5899:
6991:, especially with those constants governing the formation of comparatively minor species, but the decision as to how large is acceptable remains subjective. The decision process as to whether or not to include comparatively uncertain equilibria in a model, and for the comparison of competing models in general, can be made objective and has been outlined by Hamilton.
5332:
2379:
1698:, as shown above. Electrical charges are not shown in general expressions such as this and are often omitted from specific expressions, for simplicity of notation. In fact, electrical charges have no bearing on the equilibrium processes other that there being a requirement for overall electrical neutrality in all systems.
2954:
iterations. The logarithms of the free concentrations may be refined rather than the free concentrations themselves. Refinement of the logarithms of the free concentrations has the added advantage of automatically imposing a non-negativity constraint on the free concentrations. Once the free reactant
6981:
should not be much larger than 0.01. In exploratory work where the nature of the species present is not known in advance, several different chemical models may be tested and compared. There will be models where the uncertainties in the best estimate of an equilibrium constant may be somewhat or even
5317:
When cumulative constants have been refined it is often useful to derive stepwise constants from them. The general procedure is to write down the defining expressions for all the constants involved and then to equate concentrations. For example, suppose that one wishes to derive the pKa for removing
7429:
If the model is not acceptable, a variety of other models should be examined to find one that best fits the experimental data, within experimental error. The main difficulty is with the so-called minor species. These are species whose concentration is so low that the effect on the measured quantity
7392:
When determining the stability constants of metal-ligand complexes, it is common practice to fix ligand protonation constants at values that have been determined using data obtained from metal-free solutions. Hydrolysis constants of metal ions are usually fixed at values which were obtained using
2079:
It is quite usual to omit from the model those species whose concentrations are considered negligible. For example, it is usually assumed then there is no interaction between the reactants and/or complexes and the electrolyte used to maintain constant ionic strength or the buffer used to maintain
4357:
The unknown molar absorbances of all "coloured" species are found by using the non-iterative method of linear least-squares, one wavelength at a time. The calculations are performed once every refinement cycle, using the stability constant values obtaining at that refinement cycle to calculate
664:
may be higher than the value of the molar extinction coefficient, ε, for a species. When this is so, the detection limit for that species will be lower. At high solute concentrations, fluorescence intensity becomes non-linear with respect to concentration due to self-absorption of the scattered
165:
is performed with one or more reactants in the titration vessel and one or more reactants in the burette. Knowing the analytical concentrations of reactants initially in the reaction vessel and in the burette, all analytical concentrations can be derived as a function of the volume (or mass) of
6883:
Once a refinement has been completed the results should be checked to verify that the chosen model is acceptable. generally speaking, a model is acceptable when the data are fitted within experimental error, but there is no single criterion to use to make the judgement. The following should be
2743:
4672:
6807:
5236:
2958:
Note that the free reactant concentrations can be regarded as implicit parameters in the equilibrium constant refinement process. In that context the values of the free concentrations are constrained by forcing the conditions of mass-balance to apply at all stages of the process.
3737:
is achieved, that is, until a convergence criterion is satisfied. If, however, the updated parameters do not result in a decrease of the objective function, that is, if divergence occurs, the increment calculation must be modified. The simplest modification is to use a fraction,
7368:). The observed values are shown a blue diamonds and the species concentrations, as a percentage of the total nickel, are superimposed. The residuals are shown in the lower box. The residuals are not distributed as randomly as would be expected. This is due to the variation of
2949:
In general, solving these nonlinear equations presents a formidable challenge because of the huge range over which the free concentrations may vary. At the beginning, values for the free concentrations must be estimated. Then, these values are refined, usually by means of
1416:
The value of the equilibrium constant for the formation of a 1:1 complex, such as a host-guest species, may be calculated with a dedicated spreadsheet application, Bindfit: In this case step 2 can be performed with a non-iterative procedure and the pre-programmed routine
137:
5894:{\displaystyle {\begin{aligned}{\ce {L^3-}}+{\ce {H+ <=>}}\ {\ce {LH^2-}}&:\ =\beta _{11}\\{\ce {L^3-}}+{\ce {2H+ <=>}}\ {\ce {LH2^-}}&:\ =\beta _{12}^{2}\\{\ce {L^3-}}+{\ce {3H+ <=>}}\ {\ce {LH3}}&:\ =\beta _{13}^{3}\end{aligned}}}
4899:
7125:
157:, etc. of the chemical species in equilibrium. The equilibrium constant value can be determined if any one of these concentrations can be measured. The general procedure is that the concentration in question is measured for a series of solutions with known
2071:
In this case the equilibrium constant for the formation of hydroxide has the stoichiometric coefficients −1 in regard to the proton and zero for the other reactants. This has important implications for all protonation equilibria in aqueous solution and for
2113:
4209:
1570:
4351:
4490:
7421:, at values which have been determined in separate experiments. Use of such constraints reduces the number of parameters to be determined, but may result in the calculated errors on refined stability constant values being under-estimated.
3911:
1989:
When both H and OH must be considered as reactants, one of them is eliminated from the model by specifying that its concentration be derived from the concentration of the other. Usually the concentration of the hydroxide ion is given by
2806:
In a titration system the analytical concentrations of the reactants at each titration point are obtained from the initial conditions, the burette concentrations and volumes. The analytical (total) concentration of a reactant R at the
6583:
6485:
3503:
3358:
503:. This enormous range of stability constant values (ca. 100 to 10) is possible because of the logarithmic response of the electrode. The limitations arise because the Nernst equation breaks down at very low or very high pH.
2913:
6351:
3644:
760:
4759:
4021:
7380:
Some physical constraints are usually incorporated in the calculations. For example, all the concentrations of free reactants and species must have positive values and association constants must have positive values.
5101:
2543:
583:
is a concentration. More than one of the species may contribute to the absorbance. In principle absorbance may be measured at one wavelength only, but in present-day practice it is common to record complete spectra.
3807:
4533:
4111:
3718:
7314:
6673:
5134:
377:
506:
When a glass electrode is used to obtain the measurements on which the calculated equilibrium constants depend, the precision of the calculated parameters is limited by secondary effects such as variation of
3525:
are not expressed in terms of the equilibrium constants, but in terms of the species concentrations, which are implicit functions of these parameters. Therefore, the
Jacobian elements must be obtained using
7228:
6655:
6873:
4290:
Golub and
Pereyra showed how the pseudo-inverse can be differentiated so that parameter increments for both molar absorptivities and equilibrium constants can be calculated by solving the normal equations.
1686:
6093:
7430:
is at or below the level of error in the experimental measurement. The constant for a minor species may prove impossible to determine if there is no means to increase the concentration of the species. .
3043:
2521:
279:
2548:
2537:
constants for which the stoichiometric coefficients are 1 for the reactant concerned and zero for all other reactants. For example, with 2 reagents, the mass-balance equations assume the simpler form.
602:
of 4 is usually quoted, corresponding to the precision of the measurements, but it also depends on how intense the effect is. Spectra of contributing species should be clearly distinct from each other
4277:
3126:
1903:
6971:
One would want the errors on the stability constants to be roughly commensurate with experimental error. For example, with pH titration data, if pH is measured to 2 decimal places, the errors of
33:
5337:
2118:
6268:
7341:
are the variance-covariance matrices of the residuals and observations, respectively. This shows that even though the observations may be uncorrelated, the residuals are always correlated.
1800:
449:
4792:
7016:
2066:
2374:{\displaystyle {\begin{aligned}{\ce {T_{A}}}&=+\sum _{1,nk}p\beta _{pq\cdots }^{p}^{q}\cdots \\{\ce {T_{B}}}&=+\sum _{1,nk}q\beta _{pq\cdots }^{p}^{q}\cdots \\etc.\end{aligned}}}
1984:
1146:
3211:
7384:
With spectrophotometric data the calculated molar absorptivity (or emissivity) values should all be positive. Most computer programs do not impose this constraint on the calculations.
2790:
3519:
th parameter of the refinement, are equal to zero. One or more equilibrium constants may be parameters of the refinement. However, the measured quantities (see above) represented by
566:
7000:
1283:
2434:
1092:
984:
4137:
2080:
constant pH. These assumptions may or may not be justified. Also, it is implicitly assumed that there are no other complex species present. When complexes are wrongly ignored a
1455:
4298:
7899:
Gampp, H.; Maeder, M.; Mayer, C. J.; Zuberbühler, A. (1985). "Calculation of equilibrium constants from multiwavelength spectroscopic data—IMathematical considerations".
643:
7441:
A large number of general-purpose computer programs for equilibrium constant calculation have been published. See for a bibliography. The most frequently used programs are:
2526:
and only the free concentrations of the other reactants are calculated. The concentrations of the complexes are derived from the free concentrations via the chemical model.
1225:
385:
When the electrode is calibrated with solutions of known concentration, by means of a strong acid–strong base titration, for example, a modified Nernst equation is assumed.
4378:
2967:
The objective of the refinement process is to find equilibrium constant values that give the best fit to the experimental data. This is usually achieved by minimising an
2384:
for the free concentrations , etc. When the pH (or equivalent e.m.f., E).is measured, the free concentration of hydrogen ions, , is obtained from the measured value as
7627:"The death of the Job plot, transparency, open science and online tools, uncertainty estimation methods and other developments in supramolecular chemistry data analysis"
4399:
6928:
is 1. It is therefore very useful to estimate experimental errors and derive some reasonable weights from them as this is an absolute indicator of the goodness of fit.
4048:
1350:
1316:
898:
3833:
7811:
Gans, P.; Sabatini, A.; Vacca, A. (1996). "Investigation of equilibria in solution. Determination of equilibrium constants with the HYPERQUAD suite of programs".
7666:
Motekaitis, R. J.; Martell, A. E. (1982). "BEST — A new program for rigorous calculation of equilibrium parameters of complex multicomponent systems".
4068:
2099:
A speciation calculation is one in which concentrations of all the species in an equilibrium system are calculated, knowing the analytical concentrations, T
7579:
value of the hydroxyl group in the α-hydroxycarboxylates citrate, malate and lactate by C NMR: implications for metal coordination in biological systems".
6491:
6393:
177:
There are four main experimental methods. For less commonly used methods, see
Rossotti and Rossotti. In all cases the range can be extended by using the
3425:
3277:
6279:
2817:
2738:{\displaystyle {\begin{aligned}T_{\ce {A}}&=\sum _{0,nk}p\beta _{pq}^{p}^{q}\\T_{\ce {B}}&=\sum _{0,nk}q\beta _{pq}^{p}^{q}\\\end{aligned}}}
689:
4667:{\displaystyle \mathbf {\Sigma ^{p}=\left(J^{\mathrm {T} }WJ\right)^{-1}J^{\mathrm {T} }W\Sigma ^{y}W^{\mathrm {T} }J(J^{\mathrm {T} }WJ)^{-1}} }
3575:
4694:
3939:
7729:
Golub, G. H.; Pereyra, V. (1973). "The
Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate".
5306:
5040:
6802:{\displaystyle \sigma _{K}^{2}=\sigma _{\beta _{12}}^{2}+\sigma _{\beta _{13}}^{2}-2\sigma _{\beta _{12}}\sigma _{\beta _{13}}\rho _{12,13}\,}
5231:{\displaystyle \mathbf {\Sigma ^{p}} ={\frac {U}{n_{\mathrm {d} }-n_{\mathrm {p} }}}\left(\mathbf {J} ^{\mathrm {T} }\mathbf {J} \right)^{-1}}
7951:
6660:
Note the reverse numbering for pK and log β. When calculating the error on the stepwise constant, the fact that the cumulative constants are
3748:
4080:
3662:
7239:
859:
The competition method may be used when a stability constant value is too large to be determined by a direct method. It was first used by
1701:
With aqueous solutions the concentrations of proton (hydronium ion) and hydroxide ion are constrained by the self-dissociation of water.
7372:
and other effects at the glass/liquid interfaces. Those effects are very slow compared to the rate at which equilibrium is established.
308:
1692:
4495:
Therefore, the parameter values are (approximately) linear combinations of the observed data values and the errors on the parameters,
7171:
6818:
6589:
1575:
169:
The equilibrium constants may be derived by best-fitting of the experimental data with a chemical model of the equilibrium system.
5918:
2439:
1908:
With dilute solutions the concentration of water is assumed constant, so the equilibrium expression is written in the form of the
1695:
211:
2985:
146:
quotient is constant. For this assumption to be valid, equilibrium constants must be determined in a medium of relatively high
2955:
concentrations have been calculated, the concentrations of the complexes are derived from them and the equilibrium constants.
1399:, that depends on one or more of these analytical concentrations. A general computational procedure has four main components:
7883:
7795:
7010:
the system can be approximated to a linear one, the residuals in the case of unit weights are related to the observations by
4220:
132:{\displaystyle K={\frac {\mathrm {} ^{\sigma }\mathrm {} ^{\tau }\cdots }{\mathrm {} ^{\alpha }\mathrm {} ^{\beta }\cdots }}}
7473:
7499:
for the formation of a 1:1 adduct can be obtained. Some software for handling more complex equilibria may also be supplied.
3054:
789:
481:
The most widely used electrode is the glass electrode, which is selective for the hydrogen ion. This is suitable for all
4894:{\displaystyle W_{k}={\frac {1}{\sigma _{E}^{2}+\left({\frac {\partial E}{\partial v}}\right)_{k}^{2}\sigma _{v}^{2}}}}
1805:
7120:{\displaystyle \mathbf {r=y^{\mathrm {obs} }-J\left(J^{\mathrm {T} }T\right)^{-1}J^{\mathrm {T} }y^{\mathrm {obs} }} }
7944:
6120:
3929:
A particular issue arises with NMR and spectrophotometric data. For the latter, the observed quantity is absorbance,
3820:
811:
458:
is an empirical slope factor. A solution of known hydrogen ion concentration may be prepared by standardization of a
391:
8145:
7492:
1707:
840:
4126:
values are considered parameters of the minimization and the
Jacobian is constructed on that basis. However, the
3411:
1996:
1107:
2073:
1918:
8231:
7855:
3155:
7698:"Modelling complex solution equilibria. I. Fast, worry-free least-squares refinement of equilibrium constants"
2749:
8236:
2946:
is the volume added. The burette concentration of a reactant not present in the burette is taken to be zero.
