749:
3733:
3729:
3717:
3691:. Retrieved
3684:the original
3645:
3641:
3628:
3603:
3599:
3586:
3574:
3553:
3542:
3478:
3274:
3269:
3265:
3209:
3204:
3134:
3103:
3101:
3011:
3007:
3003:
2999:
2993:
2990:
2984:
2979:
2973:
2901:
2895:
2865:
2853:
2845:
2837:
2806:
2639:
2627:
2614:
2611:p–n junction
2595:Fermi energy
2584:
2580:electronvolt
2576:
2555:
2547:
2533:
2340:
2332:
2325:
2321:
2317:
2315:
2245:
2181:
2177:
2165:
2163:
2149:
2141:
2133:
2131:
2124:
2122:
2059:
2055:
2053:
2034:
2032:
2019:
2004:
1998:
1972:
1964:
1955:Raoult's law
1874:
1789:
1786:Applications
1781:
1496:
1494:
1487:
1365:
1281:
1148:
980:product rule
774:
772:
767:
760:
753:
746:
744:
629:
627:
521:
516:
512:
504:
496:
492:
487:
483:
478:
474:
466:
459:
455:
448:
444:
437:
435:
320:irreversible
300:
295:
291:
289:
271:
265:
261:
257:
247:
213:, including
208:
197:
192:
188:
176:
168:
161:dissociation
154:
135:
115:
96:
80:
57:
24:
18:
3881:p. 323
3824:p. 240
3808:Kittel 1980
3791:Atkins 2006
3766:Atkins 2006
3754:Atkins 2006
3722:Kittel 1980
3710:Atkins 2002
3579:Atkins 2002
3567:p. 105
3547:Atkins 2006
2623:Fermi level
2599:Fermi level
2573:Fermi level
1975:homogeneous
1951:Henry's law
785:to another
491:of species
299:of species
223:evaporation
173:acetic acid
150:equilibrium
87:Fermi level
45:free energy
4231:Potentials
4220:Categories
4049:References
3744:, page 88.
3693:2009-02-12
3563:1461487978
2821:QCD matter
2022:(see also
1977:mass in a
1973:If to any
316:reversible
227:solubility
4197:0022-4715
4160:0002-9505
3957:1007.4088
3865:pages 4–5
3658:CiteSeerX
3535:Citations
3452:γ
3435:
3396:μ
3380:γ
3373:
3342:
3303:μ
3290:μ
3219:μ
3068:
3044:μ
3040:≈
3026:μ
2945:μ
2927:μ
2914:μ
2856:positrons
2819:or other
2755:δ
2738:δ
2700:−
2688:χ
2684:−
2672:μ
2513:μ
2449:⋯
2298:⋯
2258:μ
2222:μ
2209:μ
2196:μ
2098:ν
2089:η
2083:−
2080:ϵ
2074:δ
1999:potential
1854:μ
1822:−
1810:μ
1757:≠
1716:∂
1708:∂
1688:μ
1656:≠
1615:∂
1607:∂
1587:μ
1557:−
1467:⋯
1439:μ
1410:μ
1326:μ
1305:∑
1257:≠
1216:∂
1208:∂
1188:μ
1176:results:
1158:μ
1109:μ
1088:∑
1054:−
908:−
893:−
818:−
698:∂
690:∂
663:∑
603:≠
562:∂
554:∂
534:μ
396:μ
375:∑
356:−
305:intensive
171:(such as
165:weak acid
118:diffusing
4027:39633743
3982:21107426
3926:Archived
3897:page 140
3680:16146320
3509:Fugacity
3483:See also
2872:fermions
2692:Mulliken
2676:Mulliken
2615:internal
2548:internal
2335:are the
1945:such as
1501:enthalpy
759:, ... ,
640:such as
101:such as
93:Overview
58:partial
4177:Bibcode
4140:Bibcode
3990:4349640
3962:Bibcode
3650:Bibcode
3608:Bibcode
3530:Sources
3002:,
2825:photons
2343:is the
2152:is the
2146:entropy
2043:entropy
2016:physics
1987:entropy
1961:History
509:entropy
453:entropy
436:where d
278:. (See
231:osmosis
219:boiling
215:melting
199:Vinegar
31:is the
29:species
4195:
4158:
4111:
4083:
4064:
4025:
4015:
3988:
3980:
3944:Nature
3740:
3678:
3660:
3561:
3102:where
2974:In an
2954:excess
2868:bosons
2840:bosons
2607:p-type
2603:n-type
2597:, and
2582:(eV).
2542:, and
2246:where
2154:volume
2123:Where
2039:volume
1995:energy
1991:volume
1875:where
978:, the
789:: the
469:for a
464:volume
272:except
191:⇌ H +
183:COO):
70:or in
33:energy
23:, the
3986:S2CID
3952:arXiv
3687:(PDF)
3676:S2CID
3638:(PDF)
3596:(PDF)
3035:ideal
2936:ideal
2556:total
1979:state
932:(for
632:is a
280:below
163:of a
66:. At
60:molar
47:of a
27:of a
4193:ISSN
4156:ISSN
4109:ISBN
4081:ISBN
4062:ISBN
4023:OCLC
4013:ISBN
3978:PMID
3738:ISBN
3559:ISBN
2878:and
2870:and
2656:and
2331:and
2320:and
2180:and
2028:mole
2014:and
1989:and
1790:The
1537:and
955:and
318:and
274:the
254:ions
241:and
179:= CH
4185:doi
4173:122
4148:doi
3970:doi
3948:468
3668:doi
3616:doi
2517:tot
2490:ext
2477:int
2429:ele
2407:ext
2380:int
2329:ele
2278:ele
2262:ext
2226:ext
2213:int
2200:tot
1981:of
770:.