534:
511:
in the electrode. In practice it is virtually impossible to obtain a precision for log β better than ±0.001.
4204:{\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {\left(C^{\mathrm {T} }C\right)^{-1}C^{\mathrm {T} }A} }
3733:
values are recalculated at every data point. The iterations are continued until no significant reduction in
1565:{\displaystyle {\ce {{{\mathit {p}}A}+{\mathit {q}}B\cdots <=>A_{\mathit {p}}B_{\mathit {q}}\cdots }}}
989:
where the represents the concentration, at equilibrium, of a species X in a solution of given composition.
8203:
8107:
7937:
4346:{\displaystyle \mathbf {\boldsymbol {\varepsilon }} _{\lambda }=\mathbf {A} _{\lambda }^{-1}\mathbf {C} \,}
1236:
291:
2390:
1288:
where K is the stability constant for the competition reaction. Thus, the value of the stability constant
8016:
1026:
918:
2088:
8123:
4512:
1442:
3149:. Most often only the diagonal elements are known, in which case the objective function simplifies to
616:
8031:
7999:
7369:
508:
482:
4485:{\displaystyle \mathbf {p=\left(J^{\mathrm {T} }WJ\right)^{-1}J^{\mathrm {T} }Wy^{\mathrm {obs} }} }
1157:
8172:
8167:
8128:
4779:
3527:
1909:
158:
1004:, is small enough to be determined by a direct method. For example, in the case of EDTA complexes
610:
It is assumed that the scattered light intensity is a linear function of species’ concentrations.
7532:"Definitions of pH scales, standard reference values, measurement of pH, and related terminology"
4361:
3258:
785:
194:
2107:
etc. of the reactants A, B etc. This means solving a set of nonlinear equations of mass-balance
8191:
8081:
7531:
4775:
3906:{\displaystyle \mathbf {\left(J^{\mathrm {T} }WJ+\lambda I\right)\delta p=J^{\mathrm {T} }Wr} }
8184:
8179:
8157:
8102:
8009:
7989:
5298:
in the measurements. The true uncertainty in the parameters is larger due to the presence of
4033:
673:
Chemical exchange is assumed to be rapid on the NMR time-scale. An individual chemical shift
182:
1325:
1291:
873:
150:. Where this is not possible, consideration should be given to possible activity variation.
8066:
8049:
7960:
7738:
7460:
4215:
using the refined values of the equilibrium constants to obtain the speciation. The matrix
860:
525:
143:
16:
1360:
It is assumed that the collected experimental data comprise a set of data points. At each
8:
8162:
8140:
8026:
7984:
4132:
values themselves are calculated at each step of the refinement by linear least-squares:
843:. Extension to more complex systems is limited by the availability of suitable software.
20:
7742:
7393:
ligand-free solutions. When determining the stability constants for ternary complexes, M
7344:
The diagram at the right shows the result of a refinement of the stability constants of
6931:
When unit weights are used, it is implied that all observations have the same variance.
2803:
from the combination of reactants that is specified by the stoichiometric coefficients.
800:
Limited precision of chemical shift measurements also puts an upper limit of about 4 on
8150:
8041:
7974:
7769:
7604:
7554:
6892:
When the weights have been correctly derived from estimates of experimental error, the
5291:
2968:
2951:
1429:
The chemical model consists of a set of chemical species present in solution, both the
1017:
5305:
Note that even though the observations may be uncorrelated, the parameters are always
4053:
4026:
It can be seen that, assuming that the concentrations, c, are known, that absorbance,
8196:
8076:
8004:
7994:
7916:
7912:
7879:
7851:
7828:
7824:
7791:
7648:
7596:
6893:
6665:
6578:{\displaystyle {\ce {p}}K_{{\ce {a2}}}=\log _{10}\beta _{12}-\log _{10}\beta _{11}\,}
6480:{\displaystyle {\ce {p}}K_{{\ce {a1}}}=\log _{10}\beta _{13}-\log _{10}\beta _{12}\,}
5299:
4502:
3142:
3138:
470:
7608:
7558:
3261:. Firstly the objective function is linearised by approximating it as a first-order
8135:
8021:
7908:
7820:
7746:
7709:
7675:
7638:
7588:
7546:
7450:
7131:
3824:
3541:
3251:
2081:
7572:
Silva, Andre M. N.; Kong, Xiaole; Hider, Robert C. (2009). "Determination of the p
8097:
7477:
7470:
7405:
it is common practice the fix the values for the corresponding binary complexes M
7326:
5025:
4956:
4771:
3498:{\displaystyle J_{jk}={\frac {\partial y_{j}^{\mathrm {calc} }}{\partial p_{k}}}}
3353:{\displaystyle U=U^{0}+\sum _{i}{\frac {\partial U}{\partial p_{i}}}\delta p_{i}}
500:
295:
202:
201:. If the electrode is calibrated using activity standards it is assumed that the
198:
4116:
There are two approaches to the calculation of the unknown molar absorptivities
3926:
is an identity matrix. Other methods of handling divergence have been proposed.
2908:{\displaystyle T_{{\ce {R}}}={\frac {{\ce {R}}_{0}+v_{i}{\ce {}}}{v_{0}+v_{i}}}}
193:
A free concentration or activity {A} of a species A is measured by means of an
8208:
6346:{\displaystyle \beta _{13}=K\beta _{12};K={\frac {\beta _{13}}{\beta _{12}}}\,}
4284:
3141:
of the observations. It is rare for this to be known. However, when it is, the
147:
7592:
1406:
Calculation of the concentrations of all the chemical species in each solution
8225:
8061:
5295:
4999:) it is implied that the experimental errors are uncorrelated and all equal:
3262:
2976:
781:
154:
7550:
3639:{\displaystyle {\left(J^{\mathrm {T} }WJ\right)\delta p=J^{\mathrm {T} }Wr}}
2087:
Equilibrium constant values are usually estimated initially by reference to
755:{\displaystyle {\bar {\delta }}={\frac {\sum x_{i}\delta _{i}}{\sum x_{i}}}}
7920:
7832:
7652:
7600:
7483:
4754:{\displaystyle \mathbf {\Sigma ^{p}=\left(J^{\mathrm {T} }WJ\right)^{-1}} }
4016:{\displaystyle A_{\lambda }=l\sum (\varepsilon _{pq..})_{\lambda }c_{pq..}}
7495:
are usually supplied with software with which an equilibrium constant and
5096:{\displaystyle \sigma ^{2}={\frac {U}{n_{\mathrm {d} }-n_{\mathrm {p} }}}}
3823:, on the other hand, the shift vector is rotated towards the direction of
7456:
5323:
777:
495:
values between about 2 and 11 can be measured directly by potentiometric
459:
7487:
7466:
7446:
4764:
In most cases the errors on the observations are un-correlated, so that
3802:{\displaystyle \mathbf {p} ^{n+1}=\mathbf {p} ^{n}+f\mathbf {\delta p} }
8071:
7643:
7626:
7156:
7135:
6661:
4106:{\displaystyle \mathbf {A} ={\boldsymbol {\varepsilon }}\mathbf {C} \,}
3713:{\displaystyle \mathbf {p} ^{n+1}=\mathbf {p} ^{n}+\delta \mathbf {p} }
473:
may also be used as a primary standard for hydrogen ion concentration.
7929:
7309:{\displaystyle \mathbf {M^{r}=\left(I-H\right)M^{y}\left(I-H\right)} }
4393:, the system approximates to a linear least-squares system, for which
2529:
Some authors include the free reactant terms in the sums by declaring
7979:
1013:
496:
467:
162:
7750:
7714:
7697:
7679:
4778:
of the variance of the corresponding observation. For example, in a
870:
For simplicity consider the determination of the stability constant
382:
At 298 K, 1 pH unit is approximately equal to 59 mV.
7496:
5294:
is the square root of variance. These error estimates reflect only
5242:
832:
773:
299:
7876:
Computational
Methods for the Determination of Formation Constants
6999:
372:{\displaystyle \mathrm {pH} ={\frac {nF}{RT}}\left(E^{0}-E\right)}
7365:
1691:
When using general-purpose computer programs, it is usual to use
863:
in the determination of the stability constants of complexes of
7345:
7223:{\displaystyle \mathbf {r=\left(I-H\right)y^{\mathrm {obs} }} }
6650:{\displaystyle {\ce {p}}K_{{\ce {a3}}}=\log _{10}\beta _{11}\,}
5018:
is known as the variance of an observation of unit weight, and
1418:
1364:
th data point, the analytical concentrations of the reactants,
181:. An example of the application of this method can be found in
7438:
Some simple systems are amenable to spreadsheet calculations.
6868:{\displaystyle \sigma _{\log _{10}K}={\frac {\sigma _{K}}{K}}}
4389:
In the region close to the minimum of the objective function,
1681:{\displaystyle \beta _{pq\cdots }={\frac {}{^{p}^{q}\cdots }}}
6088:{\displaystyle {\ce {{LH2^{-}}+H+<=>LH3\ ;\quad \ }}=K}
3246:, are often used but, in that case, the expectation value of
463:
3373:
are added to the corresponding initial parameters such that
3038:{\displaystyle r_{i}=y_{i}^{\text{obs}}-y_{i}^{\text{calc}}}
2516:{\displaystyle =e^{\mathrm {{-{\frac {nF}{RT}}}(E-E^{0})} }}
1318:
may be derived from the experimentally determined values of
274:{\displaystyle E=E^{0}+{\frac {RT}{nF}}\ln \mathrm {\{A\}} }
7898:
864:
1437:
formed from them. Denoting the reactants by A, B..., each
6614:
6516:
6418:
6030:
6003:
5821:
5793:
2249:
2132:
1722:
1555:
1540:
4272:{\displaystyle \mathbf {\left(C^{T}C\right)^{-1}C^{T}} }
605:
6098:
Substitute the expressions for the concentrations of LH
3727:
is an iteration number. The species concentrations and
5974:
5759:
5569:
5390:
3121:{\displaystyle U=\sum _{i}\sum _{j}r_{i}W_{ij}r_{j}\,}
1749:
1509:
153:
The equilibrium expression above is a function of the
7242:
7174:
7019:
6821:
6676:
6592:
6494:
6396:
6282:
6123:
5921:
5335:
5137:
5128:
are the number of data and parameters, respectively.
5043:
4795:
4697:
4536:
4402:
4364:
4301:
4223:
4140:
4083:
4056:
4036:
3942:
3836:
3751:
3665:
3578:
3428:
3280:
3158:
3057:
3048:
Then the most general objective function is given by
2988:
2820:
2752:
2546:
2442:
2393:
2116:
1999:
1921:
1808:
1710:
1578:
1458:
1328:
1294:
1239:
1160:
1110:
1029:
921:
876:
692:
619:
537:
394:
311:
214:
36:
7846:
Martell, A. E.; Motekaitis, R. J. (1992).
4070:, is a linear function of the molar absorptivities,
3742:, of calculated increment, so-called shift-cutting.
679:
is the mole-fraction-weighted average of the shifts
7516:Rossotti, F. J. C.; Rossotti, H. (1961).
5110:is the minimum value of the objective function and
4384:
298:solutions of known pH are used for calibration the
7768:
7308:
7222:
7119:
6867:
6801:
6649:
6577:
6479:
6345:
6262:
6087:
5893:
5230:
5095:
4893:
4753:
4666:
4484:
4372:
4345:
4271:
4203:
4105:
4062:
4042:
4015:
3905:
3801:
3712:
3638:
3497:
3410:, which are simply related to the elements of the
3352:
3205:
3145:of U is one, which means that the data are fitted
3120:
3037:
2962:
2907:
2784:
2737:
2515:
2428:
2373:
2060:
1978:
1897:
1794:
1680:
1564:
1344:
1310:
1277:
1219:
1140:
1086:
978:
892:
754:
668:
637:
560:
514:
443:
371:
273:
131:
7845:
7810:
7665:
7624:
5982:
5981:
5964:
5963:
5767:
5766:
5749:
5748:
5577:
5576:
5559:
5558:
5398:
5397:
5380:
5379:
4651:
4629:
4074:. With 1 cm path-length, in matrix notation
3812:In this case, the direction of the shift vector,
1898:{\displaystyle K_{\mathrm {W} }^{'}={\frac {}{}}}
1757:
1756:
1739:
1738:
1517:
1516:
1499:
1498:
8223:
7848:The Determination and Use of Stability Constants
7515:
6263:{\displaystyle \beta _{13}^{3}=K\beta _{12}^{2}}
1403:Definition of a chemical model of the equilibria
1392:etc. are known along with a measured quantity,
839:for 1:1 adducts is routinely carried out using
814:cannot be used with solutions of compounds in H
7571:
1795:{\displaystyle {\ce {H2O <=> H+ + OH-}}}
851:Insufficient evidence is currently available.
579:is a molar absorbance at unit path length and
444:{\displaystyle E=E^{0}+s\log _{10}\mathrm {} }
188:
7945:
7155:is known in the statistics literature as the
5302:—which, by definition, cannot be quantified.
3933:, and the Beer–Lambert law can be written as
7728:
6994:
4358:species' concentration values in the matrix
3257:The minimization may be performed using the
2061:{\displaystyle ={\frac {K_{{\ce {W}}}}{}}\,}
996:is chosen which forms a weaker complex with
821:
267:
261:
27:is expressed as a concentration quotient,
4911:is the error in electrode potential or pH,
2799:, are treated in the same way, having been
1979:{\displaystyle K_{\mathrm {W} }={\ce {}}\,}
1141:{\displaystyle AC+B\leftrightharpoons AB+C}
7952:
7938:
7869:
7867:
7762:
7760:
7691:
7689:
6963:is expected to be equal to that variance.
3265:expansion about an initial parameter set,
3206:{\displaystyle U=\sum _{i}W_{ii}r_{i}^{2}}
2094:
1445:that relate the particular combination of
846:
795:
655:
587:
7788:Excel for Chemists: A Comprehensive Guide
7713:
7642:
7620:
7618:
6887:
6798:
6646:
6574:
6476:
6342:
5727:
5537:
4342:
4102:
3137:, should be, ideally, the inverse of the
3117:
2785:{\displaystyle \beta _{10}=\beta _{01}=1}
2057:
1975:
854:
7766:
7518:The Determination of Stability Constants
3656:are added iteratively to the parameters
1355:
476:
7959:
7873:
7864:
7757:
7686:
7387:
7375:
5957:
5742:
5552:
5373:
4304:
4293:(2) The Beer–Lambert law is written as
4142:
4093:
3819:, is unchanged. With the more powerful
1732:
1492:
1409:Refinement of the equilibrium constants
561:{\displaystyle A=l\sum {\varepsilon c}}
178:
172:
8224:
7695:
7625:Hibbert, D.B.; Thordarson, P. (2017).
7615:
5264:and the covariance between parameters
3827:, by modifying the normal equations,
2795:In this manner, all chemical species,
1433:added to the reaction mixture and the
792:could be used for this determination.
7933:
7785:
2979:. First the residuals are defined as
2084:is introduced into the calculations.
1549:
1534:
1477:
1463:
1424:
1278:{\displaystyle K_{AB}=K\times K_{AC}}
606:Fluorescence (luminescence) intensity
7722:
5312:
4511:, using the linear formula. Let the
2429:{\displaystyle =10^{-\mathrm {pH} }}
19:are determined in order to quantify
6966:
5318:one proton from a tribasic acid, LH
4782:, the weight at a titration point,
4774:. If so, each weight should be the
4515:for the observations be denoted by
3544:, derived from the conditions that
1087:{\displaystyle K_{AC}={\frac {}{}}}
979:{\displaystyle K_{AB}={\frac {}{}}}
683:of nuclei in contributing species.
13:
7804:
7433:
7213:
7210:
7207:
7110:
7107:
7104:
7092:
7063:
7039:
7036:
7033:
6998:
6878:
5203:
5180:
5165:
5084:
5069:
4849:
4841:
4724:
4638:
4620:
4595:
4563:
4475:
4472:
4469:
4454:
4422:
4191:
4162:
3890:
3849:
3623:
3591:
3479:
3471:
3468:
3465:
3462:
3448:
3321:
3313:
2498:
2491:
2481:
2478:
2473:
2470:
2447:
2420:
2417:
2398:
1928:
1815:
810:. Limited to diamagnetic systems.
434:
316:
313:
264:
110:
92:
69:
51:
14:
8248:
7874:Leggett, D. J., ed. (1985).
7493:Isothermal titration calorimeters
3383:. At the minimum the derivatives
7297:
7294:
7291:
7281:
7277:
7268:
7265:
7262:
7254:
7249:
7245:
7202:
7193:
7190:
7187:
7179:
7176:
7099:
7087:
7081:
7078:
7069:
7058:
7048:
7045:
7028:
7024:
7021:
5210:
5197:
5144:
5140:
4745:
4742:
4733:
4730:
4719:
4709:
4704:
4700:
4658:
4655:
4647:
4644:
4633:
4626:
4615:
4609:
4605:
4601:
4590:
4584:
4581:
4572:
4569:
4558:
4548:
4543:
4539:
4464:
4460:
4449:
4443:
4440:
4431:
4428:
4417:
4407:
4404:
4385:Parameter errors and correlation
4366:
4338:
4319:
4263:
4259:
4253:
4250:
4241:
4236:
4232:
4197:
4186:
4180:
4177:
4168:
4157:
4098:
4085:
3899:
3896:
3885:
3881:
3878:
3867:
3861:
3858:
3855:
3844:
3795:
3775:
3754:
3706:
3689:
3668:
841:isothermal titration calorimetry
790:ultraviolet–visible spectroscopy
638:{\displaystyle I=\sum \varphi c}
7892:
7839:
7424:
6012:
3920:is the Marquardt parameter and
2963:Equilibrium constant refinement
2935:is the initial volume, is the
2811:th titration point is given by
669:NMR chemical shift measurements
652:is a proportionality constant.
515:Spectrophotometric measurements
161:of the reactants. Typically, a
23:. When an equilibrium constant
7779:
7771:Statistics in Physical Science
7659:
7565:
7524:
7509:
6257:
6242:
6233:
6217:
6214:
6196:
6171:
6155:
6152:
6134:
6082:
6067:
6064:
6044:
6033:
6017:
5984:
5959:
5878:
5862:
5859:
5841:
5825:
5807:
5769:
5744:
5692:
5676:
5673:
5655:
5639:
5619:
5579:
5554:
5508:
5493:
5490:
5472:
5456:
5438:
5400:
5375:
4966:is the error on added volume.
4521:and that of the parameters by
3985:
3962:
3540:are calculated by solving the
2975:, by the method of non-linear
2926:of R in the titration vessel,
2872:
2866:
2722:
2713:
2704:
2695:
2630:
2621:
2612:
2603:
2507:
2488:
2451:
2443:
2402:
2394:
2339:
2330:
2321:
2312:
2268:
2260:
2222:
2213:
2204:
2195:
2151:
2143:
2051:
2036:
2015:
2000:
1972:
1957:
1952:
1939:
1889:
1873:
1868:
1852:
1849:
1836:
1759:
1734:
1663:
1654:
1645:
1636:
1631:
1601:
1519:
1494:
1220:{\displaystyle K={\frac {}{}}}
1211:
1205:
1202:
1193:
1188:
1182:
1179:
1170:
1123:
1101:for the competition reaction
1078:
1072:
1069:
1063:
1058:
1049:
970:
964:
961:
955:
950:
941:
699:
660:The magnitude of the constant
437:
431:
113:
107:
95:
89:
72:
66:
54:
48:
1:
7775:. New York, NY: Ronald Press.
7767:Hamilton, W. C. (1964).
7503:
3821:Levenberg–Marquardt algorithm
519:
7913:10.1016/0039-9140(85)80035-7
7825:10.1016/0039-9140(96)01958-3
5908:constant for formation of LH
4969:When unit weights are used (
4373:{\displaystyle \mathbf {C} }
3254:of the experimental errors.
2797:including the free reactants
826:Simultaneous measurement of
575:is the optical path length,
292:standard electrode potential
7:
7790:(3rd ed.). Wiley-VCH.
7696:Potvin, P. G. (1990).
7497:standard formation enthalpy
1443:stoichiometric coefficients
189:Potentiometric measurements
10:
8253:
7370:liquid junction potentials
6982:significantly larger than
4513:variance-covariance matrix
3139:variance-covariance matrix
509:liquid junction potentials
8116:
8090:
8040:
8032:Thermodynamic equilibrium
7967:
7786:Billo, E. Joseph (2011).
7593:10.1007/s10534-009-9224-5
6995:Distribution of residuals
4030:, at a given wavelength,
3533:The parameter increments
3147:within experimental error
1000:The stability constant, K
822:Calorimetric measurements
784:data to be 14.4. Neither
780:has been determined from
159:analytical concentrations
8185:Distribution coefficient
8129:Hammett acidity function
8108:Liquid–liquid extraction
8017:Le Chatelier's principle
7455:Spectrophotometric data:
4780:potentiometric titration
4043:{\displaystyle \lambda }
3528:implicit differentiation
2939:of R in the burette and
1421:can be used for step 3.
1097:The stability constant,
7631:Chemical Communications
7551:10.1351/pac198557030531
6664:must accounted for. By
4505:from the observations,
3131:The matrix of weights,
2095:Speciation calculations
524:It is assumed that the
195:ion selective electrode
142:it is implied that the
8146:Coordination complexes
8082:Thermodynamic activity
7310:
7224:
7121:
7003:
6888:The objective function
6869:
6803:
6651:
6579:
6481:
6387:its value is given by
6347:
6264:
6089:
5895:
5232:
5097:
4895:
4755:
4668:
4486:
4374:
4347:
4273:
4205:
4107:
4064:
4044:
4017:
3907:
3803:
3714:
3640:
3499:
3354:
3207:
3122:
3039:
2909:
2786:
2739:
2524:
2517:
2430:
2375:
2062:
1980:
1899:
1796:
1682:
1566:
1346:
1345:{\displaystyle K_{AC}}
1312:
1311:{\displaystyle K_{AB}}
1279:
1221:
1142:
1088:
980:
894:
893:{\displaystyle K_{AB}}
855:The competition method
756:
639:
562:
445:
373:
302:reading will be a pH.
275:
133:
8232:Equilibrium chemistry
8158:Dissociation constant
8103:Equilibrium unfolding
7990:Equilibrium chemistry
7445:Potentiometric data:
7311:
7225:
7122:
7002:
6870:
6804:
6652:
6580:
6482:
6348:
6265:
6090:
5896:
5233:
5098:
4896:
4756:
4688:, this simplifies to
4669:
4501:, can be obtained by
4487:
4375:
4348:
4274:
4206:
4108:
4065:
4045:
4018:
3908:
3804:
3715:
3641:
3500:
3355:
3208:
3123:
3040:
2910:
2787:
2740:
2518:
2431:
2386:
2376:
2063:
1981:
1900:
1797:
1696:association constants
1683:
1567:
1356:Computational methods
1347:
1313:
1280:
1222:
1143:
1089:
1008:is a metal ion and
981:
908:with another reagent
900:of a binary complex,
895:
847:Range and limitations
796:Range and limitations
757:
656:Range and limitations
640:
588:Range and limitations
563:
477:Range and limitations
446:
374:
276:
183:palladium(II) cyanide
134:
17:Equilibrium constants
8237:Analytical chemistry
8067:Predominance diagram
8050:Equilibrium constant
7388:Chemical constraints
7376:Physical constraints
7240:
7172:
7017:
6819:
6674:
6590:
6492:
6394:
6280:
6121:
5919:
5333:
5135:
5041:
4955:is the slope of the
4793:
4695:
4534:
4400:
4362:
4299:
4221:
4138:
4081:
4054:
4034:
3940:
3834:
3749:
3663:
3576:
3426:
3278:
3156:
3055:
2986:
2818:
2750:
2544:
2440:
2391:
2114:
2074:hydrolysis constants
1997:
1919:
1806:
1708:
1576:
1456:
1441:is specified by the
1326:
1292:
1237:
1158:
1151:can be expressed as
1108:
1027:
919:
874:
690:
617:
535:
483:acid–base equilibria
392:
309:
212:
205:applies in the form
173:Experimental methods
34:
8141:Binding selectivity
8117:Specific equilibria
8027:Reversible reaction
7985:Dynamic equilibrium
7961:Chemical equilibria
7743:1973SJNA...10..413G
7731:SIAM J. Numer. Anal
7637:(87): 12792–12805.
7461:ReactLab EQUILIBRIA
6741:
6716:
6691:
6616:
6518:
6420:
6114:into this equation
6062:
6032:
6005:
5970:
5938:
5823:
5795:
5755:
5637:
5607:
5565:
5386:
5034:is approximated by
4887:
4872:
4829:
4336:
4283:is an example of a
3476:
3259:Gauss–Newton method
3202:
3034:
3016:
2251:
2134:
1829:
1745:
1724:
1557:
1542:
1505:
21:chemical equilibria
8151:Macrocyclic effect
7975:Chemical stability
7644:10.1039/c6cc03888c
7482:Calorimetric data
7476:2019-07-14 at the
7459:, SQUAD, Specfit,
7306:
7220:
7117:
7006:At the minimum in
7004:
6865:
6799:
6720:
6695:
6677:
6647:
6604:
6575:
6506:
6477:
6408:
6343:
6260:
6085:
6048:
6020:
5993:
5989:
5924:
5891:
5889:
5811:
5783:
5774:
5623:
5593:
5584:
5405:
5292:Standard deviation
5241:In all cases, the
5228:
5093:
4891:
4873:
4833:
4815:
4786:, can be given by
4751:
4664:
4482:
4370:
4343:
4317:
4269:
4201:
4103:
4060:
4050:, and path length
4040:
4013:
3903:
3799:
3710:
3636:
3495:
3451:
3350:
3309:
3203:
3188:
3174:
3118:
3083:
3073:
3035:
3020:
3002:
2969:objective function
2905:
2782:
2735:
2733:
2678:
2586:
2513:
2426:
2371:
2369:
2292:
2239:
2175:
2122:
2058:
1976:
1895:
1809:
1792:
1764:
1712:
1678:
1562:
1543:
1528:
1524:
1425:The chemical model
1342:
1308:
1275:
1217:
1138:
1084:
1018:diethylenetriamine
976:
890:
752:
635:
592:An upper limit on
558:
441:
369:
271:
179:competition method
129:
8219:
8218:
8197:Common-ion effect
8124:Acid dissociation
8077:Reaction quotient
7995:Equilibrium stage
7885:978-0-306-41957-7
7819:(10): 1739–1753.
7797:978-0-470-38123-6
7708:(12): 2198–2207.
7674:(19): 2403–2409.
7545:: 531–542. 1985.