766:of
507:is
499:is
458:, d
447:, d
258:not
256:do
248:In
159:of
81:In
19:In
4222::
4207:.
4191:.
4183:.
4171:.
4154:.
4146:.
4136:63
4134:.
4097:;
4021:.
3984:.
3976:.
3968:.
3960:.
3946:.
3798:^
3702:^
3674:.
3666:.
3656:.
3646:27
3644:.
3640:.
3614:.
3604:69
3602:.
3598:.
3565:,
3432:ln
3370:ln
3339:ln
3065:ln
2982:(ÎĽ
2831:,
2637:.
2593:,
2538:,
2531:.
2184::
2172:,
2030:.
1473:0.
752:,
511:,
503:,
262:do
252:,
237:,
233:,
229:,
225:,
221:,
217:,
175:,
152:.
133:.
89:.
4211:.
4199:.
4187::
4179::
4162:.
4150::
4142::
4117:.
4089:.
4070:.
4029:.
3992:.
3972::
3964::
3954::
3883:.
3867:.
3696:.
3670::
3652::
3622:.
3618::
3610::
3569:.
3464:.
3461:)
3456:i
3446:i
3442:x
3438:(
3429:T
3426:R
3423:+
3420:)
3417:P
3414:,
3411:T
3408:(
3403:0
3400:i
3392:=
3389:)
3384:i
3376:(
3367:T
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3361:+
3358:)
3353:i
3349:x
3345:(
3336:T
3333:R
3330:+
3327:)
3324:P
3321:,
3318:T
3315:(
3310:0
3307:i
3299:=
3294:i
3275:i
3270:i
3266:i
3250:i
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3223:i
3205:i
3191:0
3188:=
3183:i
3179:x
3158:0
3155:=
3150:i
3146:x
3135:i
3119:i
3115:x
3104:R
3087:,
3084:)
3079:i
3075:x
3071:(
3062:T
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3056:+
3051:0
3048:i
3030:i
3012:i
3008:i
3004:P
3000:T
2998:(
2996:0
2994:i
2991:ÎĽ
2985:i
2980:i
2959:,
2949:i
2941:+
2931:i
2923:=
2918:i
2896:i
2859:(
2787:.
2780:0
2776:N
2772:=
2769:N
2764:]
2758:N
2750:]
2747:N
2744:[
2741:E
2732:[
2727:=
2722:2
2718:A
2715:E
2712:+
2709:P
2706:I
2697:=
2681:=
2486:U
2482:+
2473:U
2452:)
2446:+
2443:h
2440:g
2437:m
2434:+
2425:V
2421:q
2418:(
2415:N
2412:=
2403:U
2376:U
2355:U
2341:g
2333:h
2326:V
2322:m
2318:q
2301:,
2295:+
2292:h
2289:g
2286:m
2283:+
2274:V
2270:q
2267:=
2231:,
2218:+
2205:=
2150:ν
2142:η
2134:ε
2125:δ
2107:0
2104:=
2101:)
2095:P
2092:+
2086:T
2077:(
2060:T
2056:P
2045:–
2041:–
1915:B
1911:n
1888:A
1884:n
1858:A
1850:d
1842:B
1838:n
1832:A
1828:n
1819:=
1814:B
1806:d
1767:.
1760:i
1754:j
1750:N
1746:,
1743:V
1740:,
1737:T
1732:)
1724:i
1720:N
1711:F
1702:(
1697:=
1692:i
1666:,
1659:i
1653:j
1649:N
1645:,
1642:P
1639:,
1636:S
1631:)
1623:i
1619:N
1610:H
1601:(
1596:=
1591:i
1563:S
1560:T
1554:U
1551:=
1548:F
1525:V
1522:P
1519:+
1516:U
1513:=
1510:H
1497:U
1470:=
1464:+
1459:2
1455:N
1450:d
1443:2
1435:+
1430:1
1426:N
1421:d
1414:1
1386:0
1383:=
1380:G
1376:d
1351:.
1346:i
1342:N
1337:d
1330:i
1320:n
1315:1
1312:=
1309:i
1301:=
1298:G
1294:d
1267:,
1260:i
1254:j
1250:N
1246:,
1243:P
1240:,
1237:T
1232:)
1224:i
1220:N
1211:G
1202:(
1197:=
1192:i
1162:i
1134:.
1129:i
1125:N
1120:d
1113:i
1103:n
1098:1
1095:=
1092:i
1084:+
1081:P
1077:d
1072:V
1069:+
1066:T
1062:d
1057:S
1051:=
1048:G
1044:d
1020:G
1016:d
995:U
991:d
966:S
963:T
943:V
940:P
920:T
916:d
911:S
905:S
901:d
896:T
890:P
886:d
881:V
878:+
875:V
871:d
866:P
863:+
860:U
856:d
852:=
849:G
845:d
824:S
821:T
815:V
812:P
809:+
806:U
803:=
800:G
775:U
768:U
763:N
761:x
757:2
754:x
750:1
747:x
728:i
724:x
719:d
714:)
706:i
702:x
693:U
684:(
678:N
673:1
670:=
667:i
659:=
656:U
652:d
630:U
613:.
606:i
600:j
596:N
592:,
589:V
586:,
583:S
578:)
570:i
566:N
557:U
548:(
543:=
538:i
517:V
513:P
505:S
497:T
493:i
488:i
484:N
479:i
475:N
467:V
460:V
456:S
449:S
445:U
438:U
421:,
416:i
412:N
407:d
400:i
390:n
385:1
382:=
379:i
371:+
368:V
364:d
359:P
353:S
349:d
344:T
341:=
338:U
334:d
301:i
296:i
292:ÎĽ
193:A
189:A
187:H
181:3
177:A
169:A
167:H
142:2
138:2
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