6894:expectation value
6863:
6666:error propagation
6607:
6596:
6509:
6498:
6411:
6400:
6340:
6249:
6224:
6203:
6162:
6141:
6074:
6051:
6023:
6015:
6008:
5996:
5991:
5946:
5927:
5869:
5848:
5814:
5806:
5786:
5781:
5776:
5731:
5709:
5683:
5662:
5626:
5618:
5596:
5591:
5586:
5541:
5519:
5500:
5479:
5445:
5437:
5417:
5412:
5407:
5362:
5344:
5313:Derived constants
5300:systematic errors
5245:of the parameter
5187:
5091:
4889:
4856:
4503:error propagation
4063:{\displaystyle l}
3493:
3335:
3300:
3165:
3143:expectation value
3074:
3064:
3032:
3014:
2903:
2871:
2843:
2829:
2719:
2701:
2660:
2650:
2627:
2609:
2568:
2558:
2485:
2336:
2318:
2274:
2266:
2246:
2242:
2219:
2201:
2157:
2149:
2129:
2125:
2055:
2043:
2031:
2007:
1964:
1945:
1893:
1784:
1771:
1766:
1727:
1715:
1676:
1660:
1642:
1620:
1608:
1551:
1546:
1536:
1531:
1526:
1484:
1479:
1470:
1465:
1215:
1082:
974:
867:with metal ions.
750:
702:
471:hydrochloric acid
341:
252:
127:
8244:
8136:Binding constant
8022:Phase separation
7954:
7947:
7940:
7931:
7930:
7925:
7924:
7896:
7890:
7889:
7878:. Plenum Press.
7871:
7862:
7861:
7843:
7837:
7836:
7808:
7802:
7801:
7783:
7777:
7776:
7774:
7764:
7755:
7754:
7726:
7720:
7719:
7717:
7693:
7684:
7683:
7663:
7657:
7656:
7646:
7622:
7613:
7612:
7569:
7563:
7562:
7536:
7528:
7522:
7521:
7513:
7449:, BEST PSEQUAD,
7363:
7362:
7361:
7340:
7334:
7324:
7315:
7313:
7312:
7307:
7305:
7304:
7300:
7285:
7284:
7275:
7271:
7253:
7252:
7229:
7227:
7226:
7221:
7219:
7218:
7217:
7216:
7200:
7196:
7164:
7154:
7126:
7124:
7123:
7118:
7116:
7115:
7114:
7113:
7097:
7096:
7095:
7085:
7084:
7076:
7072:
7068:
7067:
7066:
7044:
7043:
7042:
7009:
6990:
6980:
6967:Parameter errors
6962:
6961:
6959:
6958:
6943:
6940:
6927:
6926:
6924:
6923:
6908:
6905:
6874:
6872:
6871:
6866:
6864:
6859:
6858:
6849:
6844:
6843:
6836:
6835:
6808:
6806:
6805:
6800:
6797:
6796:
6781:
6780:
6779:
6778:
6764:
6763:
6762:
6761:
6740:
6735:
6734:
6733:
6715:
6710:
6709:
6708:
6690:
6685:
6656:
6654:
6653:
6648:
6645:
6644:
6632:
6631:
6619:
6618:
6617:
6615:
6612:
6605:
6597:
6594:
6584:
6582:
6581:
6576:
6573:
6572:
6560:
6559:
6547:
6546:
6534:
6533:
6521:
6520:
6519:
6517:
6514:
6507:
6499:
6496:
6486:
6484:
6483:
6478:
6475:
6474:
6462:
6461:
6449:
6448:
6436:
6435:
6423:
6422:
6421:
6419:
6416:
6409:
6401:
6398:
6386:
6385:
6383:
6382:
6377:
6374:
6352:
6350:
6349:
6344:
6341:
6339:
6338:
6329:
6328:
6319:
6308:
6307:
6292:
6291:
6269:
6267:
6266:
6261:
6256:
6255:
6254:
6247:
6241:
6240:
6231:
6230:
6229:
6222:
6213:
6212:
6211:
6201:
6195:
6194:
6179:
6178:
6169:
6168:
6167:
6160:
6151:
6150:
6149:
6139:
6133:
6132:
6113:
6112:
6111:
6094:
6092:
6091:
6086:
6081:
6080:
6079:
6072:
6063:
6061:
6056:
6049:
6037:
6036:
6031:
6028:
6021:
6013:
6006:
6004:
6001:
5994:
5992:
5990:
5988:
5987:
5980:
5972:
5971:
5969:
5962:
5954:
5952:
5951:
5944:
5939:
5937:
5932:
5925:
5900:
5898:
5897:
5892:
5890:
5886:
5885:
5876:
5875:
5874:
5867:
5858:
5857:
5856:
5846:
5840:
5839:
5824:
5822:
5819:
5812:
5804:
5796:
5794:
5791:
5784:
5779:
5778:
5777:
5775:
5773:
5772:
5765:
5757:
5756:
5754:
5747:
5739:
5737:
5736:
5729:
5719:
5718:
5717:
5707:
5700:
5699:
5690:
5689:
5688:
5681:
5672:
5671:
5670:
5660:
5654:
5653:
5638:
5636:
5631:
5624:
5616:
5608:
5606:
5601:
5594:
5589:
5588:
5587:
5585:
5583:
5582:
5575:
5567:
5566:
5564:
5557:
5549:
5547:
5546:
5539:
5529:
5528:
5527:
5517:
5507:
5506:
5505:
5498:
5489:
5488:
5487:
5477:
5471:
5470:
5455:
5454:
5453:
5443:
5435:
5427:
5426:
5425:
5415:
5410:
5409:
5408:
5406:
5404:
5403:
5396:
5388:
5387:
5385:
5378:
5370:
5368:
5367:
5360:
5354:
5353:
5352:
5342:
5289:
5288:
5287:
5277:
5270:
5263:
5262:
5261:
5251:
5237:
5235:
5234:
5229:
5227:
5226:
5218:
5214:
5213:
5208:
5207:
5206:
5200:
5188:
5186:
5185:
5184:
5183:
5170:
5169:
5168:
5154:
5149:
5148:
5147:
5127:
5118:
5109:
5102:
5100:
5099:
5094:
5092:
5090:
5089:
5088:
5087:
5074:
5073:
5072:
5058:
5053:
5052:
5033:
5023:
5017:
5011:
4998:
4978:
4965:
4954:
4953:
4952:
4938:
4936:
4935:
4929:
4926:
4910:
4900:
4898:
4897:
4892:
4890:
4888:
4886:
4881:
4871:
4866:
4861:
4857:
4855:
4847:
4839:
4828:
4823:
4810:
4805:
4804:
4785:
4769:
4760:
4758:
4757:
4752:
4750:
4749:
4748:
4740:
4736:
4729:
4728:
4727:
4708:
4707:
4687:
4673:
4671:
4670:
4665:
4663:
4662:
4661:
4643:
4642:
4641:
4625:
4624:
4623:
4613:
4612:
4600:
4599:
4598:
4588:
4587:
4579:
4575:
4568:
4567:
4566:
4547:
4546:
4526:
4520:
4510:
4500:
4491:
4489:
4488:
4483:
4481:
4480:
4479:
4478:
4459:
4458:
4457:
4447:
4446:
4438:
4434:
4427:
4426:
4425:
4392:
4379:
4377:
4376:
4371:
4369:
4352:
4350:
4349:
4344:
4341:
4335:
4327:
4322:
4313:
4312:
4307:
4278:
4276:
4275:
4270:
4268:
4267:
4266:
4257:
4256:
4248:
4244:
4240:
4239:
4210:
4208:
4207:
4202:
4200:
4196:
4195:
4194:
4184:
4183:
4175:
4171:
4167:
4166:
4165:
4145:
4131:
4125:
4112:
4110:
4109:
4104:
4101:
4096:
4088:
4073:
4069:
4067:
4066:
4061:
4049:
4047:
4046:
4041:
4029:
4022:
4020:
4019:
4014:
4012:
4011:
3993:
3992:
3983:
3982:
3952:
3951:
3932:
3925:
3919:
3912:
3910:
3909:
3904:
3902:
3895:
3894:
3893:
3874:
3870:
3854:
3853:
3852:
3825:steepest descent
3818:
3808:
3806:
3805:
3800:
3798:
3784:
3783:
3778:
3769:
3768:
3757:
3741:
3736:
3732:
3726:
3719:
3717:
3716:
3711:
3709:
3698:
3697:
3692:
3683:
3682:
3671:
3655:
3645:
3643:
3642:
3637:
3635:
3628:
3627:
3626:
3607:
3603:
3596:
3595:
3594:
3569:at the minimum.
3568:
3566:
3564:
3563:
3557:
3554:
3542:normal equations
3539:
3524:
3518:
3514:
3504:
3502:
3501:
3496:
3494:
3492:
3491:
3490:
3477:
3475:
3474:
3459:
3446:
3441:
3440:
3419:
3409:
3408:
3406:
3405:
3396:
3393:
3382:
3376:
3372:
3359:
3357:
3356:
3351:
3349:
3348:
3336:
3334:
3333:
3332:
3319:
3311:
3308:
3296:
3295:
3270:
3252:root mean square
3249:
3245:
3236:. Unit weights,
3235:
3225:
3212:
3210:
3209:
3204:
3201:
3196:
3187:
3186:
3173:
3136:
3127:
3125:
3124:
3119:
3116:
3115:
3106:
3105:
3093:
3092:
3082:
3072:
3044:
3042:
3041:
3036:
3033:
3030:
3028:
3015:
3012:
3010:
2998:
2997:
2974:
2945:
2934:
2914:
2912:
2911:
2906:
2904:
2902:
2901:
2900:
2888:
2887:
2877:
2876:
2875:
2869:
2863:
2862:
2850:
2849:
2844:
2841:
2837:
2832:
2831:
2830:
2827:
2810:
2791:
2789:
2788:
2783:
2775:
2774:
2762:
2761:
2744:
2742:
2741:
2736:
2734:
2730:
2729:
2720:
2717:
2712:
2711:
2702:
2699:
2694:
2693:
2677:
2652:
2651:
2648:
2638:
2637:
2628:
2625:
2620:
2619:
2610:
2607:
2602:
2601:
2585:
2560:
2559:
2556:
2536:
2522:
2520:
2519:
2514:
2512:
2511:
2510:
2506:
2505:
2487:
2486:
2484:
2476:
2468:
2450:
2435:
2433:
2432:
2427:
2425:
2424:
2423:
2401:
2380:
2378:
2377:
2372:
2370:
2347:
2346:
2337:
2334:
2329:
2328:
2319:
2316:
2311:
2310:
2291:
2267:
2264:
2252:
2250:
2247:
2244:
2240:
2230:
2229:
2220:
2217:
2212:
2211:
2202:
2199:
2194:
2193:
2174:
2150:
2147:
2135:
2133:
2130:
2127:
2123:
2082:systematic error
2067:
2065:
2064:
2059:
2056:
2054:
2050:
2049:
2048:
2041:
2034:
2033:
2032:
2029:
2022:
2014:
2013:
2012:
2005:
1985:
1983:
1982:
1977:
1971:
1970:
1969:
1962:
1956:
1955:
1951:
1950:
1943:
1933:
1932:
1931:
1904:
1902:
1901:
1896:
1894:
1892:
1885:
1884:
1871:
1867:
1866:
1848:
1847:
1834:
1828:
1827:
1819:
1818:
1801:
1799:
1798:
1793:
1791:
1790:
1789:
1782:
1777:
1776:
1769:
1767:
1765:
1763:
1762:
1755:
1747:
1746:
1744:
1737:
1729:
1725:
1723:
1720:
1713:
1687:
1685:
1684:
1679:
1677:
1675:
1671:
1670:
1661:
1658:
1653:
1652:
1643:
1640:
1634:
1627:
1626:
1621:
1618:
1615:
1614:
1609:
1606:
1599:
1594:
1593:
1571:
1569:
1568:
1563:
1561:
1556:
1553:
1552:
1544:
1541:
1538:
1537:
1529:
1527:
1525:
1523:
1522:
1515:
1507:
1506:
1504:
1497:
1489:
1482:
1481:
1480:
1471:
1468:
1467:
1466:
1398:
1391:
1377:
1363:
1351:
1349:
1348:
1343:
1341:
1340:
1317:
1315:
1314:
1309:
1307:
1306:
1284:
1282:
1281:
1276:
1274:
1273:
1252:
1251:
1230:It follows that
1226:
1224:
1223:
1218:
1216:
1214:
1191:
1168:
1147:
1145:
1144:
1139:
1093:
1091:
1090:
1085:
1083:
1081:
1061:
1047:
1042:
1041:
985:
983:
982:
977:
975:
973:
953:
939:
934:
933:
899:
897:
896:
891:
889:
888:
838:
829:
809:
782:C chemical shift
761:
759:
758:
753:
751:
749:
748:
747:
734:
733:
732:
723:
722:
709:
704:
703:
695:
682:
678:
677:
663:
651:
644:
642:
641:
636:
601:
582:
578:
574:
567:
565:
564:
559:
557:
526:Beer–Lambert law
494:
468:Constant-boiling
457:
450:
448:
447:
442:
440:
426:
425:
410:
409:
378:
376:
375:
370:
368:
364:
357:
356:
342:
340:
332:
324:
319:
289:
280:
278:
277:
272:
270:
253:
251:
243:
235:
230:
229:
138:
136:
135:
130:
128:
126:
122:
121:
116:
104:
103:
98:
85:
81:
80:
75:
63:
62:
57:
44:
26:
8252:
8251:
8247:
8246:
8245:
8243:
8242:
8241:
8222:
8221:
8220:
8215:
8168:Self-ionization
8112:
8098:Buffer solution
8086:
8036:
7963:
7958:
7928:
7897:
7893:
7886:
7872:
7865:
7858:
7844:
7840:
7809:
7805:
7798:
7784:
7780:
7765:
7758:
7751:10.1137/0710036
7727:
7723:
7715:10.1139/v90-337
7694:
7687:
7680:10.1139/v82-347
7664:
7660:
7623:
7616:
7578:
7570:
7566:
7539:Pure Appl. Chem
7534:
7530:
7529:
7525:
7514:
7510:
7506:
7490:
7478:Wayback Machine
7451:ReactLab pH PRO
7436:
7434:Implementations
7427:
7420:
7416:
7412:
7408:
7404:
7400:
7396:
7390:
7378:
7360:
7357:
7356:
7355:
7353:
7351:
7336:
7330:
7327:identity matrix
7320:
7290:
7286:
7280:
7276:
7261:
7257:
7248:
7244:
7243:
7241:
7238:
7237:
7206:
7205:
7201:
7186:
7182:
7175:
7173:
7170:
7169:
7160:
7139:
7103:
7102:
7098:
7091:
7090:
7086:
7077:
7062:
7061:
7057:
7056:
7052:
7051:
7032:
7031:
7027:
7020:
7018:
7015:
7014:
7007:
6997:
6989:
6983:
6976:
6972:
6969:
6957:
6950:
6944:
6941:
6936:
6935:
6933:
6932:
6922:
6915:
6909:
6906:
6901:
6900:
6898:
6897:
6890:
6881:
6879:Model selection
6854:
6850:
6848:
6831:
6827:
6826:
6822:
6820:
6817:
6816:
6786:
6782:
6774:
6770:
6769:
6765:
6757:
6753:
6752:
6748:
6736:
6729:
6725:
6724:
6711:
6704:
6700:
6699:
6686:
6681:
6675:
6672:
6671:
6640:
6636:
6627:
6623:
6613:
6608:
6603:
6602:
6598:
6593:
6591:
6588:
6587:
6568:
6564:
6555:
6551:
6542:
6538:
6529:
6525:
6515:
6510:
6505:
6504:
6500:
6495:
6493:
6490:
6489:
6470:
6466:
6457:
6453:
6444:
6440:
6431:
6427:
6417:
6412:
6407:
6406:
6402:
6397:
6395:
6392:
6391:
6378:
6375:
6372:
6371:
6369:
6368:
6364:
6357:
6334:
6330:
6324:
6320:
6318:
6303:
6299:
6287:
6283:
6281:
6278:
6277:
6250:
6246:
6245:
6236:
6232:
6225:
6221:
6220:
6204:
6200:
6199:
6190:
6186:
6174:
6170:
6163:
6159:
6158:
6142:
6138:
6137:
6128:
6124:
6122:
6119:
6118:
6110:
6107:
6106:
6105:
6103:
6101:
6075:
6071:
6070:
6057:
6052:
6047:
6029:
6024:
6016:
6002:
5997:
5983:
5976:
5975:
5973:
5965:
5958:
5956:
5955:
5953:
5947:
5943:
5933:
5928:
5923:
5922:
5920:
5917:
5916:
5911:
5888:
5887:
5881:
5877:
5870:
5866:
5865:
5849:
5845:
5844:
5835:
5831:
5820:
5815:
5810:
5797:
5792:
5787:
5782:
5768:
5761:
5760:
5758:
5750:
5743:
5741:
5740:
5738:
5732:
5728:
5723:
5710:
5706:
5705:
5702:
5701:
5695:
5691:
5684:
5680:
5679:
5663:
5659:
5658:
5649:
5645:
5632:
5627:
5622:
5609:
5602:
5597:
5592:
5578:
5571:
5570:
5568:
5560:
5553:
5551:
5550:
5548:
5542:
5538:
5533:
5520:
5516:
5515:
5512:
5511:
5501:
5497:
5496:
5480:
5476:
5475:
5466:
5462:
5446:
5442:
5441:
5428:
5418:
5414:
5413:
5399:
5392:
5391:
5389:
5381:
5374:
5372:
5371:
5369:
5363:
5359:
5358:
5345:
5341:
5340:
5336:
5334:
5331:
5330:
5321:
5315:
5286:
5283:
5282:
5281:
5279:
5276:
5272:
5269:
5265:
5260:
5257:
5256:
5255:
5253:
5250:
5246:
5219:
5209:
5202:
5201:
5196:
5195:
5194:
5190:
5189:
5179:
5178:
5174:
5164:
5163:
5159:
5158:
5153:
5143:
5139:
5138:
5136:
5133:
5132:
5126:
5120:
5117:
5111:
5107:
5083:
5082:
5078:
5068:
5067:
5063:
5062:
5057:
5048:
5044:
5042:
5039:
5038:
5029:
5028:. In this case
5026:identity matrix
5019:
5013:
5000:
4980:
4970:
4964:
4960:
4957:titration curve
4951:
4946:
4945:
4944:
4943:
4942:
4930:
4927:
4921:
4920:
4918:
4917:
4916:
4912:
4909:
4905:
4882:
4877:
4867:
4862:
4848:
4840:
4838:
4834:
4824:
4819:
4814:
4809:
4800:
4796:
4794:
4791:
4790:
4783:
4765:
4741:
4723:
4722:
4718:
4717:
4713:
4712:
4703:
4699:
4698:
4696:
4693:
4692:
4678:
4654:
4650:
4637:
4636:
4632:
4619:
4618:
4614:
4608:
4604:
4594:
4593:
4589:
4580:
4562:
4561:
4557:
4556:
4552:
4551:
4542:
4538:
4537:
4535:
4532:
4531:
4522:
4516:
4506:
4496:
4468:
4467:
4463:
4453:
4452:
4448:
4439:
4421:
4420:
4416:
4415:
4411:
4410:
4403:
4401:
4398:
4397:
4390:
4387:
4365:
4363:
4360:
4359:
4337:
4328:
4323:
4318:
4308:
4303:
4302:
4300:
4297:
4296:
4262:
4258:
4249:
4235:
4231:
4230:
4226:
4225:
4224:
4222:
4219:
4218:
4190:
4189:
4185:
4176:
4161:
4160:
4156:
4155:
4151:
4150:
4149:
4141:
4139:
4136:
4135:
4127:
4121:
4097:
4092:
4084:
4082:
4079:
4078:
4071:
4055:
4052:
4051:
4035:
4032:
4031:
4027:
3998:
3994:
3988:
3984:
3969:
3965:
3947:
3943:
3941:
3938:
3937:
3930:
3921:
3917:
3889:
3888:
3884:
3848:
3847:
3843:
3842:
3838:
3837:
3835:
3832:
3831:
3813:
3791:
3779:
3774:
3773:
3758:
3753:
3752:
3750:
3747:
3746:
3739:
3734:
3728:
3724:
3705:
3693:
3688:
3687:
3672:
3667:
3666:
3664:
3661:
3660:
3650:
3649:The increments
3622:
3621:
3617:
3590:
3589:
3585:
3584:
3580:
3579:
3577:
3574:
3573:
3558:
3555:
3549:
3548:
3546:
3545:
3534:
3520:
3516:
3513:
3509:
3486:
3482:
3478:
3461:
3460:
3455:
3447:
3445:
3433:
3429:
3427:
3424:
3423:
3415:
3403:
3397:
3394:
3388:
3387:
3385:
3384:
3378:
3374:
3370:
3364:
3363:The increments
3344:
3340:
3328:
3324:
3320:
3312:
3310:
3304:
3291:
3287:
3279:
3276:
3275:
3266:
3247:
3242:
3237:
3227:
3222:
3217:
3197:
3192:
3179:
3175:
3169:
3157:
3154:
3153:
3132:
3111:
3107:
3098:
3094:
3088:
3084:
3078:
3068:
3056:
3053:
3052:
3029:
3024:
3011:
3006:
2993:
2989:
2987:
2984:
2983:
2972:
2965:
2944:
2940:
2933:
2927:
2922:is the initial
2921:
2896:
2892:
2883:
2879:
2878:
2865:
2864:
2858:
2854:
2845:
2840:
2839:
2838:
2836:
2826:
2825:
2821:
2819:
2816:
2815:
2808:
2770:
2766:
2757:
2753:
2751:
2748:
2747:
2732:
2731:
2725:
2721:
2716:
2707:
2703:
2698:
2686:
2682:
2664:
2653:
2647:
2643:
2640:
2639:
2633:
2629:
2624:
2615:
2611:
2606:
2594:
2590:
2572:
2561:
2555:
2551:
2547:
2545:
2542:
2541:
2534:
2501:
2497:
2477:
2469:
2467:
2463:
2462:
2461:
2457:
2446:
2441:
2438:
2437:
2416:
2412:
2408:
2397:
2392:
2389:
2388:
2368:
2367:
2352:
2351:
2342:
2338:
2333:
2324:
2320:
2315:
2300:
2296:
2278:
2263:
2253:
2248:
2243:
2238:
2235:
2234:
2225:
2221:
2216:
2207:
2203:
2198:
2183:
2179:
2161:
2146:
2136:
2131:
2126:
2121:
2117:
2115:
2112:
2111:
2106:
2102:
2097:
2076:in particular.
2044:
2040:
2039:
2035:
2028:
2027:
2023:
2021:
2008:
2004:
2003:
1998:
1995:
1994:
1965:
1961:
1960:
1946:
1942:
1938:
1937:
1927:
1926:
1922:
1920:
1917:
1916:
1880:
1876:
1872:
1862:
1858:
1843:
1839:
1835:
1833:
1821:
1820:
1814:
1813:
1807:
1804:
1803:
1785:
1781:
1772:
1768:
1758:
1751:
1750:
1748:
1740:
1733:
1731:
1730:
1728:
1721:
1716:
1711:
1709:
1706:
1705:
1666:
1662:
1657:
1648:
1644:
1639:
1635:
1622:
1617:
1616:
1610:
1605:
1604:
1600:
1598:
1583:
1579:
1577:
1574:
1573:
1554:
1548:
1547:
1539:
1533:
1532:
1518:
1511:
1510:
1508:
1500:
1493:
1491:
1490:
1488:
1476:
1475:
1462:
1461:
1460:
1459:
1457:
1454:
1453:
1439:complex species
1435:complex species
1427:
1412:Model selection
1397:
1393:
1385:
1379:
1371:
1365:
1361:
1358:
1333:
1329:
1327:
1324:
1323:
1299:
1295:
1293:
1290:
1289:
1266:
1262:
1244:
1240:
1238:
1235:
1234:
1192:
1169:
1167:
1159:
1156:
1155:
1109:
1106:
1105:
1062:
1048:
1046:
1034:
1030:
1028:
1025:
1024:
1003:
954:
940:
938:
926:
922:
920:
917:
916:
904:, of a reagent
881:
877:
875:
872:
871:
857:
849:
831:
827:
824:
817:
805:
801:
798:
771:
743:
739:
735:
728:
724:
718:
714:
710:
708:
694:
693:
691:
688:
687:
680:
675:
674:
671:
661:
658:
649:
618:
615:
614:
608:
597:
593:
590:
580:
576:
572:
550:
536:
533:
532:
522:
517:
501:glass electrode
490:
486:
479:
455:
430:
421:
417:
405:
401:
393:
390:
389:
352:
348:
347:
343:
333:
325:
323:
312:
310:
307:
306:
285:
260:
244:
236:
234:
225:
221:
213:
210:
209:
203:Nernst equation
199:glass electrode
191:
175:
166:titrant added.
117:
106:
105:
99:
88:
87:
86:
76:
65:
64:
58:
47:
46:
45:
43:
35:
32:
31:
24:
12:
11:
5:
8250:
8240:
8239:
8234:
8217:
8216:
8214:
8213:
8212:
8211:
8201:
8200:
8199:
8189:
8188:
8187:
8177:
8176:
8175:
8165:
8160:
8155:
8154:
8153:
8143:
8138:
8133:
8132:
8131:
8120:
8118:
8114:
8113:
8111:
8110:
8105:
8100:
8094:
8092:
8088:
8087:
8085:
8084:
8079:
8074:
8069:
8064:
8059:
8058:
8057:
8046:
8044:
8038:
8037:
8035:
8034:
8029:
8024:
8019:
8014:
8013:
8012:
8007:
7997:
7992:
7987:
7982:
7977:
7971:
7969:
7965:
7964:
7957:
7956:
7949:
7942:
7934:
7927:
7926:
7907:(95): 95–101.
7891:
7884:
7863:
7856:
7838:
7803:
7796:
7778:
7756:
7737:(2): 413–432.
7721:
7685:
7658:
7614:
7587:(5): 771–778.
7576:
7564:
7523:
7520:. McGraw-Hill.
7507:
7505:
7502:
7501:
7500:
7480:
7463:
7453:
7435:
7432:
7426:
7423:
7418:
7414:
7410:
7406:
7402:
7398:
7394:
7389:
7386:
7377:
7374:
7364:(where GlyH =
7358:
7349:
7348:(Gly), Ni(Gly)
7317:
7316:
7303:
7299:
7296:
7293:
7289:
7283:
7279:
7274:
7270:
7267:
7264:
7260:
7256:
7251:
7247:
7231:
7230:
7215:
7212:
7209:
7204:
7199:
7195:
7192:
7189:
7185:
7181:
7178:
7128:
7127:
7112:
7109:
7106:
7101:
7094:
7089:
7083:
7080:
7075:
7071:
7065:
7060:
7055:
7050:
7047:
7041:
7038:
7035:
7030:
7026:
7023:
6996:
6993:
6987:
6974:
6968:
6965:
6955:
6948:
6920:
6913:
6889:
6886:
6880:
6877:
6876:
6875:
6862:
6857:
6853:
6847:
6842:
6839:
6834:
6830:
6825:
6810:
6809:
6795:
6792:
6789:
6785:
6777:
6773:
6768:
6760:
6756:
6751:
6747:
6744:
6739:
6732:
6728:
6723:
6719:
6714:
6707:
6703:
6698:
6694:
6689:
6684:
6680:
6658:
6657:
6643:
6639:
6635:
6630:
6626:
6622:
6611:
6601:
6585:
6571:
6567:
6563:
6558:
6554:
6550:
6545:
6541:
6537:
6532:
6528:
6524:
6513:
6503:
6487:
6473:
6469:
6465:
6460:
6456:
6452:
6447:
6443:
6439:
6434:
6430:
6426:
6415:
6405:
6366:
6362:
6354:
6353:
6337:
6333:
6327:
6323:
6317:
6314:
6311:
6306:
6302:
6298:
6295:
6290:
6286:
6271:
6270:
6259:
6253:
6244:
6239:
6235:
6228:
6219:
6216:
6210:
6207:
6198:
6193:
6189:
6185:
6182:
6177:
6173:
6166:
6157:
6154:
6148:
6145:
6136:
6131:
6127:
6108:
6099:
6096:
6095:
6084:
6078:
6069:
6066:
6060:
6055:
6046:
6043:
6040:
6035:
6027:
6019:
6011:
6000:
5986:
5979:
5968:
5961:
5950:
5942:
5936:
5931:
5909:
5902:
5901:
5884:
5880:
5873:
5864:
5861:
5855:
5852:
5843:
5838:
5834:
5830:
5827:
5818:
5809:
5803:
5800:
5798:
5790:
5771:
5764:
5753:
5746:
5735:
5726:
5722:
5716:
5713:
5704:
5703:
5698:
5694:
5687:
5678:
5675:
5669:
5666:
5657:
5652:
5648:
5644:
5641:
5635:
5630:
5621:
5615:
5612:
5610:
5605:
5600:
5581:
5574:
5563:
5556:
5545:
5536:
5532:
5526:
5523:
5514:
5513:
5510:
5504:
5495:
5492:
5486:
5483:
5474:
5469:
5465:
5461:
5458:
5452:
5449:
5440:
5434:
5431:
5429:
5424:
5421:
5402:
5395:
5384:
5377:
5366:
5357:
5351:
5348:
5339:
5338:
5319:
5314:
5311:
5284:
5274:
5267:
5258:
5248:
5239:
5238:
5225:
5222:
5217:
5212:
5205:
5199:
5193:
5182:
5177:
5173:
5167:
5162:
5157:
5152:
5146:
5142:
5124:
5115:
5104:
5103:
5086:
5081:
5077:
5071:
5066:
5061:
5056:
5051:
5047:
4962:
4947:
4940:
4939:
4914:
4913:
4907:
4902:
4901:
4885:
4880:
4876:
4870:
4865:
4860:
4854:
4851:
4846:
4843:
4837:
4832:
4827:
4822:
4818:
4813:
4808:
4803:
4799:
4762:
4761:
4747:
4744:
4739:
4735:
4732:
4726:
4721:
4716:
4711:
4706:
4702:
4675:
4674:
4660:
4657:
4653:
4649:
4646:
4640:
4635:
4631:
4628:
4622:
4617:
4611:
4607:
4603:
4597:
4592:
4586:
4583:
4578:
4574:
4571:
4565:
4560:
4555:
4550:
4545:
4541:
4493:
4492:
4477:
4474:
4471:
4466:
4462:
4456:
4451:
4445:
4442:
4437:
4433:
4430:
4424:
4419:
4414:
4409:
4406:
4386:
4383:
4382:
4381:
4368:
4355:
4354:
4353:
4340:
4334:
4331:
4326:
4321:
4316:
4311:
4306:
4291:
4288:
4285:pseudo-inverse
4281:
4280:
4279:
4265:
4261:
4255:
4252:
4247:
4243:
4238:
4234:
4229:
4213:
4212:
4211:
4199:
4193:
4188:
4182:
4179:
4174:
4170:
4164:
4159:
4154:
4148:
4144:
4114:
4113:
4100:
4095:
4091:
4087:
4059:
4039:
4024:
4023:
4010:
4007:
4004:
4001:
3997:
3991:
3987:
3981:
3978:
3975:
3972:
3968:
3964:
3961:
3958:
3955:
3950:
3946:
3914:
3913:
3901:
3898:
3892:
3887:
3883:
3880:
3877:
3873:
3869:
3866:
3863:
3860:
3857:
3851:
3846:
3841:
3810:
3809:
3797:
3794:
3790:
3787:
3782:
3777:
3772:
3767:
3764:
3761:
3756:
3721:
3720:
3708:
3704:
3701:
3696:
3691:
3686:
3681:
3678:
3675:
3670:
3647:
3646:
3634:
3631:
3625:
3620:
3616:
3613:
3610:
3606:
3602:
3599:
3593:
3588:
3583:
3511:
3506:
3505:
3489:
3485:
3481:
3473:
3470:
3467:
3464:
3458:
3454:
3450:
3444:
3439:
3436:
3432:
3401:
3368:
3361:
3360:
3347:
3343:
3339:
3331:
3327:
3323:
3318:
3315:
3307:
3303:
3299:
3294:
3290:
3286:
3283:
3240:
3220:
3214:
3213:
3200:
3195:
3191:
3185:
3182:
3178:
3172:
3168:
3164:
3161:
3129:
3128:
3114:
3110:
3104:
3101:
3097:
3091:
3087:
3081:
3077:
3071:
3067:
3063:
3060:
3046:
3045:
3027:
3023:
3019:
3009:
3005:
3001:
2996:
2992:
2964:
2961:
2952:Newton–Raphson
2942:
2931:
2919:
2916:
2915:
2899:
2895:
2891:
2886:
2882:
2874:
2868:
2861:
2857:
2853:
2848:
2835:
2824:
2793:
2792:
2781:
2778:
2773:
2769:
2765:
2760:
2756:
2745:
2728:
2724:
2715:
2710:
2706:
2697:
2692:
2689:
2685:
2681:
2676:
2673:
2670:
2667:
2663:
2659:
2656:
2654:
2646:
2642:
2641:
2636:
2632:
2623:
2618:
2614:
2605:
2600:
2597:
2593:
2589:
2584:
2581:
2578:
2575:
2571:
2567:
2564:
2562:
2554:
2550:
2549:
2509:
2504:
2500:
2496:
2493:
2490:
2483:
2480:
2475:
2472:
2466:
2460:
2456:
2453:
2449:
2445:
2422:
2419:
2415:
2411:
2407:
2404:
2400:
2396:
2382:
2381:
2366:
2363:
2360:
2357:
2354:
2353:
2350:
2345:
2341:
2332:
2327:
2323:
2314:
2309:
2306:
2303:
2299:
2295:
2290:
2287:
2284:
2281:
2277:
2273:
2270:
2262:
2259:
2256:
2254:
2237:
2236:
2233:
2228:
2224:
2215:
2210:
2206:
2197:
2192:
2189:
2186:
2182:
2178:
2173:
2170:
2167:
2164:
2160:
2156:
2153:
2145:
2142:
2139:
2137:
2120:
2119:
2104:
2100:
2096:
2093:
2069:
2068:
2053:
2047:
2038:
2026:
2020:
2017:
2011:
2002:
1987:
1986:
1974:
1968:
1959:
1954:
1949:
1941:
1936:
1930:
1925:
1906:
1905:
1891:
1888:
1883:
1879:
1875:
1870:
1865:
1861:
1857:
1854:
1851:
1846:
1842:
1838:
1832:
1826:
1823:
1817:
1812:
1788:
1780:
1775:
1761:
1754:
1743:
1736:
1719:
1689:
1688:
1674:
1669:
1665:
1656:
1651:
1647:
1638:
1633:
1630:
1625:
1613:
1603:
1597:
1592:
1589:
1586:
1582:
1560:
1521:
1514:
1503:
1496:
1487:
1474:
1449:forming them.
1426:
1423:
1414:
1413:
1410:
1407:
1404:
1395:
1383:
1369:
1357:
1354:
1339:
1336:
1332:
1305:
1302:
1298:
1286:
1285:
1272:
1269:
1265:
1261:
1258:
1255:
1250:
1247:
1243:
1228:
1227:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1166:
1163:
1149:
1148:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1095:
1094:
1080:
1077:
1074:
1071:
1068:
1065:
1060:
1057:
1054:
1051:
1045:
1040:
1037:
1033:
1001:
987:
986:
972:
969:
966:
963:
960:
957:
952:
949:
946:
943:
937:
932:
929:
925:
887:
884:
880:
856:
853:
848:
845:
823:
820:
815:
803:
797:
794:
769:
765:Example: the p
763:
762:
746:
742:
738:
731:
727:
721:
717:
713:
707:
701:
698:
670:
667:
657:
654:
646:
645:
634:
631:
628:
625:
622:
607:
604:
595:
589:
586:
569:
568:
556:
553:
549:
546:
543:
540:
521:
518:
516:
513:
488:
478:
475:
452:
451:
439:
436:
433:
429:
424:
420:
416:
413:
408:
404:
400:
397:
380:
379:
367:
363:
360:
355:
351:
346:
339:
336:
331:
328:
322:
318:
315:
282:
281:
269:
266:
263:
259:
256:
250:
247:
242:
239:
233:
228:
224:
220:
217:
190:
187:
174:
171:
155:concentrations
148:ionic strength
140:
139:
125:
120:
115:
112:
109:
102:
97:
94:
91:
84:
79:
74:
71:
68:
61:
56:
53:
50:
42:
39:
9:
6:
4:
3:
2:
8249:
8238:
8235:
8233:
8230:
8229:
8227:
8210:
8207:
8206:
8205:
8202:
8198:
8195:
8194:
8193:
8190:
8186:
8183:
8182:
8181:
8178:
8174:
8171:
8170:
8169:
8166:
8164:
8161:
8159:
8156:
8152:
8149:
8148:
8147:
8144:
8142:
8139:
8137:
8134:
8130:
8127:
8126:
8125:
8122:
8121:
8119:
8115:
8109:
8106:
8104:
8101:
8099:
8096:
8095:
8093:
8089:
8083:
8080:
8078:
8075:
8073:
8070:
8068:
8065:
8063:
8062:Phase diagram
8060:
8056:
8055:determination
8053:
8052:
8051:
8048:
8047:
8045:
8043:
8039:
8033:
8030:
8028:
8025:
8023:
8020:
8018:
8015:
8011:
8008:
8006:
8003:
8002:
8001:
7998:
7996:
7993:
7991:
7988:
7986:
7983:
7981:
7978:
7976:
7973:
7972:
7970:
7966:
7962:
7955:
7950:
7948:
7943:
7941:
7936:
7935:
7932:
7922:
7918:
7914:
7910:
7906:
7902:
7895:
7887:
7881:
7877:
7870:
7868:
7859:
7853:
7850:. Wiley-VCH.
7849:
7842:
7834:
7830:
7826:
7822:
7818:
7814:
7807:
7799:
7793:
7789:
7782:
7773:
7772:
7763:
7761:
7752:
7748:
7744:
7740:
7736:
7732:
7725:
7716:
7711:
7707:
7703:
7699:
7692:
7690:
7681:
7677:
7673:
7669:
7662:
7654:
7650:
7645:
7640:
7636:
7632:
7628:
7621:
7619:
7610:
7606:
7602:
7598:
7594:
7590:
7586:
7582:
7575:
7568:
7560:
7556:
7552:
7548:
7544:
7540:
7533:
7527:
7519:
7512:
7508:
7498:
7494:
7489:
7485:
7481:
7479:
7475:
7472:
7468:
7464:
7462:
7458:
7454:
7452:
7448:
7444:
7443:
7442:
7439:
7431:
7422:
7385:
7382:
7373:
7371:
7367:
7347:
7342:
7339:
7333:
7328:
7323:
7301:
7287:
7272:
7258:
7236:
7235:
7234:
7197:
7183:
7168:
7167:
7166:
7163:
7158:
7153:
7149:
7146:
7142:
7137:
7133:
7073:
7053:
7013:
7012:
7011:
7001:
6992:
6986:
6979:
6964:
6954:
6947:
6939:
6929:
6919:
6912:
6904:
6895:
6885:
6860:
6855:
6851:
6845:
6840:
6837:
6832:
6828:
6823:
6815:
6814:
6813:
6793:
6790:
6787:
6783:
6775:
6771:
6766:
6758:
6754:
6749:
6745:
6742:
6737:
6730:
6726:
6721:
6717:
6712:
6705:
6701:
6696:
6692:
6687:
6682:
6678:
6670:
6669:
6668:
6667:
6663:
6641:
6637:
6633:
6628:
6624:
6620:
6609:
6599:
6586:
6569:
6565:
6561:
6556:
6552:
6548:
6543:
6539:
6535:
6530:
6526:
6522:
6511:
6501:
6488:
6471:
6467:
6463:
6458:
6454:
6450:
6445:
6441:
6437:
6432:
6428:
6424:
6413:
6403:
6390:
6389:
6388:
6381:
6361:
6335:
6331:
6325:
6321:
6315:
6312:
6309:
6304:
6300:
6296:
6293:
6288:
6284:
6276:
6275:
6274:
6251:
6237:
6226:
6208:
6205:
6191:
6187:
6183:
6180:
6175:
6164:
6146:
6143:
6129:
6125:
6117:
6116:
6115:
6076:
6058:
6053:
6041:
6038:
6025:
6009:
5998:
5977:
5966:
5948:
5940:
5934:
5929:
5915:
5914:
5913:
5907:
5904:The stepwise
5882:
5871:
5853:
5850:
5836:
5832:
5828:
5816:
5801:
5799:
5788:
5762:
5751:
5733:
5724:
5720:
5714:
5711:
5696:
5685:
5667:
5664:
5650:
5646:
5642:
5633:
5628:
5613:
5611:
5603:
5598:
5572:
5561:
5543:
5534:
5530:
5524:
5521:
5502:
5484:
5481:
5467:
5463:
5459:
5450:
5447:
5432:
5430:
5422:
5419:
5393:
5382:
5364:
5355:
5349:
5346:
5329:
5328:
5327:
5325:
5310:
5308:
5303:
5301:
5297:
5296:random errors
5293:
5244:
5223:
5220:
5215:
5191:
5175:
5171:
5160:
5155:
5150:
5131:
5130:
5129:
5123:
5114:
5079:
5075:
5064:
5059:
5054:
5049:
5045:
5037:
5036:
5035:
5032:
5027:
5022:
5016:
5010:
5007:
5003:
4997:
4994:
4990:
4987:
4983:
4977:
4973:
4967:
4958:
4950:
4934:
4925:
4883:
4878:
4874:
4868:
4863:
4858:
4852:
4844:
4835:
4830:
4825:
4820:
4816:
4811:
4806:
4801:
4797:
4789:
4788:
4787:
4781:
4777:
4773:
4768:
4737:
4714:
4691:
4690:
4689:
4685:
4681:
4576:
4553:
4530:
4529:
4528:
4525:
4519:
4514:
4509:
4504:
4499:
4435:
4412:
4396:
4395:
4394:
4356:
4332:
4329:
4324:
4314:
4309:
4295:
4294:
4292:
4289:
4286:
4282:
4245:
4227:
4217:
4216:
4214:
4172:
4152:
4146:
4134:
4133:
4130:
4124:
4119:
4118:
4117:
4089:
4077:
4076:
4075:
4057:
4037:
4008:
4005:
4002:
3999:
3995:
3989:
3979:
3976:
3973:
3970:
3966:
3959:
3956:
3953:
3948:
3944:
3936:
3935:
3934:
3927:
3924:
3875:
3871:
3864:
3839:
3830:
3829:
3828:
3826:
3822:
3817:
3792:
3788:
3785:
3780:
3770:
3765:
3762:
3759:
3745:
3744:
3743:
3731:
3702:
3699:
3694:
3684:
3679:
3676:
3673:
3659:
3658:
3657:
3654:
3632:
3629:
3618:
3614:
3611:
3608:
3604:
3600:
3597:
3586:
3581:
3572:
3571:
3570:
3562:
3553:
3543:
3538:
3531:
3529:
3523:
3487:
3483:
3456:
3452:
3442:
3437:
3434:
3430:
3422:
3421:
3420:
3418:
3413:
3404:
3392:
3381:
3377:is less than
3371:
3345:
3341:
3337:
3329:
3325:
3316:
3305:
3301:
3297:
3292:
3288:
3284:
3281:
3274:
3273:
3272:
3269:
3264:
3263:Taylor series
3260:
3255:
3253:
3243:
3234:
3230:
3223:
3198:
3193:
3189:
3183:
3180:
3176:
3170:
3166:
3162:
3159:
3152:
3151:
3150:
3148:
3144:
3140:
3135:
3112:
3108:
3102:
3099:
3095:
3089:
3085:
3079:
3075:
3069:
3065:
3061:
3058:
3051:
3050:
3049:
3025:
3021:
3017:
3007:
3003:
2999:
2994:
2990:
2982:
2981:
2980:
2978:
2977:least-squares
2970:
2960:
2956:
2953:
2947:
2938:
2937:concentration
2930:
2925:
2897:
2893:
2889:
2884:
2880:
2859:
2855:
2851:
2846:
2833:
2822:
2814:
2813:
2812:
2804:
2802:
2798:
2779:
2776:
2771:
2767:
2763:
2758:
2754:
2746:
2726:
2708:
2690:
2687:
2683:
2679:
2674:
2671:
2668:
2665:
2661:
2657:
2655:
2644:
2634:
2616:
2598:
2595:
2591:
2587:
2582:
2579:
2576:
2573:
2569:
2565:
2563:
2552:
2540:
2539:
2538:
2532:
2527:
2523:
2502:
2494:
2464:
2458:
2454:
2413:
2409:
2405:
2385:
2364:
2361:
2358:
2355:
2348:
2343:
2325:
2307:
2304:
2301:
2297:
2293:
2288:
2285:
2282:
2279:
2275:
2271:
2257:
2255:
2231:
2226:
2208:
2190:
2187:
2184:
2180:
2176:
2171:
2168:
2165:
2162:
2158:
2154:
2140:
2138:
2110:
2109:
2108:
2092:
2090:
2085:
2083:
2077:
2075:
2045:
2024:
2018:
2009:
1993:
1992:
1991:
1966:
1947:
1934:
1923:
1915:
1914:
1913:
1911:
1910:ionic product
1886:
1881:
1877:
1863:
1859:
1855:
1844:
1840:
1830:
1824:
1822:
1810:
1786:
1778:
1773:
1752:
1741:
1717:
1704:
1703:
1702:
1699:
1697:
1694:
1672:
1667:
1649:
1628:
1623:
1611:
1595:
1590:
1587:
1584:
1580:
1558:
1512:
1501:
1485:
1472:
1452:
1451:
1450:
1448:
1444:
1440:
1436:
1432:
1422:
1420:
1411:
1408:
1405:
1402:
1401:
1400:
1389:
1382:
1375:
1368:
1353:
1337:
1334:
1330:
1321:
1303:
1300:
1296:
1270:
1267:
1263:
1259:
1256:
1253:
1248:
1245:
1241:
1233:
1232:
1231:
1208:
1199:
1196:
1185:
1176:
1173:
1164:
1161:
1154:
1153:
1152:
1135:
1132:
1129:
1126:
1120:
1117:
1114:
1111:
1104:
1103:
1102:
1100:
1075:
1066:
1055:
1052:
1043:
1038:
1035:
1031:
1023:
1022:
1021:
1019:
1015:
1011:
1007:
999:
995:
990:
967:
958:
947:
944:
935:
930:
927:
923:
915:
914:
913:
911:
907:
903:
885:
882:
878:
868:
866:
862:
861:Schwarzenbach
852:
844:
842:
837:
836:
819:
813:
808:
793:
791:
787:
786:potentiometry
783:
779:
775:
768:
744:
740:
736:
729:
725:
719:
715:
711:
705:
696:
686:
685:
684:
666:
653:
632:
629:
626:
623:
620:
613:
612:
611:
603:
600:
585:
554:
551:
547:
544:
541:
538:
531:
530:
529:
527:
512:
510:
504:
502:
498:
493:
484:
474:
472:
469:
465:
461:
427:
422:
418:
414:
411:
406:
402:
398:
395:
388:
387:
386:
383:
365:
361:
358:
353:
349:
344:
337:
334:
329:
326:
320:
305:
304:
303:
301:
297:
293:
288:
257:
254:
248:
245:
240:
237:
231:
226:
222:
218:
215:
208:
207:
206:
204:
200:
196:
186:
184:
180:
170:
167:
164:
160:
156:
151:
149:
145:
123:
118:
100:
82:
77:
59:
40:
37:
30:
29:
28:
22:
18:
8204:Vapor–liquid
8091:Applications
8054:
7904:
7900:
7894:
7875:
7847:
7841:
7816:
7812:
7806:
7787:
7781:
7770:
7734:
7730:
7724:
7705:
7702:Can. J. Chem
7701:
7671:
7668:Can. J. Chem
7667:
7661:
7634:
7630:
7584:
7580:
7573:
7567:
7542:
7538:
7526:
7517:
7511:
7440:
7437:
7428:
7425:Other models
7391:
7383:
7379:
7343:
7337:
7331:
7321:
7318:
7232:
7161:
7151:
7147:
7144:
7140:
7129:
7005:
6984:
6977:
6970:
6952:
6945:
6937:
6930:
6917:
6910:
6902:
6891:
6884:considered.
6882:
6811:
6659:
6379:
6359:
6355:
6272:
6097:
5912:is given by
5905:
5903:
5316:
5304:
5278:is given by
5252:is given by
5240:
5121:
5112:
5105:
5030:
5020:
5014:
5008:
5005:
5001:
4995:
4992:
4988:
4985:
4981:
4975:
4971:
4968:
4948:
4932:
4923:
4903:
4766:
4763:
4683:
4679:
4676:
4523:
4517:
4507:
4497:
4494:
4388:
4128:
4122:
4115:
4025:
3928:
3922:
3915:
3815:
3811:
3729:
3722:
3652:
3648:
3560:
3551:
3536:
3532:
3521:
3507:
3416:
3399:
3390:
3379:
3366:
3362:
3267:
3256:
3238:
3232:
3228:
3218:
3215:
3146:
3133:
3130:
3047:
2966:
2957:
2948:
2936:
2928:
2923:
2917:
2805:
2800:
2796:
2794:
2530:
2528:
2525:
2387:
2383:
2098:
2089:data sources
2086:
2078:
2070:
1988:
1907:
1700:
1690:
1446:
1438:
1434:
1430:
1428:
1415:
1387:
1380:
1373:
1366:
1359:
1319:
1287:
1229:
1150:
1098:
1096:
1009:
1005:
997:
993:
991:
988:
909:
905:
901:
869:
858:
850:
834:
825:
806:
799:
766:
764:
672:
659:
647:
609:
598:
591:
570:
523:
505:
491:
480:
453:
384:
381:
286:
283:
197:such as the
192:
176:
168:
152:
141:
15:
8209:Henry's law
8000:Free energy
7491:Commercial
7488:Affinimeter
5906:association
5324:citric acid
778:citric acid
665:radiation.
460:strong acid
8226:Categories
8192:Solubility
8163:Hydrolysis
8072:Phase rule
7857:0471188174
7504:References
7157:hat matrix
7136:idempotent
6662:correlated
6356:and since
5322:, such as
5307:correlated
4776:reciprocal
1912:of water.
1693:cumulative
812:H NMR
520:Absorbance
8180:Partition
8010:Helmholtz
7980:Chelation
7581:Biometals
7465:NMR data
7447:Hyperquad
7295:−
7266:−
7191:−
7132:symmetric
7079:−
7046:−
6852:σ
6838:
6824:σ
6784:ρ
6772:β
6767:σ
6755:β
6750:σ
6743:−
6727:β
6722:σ
6702:β
6697:σ
6679:σ
6638:β
6634:
6566:β
6562:
6549:−
6540:β
6536:
6468:β
6464:
6451:−
6442:β
6438:
6332:β
6322:β
6301:β
6285:β
6209:−
6188:β
6147:−
6126:β
6059:−
5985:⇀
5978:−
5967:−
5960:↽
5935:−
5854:−
5833:β
5770:⇀
5763:−
5752:−
5745:↽
5715:−
5668:−
5647:β
5634:−
5604:−
5580:⇀
5573:−
5562:−
5555:↽
5525:−
5485:−
5464:β
5451:−
5423:−
5401:⇀
5394:−
5383:−
5376:↽
5350:−
5221:−
5172:−
5141:Σ
5076:−
5046:σ
4875:σ
4850:∂
4842:∂
4817:σ
4743:−
4701:Σ
4656:−
4606:Σ
4582:−
4540:Σ
4441:−
4330:−
4325:λ
4310:λ
4305:ε
4251:−
4178:−
4143:ε
4094:ε
4038:λ
3990:λ
3967:ε
3960:∑
3949:λ
3876:δ
3865:λ
3793:δ
3703:δ
3609:δ
3480:∂
3449:∂
3338:δ
3322:∂
3314:∂
3302:∑
3167:∑
3076:∑
3066:∑
3018:−
2768:β
2755:β
2684:β
2662:∑
2592:β
2570:∑
2495:−
2465:−
2414:−
2349:⋯
2308:⋯
2298:β
2276:∑
2232:⋯
2191:⋯
2181:β
2159:∑
2010:−
1967:−
1864:−
1787:−
1760:⇀
1753:−
1742:−
1735:↽
1673:⋯
1629:⋯
1591:⋯
1581:β
1559:⋯
1520:⇀
1513:−
1502:−
1495:↽
1486:⋯
1447:reactants
1431:reactants
1260:×
1124:⇋
1014:polyamine
1012:may be a
992:A ligand
776:group in
737:∑
726:δ
712:∑
700:¯
697:δ
630:φ
627:∑
552:ε
548:∑
528:applies.
497:titration
428:
359:−
258:
163:titration
124:⋯
119:β
101:α
83:⋯
78:τ
60:σ
8173:of water
7968:Concepts
7921:18963802
7833:18966661
7653:27779264
7609:11615864
7601:19288211
7559:14182410
7474:Archived
7165:. Thus,
5243:variance
5012:, where
4772:diagonal
4527:. Then,
4120:(1) The
3414:matrix,
3412:Jacobian
2531:identity
1825:′
1016:such as
774:hydroxyl
499:using a
462:against
144:activity
7901:Talanta
7813:Talanta
7739:Bibcode
7457:HypSpec
7366:glycine
7354:Ni(Gly)
7138:matrix
6960:
6934:
6925:
6899:
6384:
6370:
6273:whence
4937:
4919:
3565:
3547:
3515:is the
3407:
3386:
3250:is the
2918:where R
2533:(unit)
772:of the
294:. When
290:is the
8042:Models
7919:
7882:
7854:
7831:
7794:
7651:
7607:
7599:
7557:
7471:EQNMR
7467:HypNMR
7325:is an
7319:where
6365:= −log
6014:
6007:
5805:
5780:
5617:
5590:
5436:
5411:
5106:where
5024:is an
4904:where
3916:where
3723:where
3508:where
2924:amount
2801:formed
1419:Solver
648:where
571:where
454:where
296:buffer
284:where
8005:Gibbs
7605:S2CID
7555:S2CID
7535:(PDF)
7484:HypΔH
7413:and M
4677:When
3226:when
3216:with
464:borax
300:meter
7917:PMID
7880:ISBN
7852:ISBN
7829:PMID
7792:ISBN
7649:PMID
7597:PMID
7352:and
7335:and
7329:and
7233:and
7130:The
6812:and
6102:and
5271:and
5119:and
4959:and
3031:calc
1322:and
865:EDTA
830:and
788:nor
7909:doi
7821:doi
7747:doi
7710:doi
7676:doi
7639:doi
7589:doi
7547:doi
7419:q′′
7415:p′′
6973:log
6896:of
6829:log
6625:log
6553:log
6527:log
6455:log
6429:log
4984:= (
4770:is
4682:= (
3567:= 0
3244:= 1
3224:= 0
3013:obs
2436:or
2103:, T
818:O.
802:log
594:log
487:log
419:log
8228::
7915:.
7905:32
7903:.
7866:^
7827:.
7817:43
7815:.
7759:^
7745:.
7735:10
7733:.
7706:68
7704:.
7700:.
7688:^
7672:60
7670:.
7647:.
7635:52
7633:.
7629:.
7617:^
7603:.
7595:.
7585:22
7583:.
7553:.
7543:57
7541:.
7537:.
7486:.
7469:,
7411:q′
7407:p′
7346:Ni
7159:,
7134:,
6988:pH
6975:10
6951:−
6916:−
6833:10
6794:13
6788:12
6776:13
6759:12
6731:13
6706:12
6642:11
6629:10
6570:11
6557:10
6544:12
6531:10
6472:12
6459:10
6446:13
6433:10
6367:10
6336:12
6326:13
6305:12
6289:13
6192:12
6130:13
6104:LH
6050:LH
6022:LH
5995:LH
5926:LH
5837:13
5813:LH
5785:LH
5651:12
5625:LH
5595:LH
5468:11
5444:LH
5416:LH
5326:.
5309:.
5290:.
5285:ij
5259:ii
5004:=
4979:,
4974:=
3530:.
3271:.
3241:ii
3231:≠
3221:ij
2971:,
2772:01
2759:10
2410:10
2091:.
2006:OH
1963:OH
1802::
1783:OH
1572::
1378:,
1352:.
1020:.
1002:AC
912:.
902:AB
804:10
596:10
489:10
485:.
466:.
423:10
255:ln
185:.
7953:e
7946:t
7939:v
7923:.
7911::
7888:.
7860:.
7835:.
7823::
7800:.
7753:.
7749::
7741::
7718:.
7712::
7682:.
7678::
7655:.
7641::
7611:.
7591::
7577:a
7574:K
7561:.
7549::
7417:B
7409:A
7403:r
7401:B
7399:q
7397:A
7395:p
7359:3
7350:2
7338:M
7332:M
7322:I
7302:)
7298:H
7292:I
7288:(
7282:y
7278:M
7273:)
7269:H
7263:I
7259:(
7255:=
7250:r
7246:M
7214:s
7211:b
7208:o
7203:y
7198:)
7194:H
7188:I
7184:(
7180:=
7177:r
7162:H
7152:J
7150:)
7148:T
7145:J
7143:(
7141:J
7111:s
7108:b
7105:o
7100:y
7093:T
7088:J
7082:1
7074:)
7070:T
7064:T
7059:J
7054:(
7049:J
7040:s
7037:b
7034:o
7029:y
7025:=
7022:r
7008:U
6985:σ
6978:β
6956:p
6953:n
6949:d
6946:n
6942:/
6938:U
6921:p
6918:n
6914:d
6911:n
6907:/
6903:U
6861:K
6856:K
6846:=
6841:K
6791:,
6746:2
6738:2
6718:+
6713:2
6693:=
6688:2
6683:K
6621:=
6610:3
6606:a
6600:K
6595:p
6523:=
6512:2
6508:a
6502:K
6497:p
6425:=
6414:1
6410:a
6404:K
6399:p
6380:K
6376:/
6373:1
6363:a
6360:K
6358:p
6316:=
6313:K
6310:;
6297:K
6294:=
6258:]
6252:+
6248:H
6243:[
6238:2
6234:]
6227:+
6223:H
6218:[
6215:]
6206:3
6202:L
6197:[
6184:K
6181:=
6176:3
6172:]
6165:+
6161:H
6156:[
6153:]
6144:3
6140:L
6135:[
6109:2
6100:3
6083:]
6077:+
6073:H
6068:[
6065:]
6054:2
6045:[
6042:K
6039:=
6034:]
6026:3
6018:[
6010:;
5999:3
5949:+
5945:H
5941:+
5930:2
5910:3
5883:3
5879:]
5872:+
5868:H
5863:[
5860:]
5851:3
5847:L
5842:[
5829:=
5826:]
5817:3
5808:[
5802::
5789:3
5734:+
5730:H
5725:3
5721:+
5712:3
5708:L
5697:2
5693:]
5686:+
5682:H
5677:[
5674:]
5665:3
5661:L
5656:[
5643:=
5640:]
5629:2
5620:[
5614::
5599:2
5544:+
5540:H
5535:2
5531:+
5522:3
5518:L
5509:]
5503:+
5499:H
5494:[
5491:]
5482:3
5478:L
5473:[
5460:=
5457:]
5448:2
5439:[
5433::
5420:2
5365:+
5361:H
5356:+
5347:3
5343:L
5320:3
5280:Σ
5275:j
5273:p
5268:i
5266:p
5254:Σ
5249:i
5247:p
5224:1
5216:)
5211:J
5204:T
5198:J
5192:(
5181:p
5176:n
5166:d
5161:n
5156:U
5151:=
5145:p
5125:p
5122:n
5116:d
5113:n
5108:U
5085:p
5080:n
5070:d
5065:n
5060:U
5055:=
5050:2
5031:σ
5021:I
5015:σ
5009:I
5006:σ
5002:Σ
4996:y
4993:J
4991:)
4989:J
4986:J
4982:p
4976:I
4972:W
4963:v
4961:σ
4949:k
4941:)
4933:v
4931:∂
4928:/
4924:E
4922:∂
4915:(
4908:E
4906:σ
4884:2
4879:v
4869:2
4864:k
4859:)
4853:v
4845:E
4836:(
4831:+
4826:2
4821:E
4812:1
4807:=
4802:k
4798:W
4784:k
4767:Σ
4746:1
4738:)
4734:J
4731:W
4725:T
4720:J
4715:(
4710:=
4705:p
4686:)
4684:Σ
4680:W
4659:1
4652:)
4648:J
4645:W
4639:T
4634:J
4630:(
4627:J
4621:T
4616:W
4610:y
4602:W
4596:T
4591:J
4585:1
4577:)
4573:J
4570:W
4564:T
4559:J
4554:(
4549:=
4544:p
4524:Σ
4518:Σ
4508:y
4498:p
4476:s
4473:b
4470:o
4465:y
4461:W
4455:T
4450:J
4444:1
4436:)
4432:J
4429:W
4423:T
4418:J
4413:(
4408:=
4405:p
4391:U
4380:.
4367:C
4339:C
4333:1
4320:A
4315:=
4287:.
4264:T
4260:C
4254:1
4246:)
4242:C
4237:T
4233:C
4228:(
4198:A
4192:T
4187:C
4181:1
4173:)
4169:C
4163:T
4158:C
4153:(
4147:=
4129:ε
4123:ε
4099:C
4090:=
4086:A
4072:ε
4058:l
4028:A
4009:.
4006:.
4003:q
4000:p
3996:c
3986:)
3980:.
3977:.
3974:q
3971:p
3963:(
3957:l
3954:=
3945:A
3931:A
3923:I
3918:λ
3900:r
3897:W
3891:T
3886:J
3882:=
3879:p
3872:)
3868:I
3862:+
3859:J
3856:W
3850:T
3845:J
3840:(
3816:p
3814:δ
3796:p
3789:f
3786:+
3781:n
3776:p
3771:=
3766:1
3763:+
3760:n
3755:p
3740:f
3735:U
3730:y
3725:n
3707:p
3700:+
3695:n
3690:p
3685:=
3680:1
3677:+
3674:n
3669:p
3653:p
3651:δ
3633:r
3630:W
3624:T
3619:J
3615:=
3612:p
3605:)
3601:J
3598:W
3592:T
3587:J
3582:(
3561:p
3559:∂
3556:/
3552:U
3550:∂
3537:p
3535:δ
3522:y
3517:k
3512:k
3510:p
3488:k
3484:p
3472:c
3469:l
3466:a
3463:c
3457:j
3453:y
3443:=
3438:k
3435:j
3431:J
3417:J
3402:i
3400:p
3398:∂
3395:/
3391:U
3389:∂
3380:U
3375:U
3369:i
3367:p
3365:δ
3346:i
3342:p
3330:i
3326:p
3317:U
3306:i
3298:+
3293:0
3289:U
3285:=
3282:U
3268:p
3248:U
3239:W
3233:i
3229:j
3219:W
3199:2
3194:i
3190:r
3184:i
3181:i
3177:W
3171:i
3163:=
3160:U
3134:W
3113:j
3109:r
3103:j
3100:i
3096:W
3090:i
3086:r
3080:j
3070:i
3062:=
3059:U
3026:i
3022:y
3008:i
3004:y
3000:=
2995:i
2991:r
2973:U
2943:i
2941:v
2932:0
2929:v
2920:0
2898:i
2894:v
2890:+
2885:0
2881:v
2873:]
2870:R
2867:[
2860:i
2856:v
2852:+
2847:0
2842:R
2834:=
2828:R
2823:T
2809:i
2780:1
2777:=
2764:=
2727:q
2723:]
2718:B
2714:[
2709:p
2705:]
2700:A
2696:[
2691:q
2688:p
2680:q
2675:k
2672:n
2669:,
2666:0
2658:=
2649:B
2645:T
2635:q
2631:]
2626:B
2622:[
2617:p
2613:]
2608:A
2604:[
2599:q
2596:p
2588:p
2583:k
2580:n
2577:,
2574:0
2566:=
2557:A
2553:T
2535:β
2508:)
2503:0
2499:E
2492:E
2489:(
2482:T
2479:R
2474:F
2471:n
2459:e
2455:=
2452:]
2448:H
2444:[
2421:H
2418:p
2406:=
2403:]
2399:H
2395:[
2365:.
2362:c
2359:t
2356:e
2344:q
2340:]
2335:B
2331:[
2326:p
2322:]
2317:A
2313:[
2305:q
2302:p
2294:q
2289:k
2286:n
2283:,
2280:1
2272:+
2269:]
2265:B
2261:[
2258:=
2245:B
2241:T
2227:q
2223:]
2218:B
2214:[
2209:p
2205:]
2200:A
2196:[
2188:q
2185:p
2177:p
2172:k
2169:n
2166:,
2163:1
2155:+
2152:]
2148:A
2144:[
2141:=
2128:A
2124:T
2105:B
2101:A
2052:]
2046:+
2042:H
2037:[
2030:W
2025:K
2019:=
2016:]
2001:[
1973:]
1958:[
1953:]
1948:+
1944:H
1940:[
1935:=
1929:W
1924:K
1890:]
1887:O
1882:2
1878:H
1874:[
1869:]
1860:H
1856:O
1853:[
1850:]
1845:+
1841:H
1837:[
1831:=
1816:W
1811:K
1779:+
1774:+
1770:H
1726:O
1718:2
1714:H
1668:q
1664:]
1659:B
1655:[
1650:p
1646:]
1641:A
1637:[
1632:]
1624:q
1619:B
1612:p
1607:A
1602:[
1596:=
1588:q
1585:p
1550:q
1545:B
1535:p
1530:A
1483:B
1478:q
1473:+
1469:A
1464:p
1396:i
1394:y
1390:)
1388:i
1386:(
1384:B
1381:T
1376:)
1374:i
1372:(
1370:A
1367:T
1362:i
1338:C
1335:A
1331:K
1320:K
1304:B
1301:A
1297:K
1271:C
1268:A
1264:K
1257:K
1254:=
1249:B
1246:A
1242:K
1212:]
1209:B
1206:[
1203:]
1200:C
1197:A
1194:[
1189:]
1186:C
1183:[
1180:]
1177:B
1174:A
1171:[
1165:=
1162:K
1136:C
1133:+
1130:B
1127:A
1121:B
1118:+
1115:C
1112:A
1099:K
1079:]
1076:C
1073:[
1070:]
1067:A
1064:[
1059:]
1056:C
1053:A
1050:[
1044:=
1039:C
1036:A
1032:K
1010:C
1006:A
998:A
994:C
971:]
968:B
965:[
962:]
959:A
956:[
951:]
948:B
945:A
942:[
936:=
931:B
928:A
924:K
910:B
906:A
886:B
883:A
879:K
835:H
833:Δ
828:K
816:2
807:β
770:a
767:K
745:i
741:x
730:i
720:i
716:x
706:=
681:δ
676:δ
662:φ
650:φ
633:c
624:=
621:I
599:β
581:c
577:ε
573:l
555:c
545:l
542:=
539:A
492:β
456:s
438:]
435:A
432:[
415:s
412:+
407:0
403:E
399:=
396:E
366:)
362:E
354:0
350:E
345:(
338:T
335:R
330:F
327:n
321:=
317:H
314:p
287:E
268:}
265:A
262:{
249:F
246:n
241:T
238:R
232:+
227:0
223:E
219:=
216:E
114:]
111:B
108:[
96:]
93:A
90:[
73:]
70:T
67:[
55:]
52:S
49:[
41:=
38:K
25:K
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.