Determination of equilibrium constants

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: 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".

5306: 7666:

Motekaitis, R. J.; Martell, A. E. (1982). "BEST — A new program for rigorous calculation of equilibrium parameters of complex multicomponent systems".

7951: 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".

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}} 7883: 7795: 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: 789: 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.

7000: 308: 4495:

Therefore, the parameter values are (approximately) linear combinations of the observed data values and the errors on the parameters,

3820: 7944: 7171: 6818: 6589: 811: 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

211: 8145: 7492: 2985: 840: 3411: 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: 8231: 7855: 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: 8236: 8203: 8107: 481:

The most widely used electrode is the glass electrode, which is selective for the hydrogen ion. This is suitable for all

7937: 291: 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} }} } 8016: 6120: 3929:

A particular issue arises with NMR and spectrophotometric data. For the latter, the observed quantity is absorbance,

458:

is an empirical slope factor. A solution of known hydrogen ion concentration may be prepared by standardization of a

391: 8123: 4512: 1707: 1442: 482: 8031: 7999: 7369: 4126:

values are considered parameters of the minimization and the Jacobian is constructed on that basis. However, the

1996: 1107: 508: 1918: 8172: 8167: 8128: 4779: 3527: 3155: 1909: 158: 7698:"Modelling complex solution equilibria. I. Fast, worry-free least-squares refinement of equilibrium constants" 3258: 2749: 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.

785: 194: 4346:{\displaystyle \mathbf {\boldsymbol {\varepsilon }} _{\lambda }=\mathbf {A} _{\lambda }^{-1}\mathbf {C} \,} 1236: 8191: 8081: 4775: 2390: 1288:

where K is the stability constant for the competition reaction. Thus, the value of the stability constant

1026: 918: 8184: 8179: 8157: 8102: 8009: 7989: 182: 525: 3149:. Most often only the diagonal elements are known, in which case the objective function simplifies to 616: 8066: 8049: 7960: 7738: 2088: 2073: 1695: 1692: 860: 143: 16: 4485:{\displaystyle \mathbf {p=\left(J^{\mathrm {T} }WJ\right)^{-1}J^{\mathrm {T} }Wy^{\mathrm {obs} }} } 1157: 8162: 8140: 8026: 7984: 20: 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.

8150: 8041: 7974: 7604: 7554: 7532:"Definitions of pH scales, standard reference values, measurement of pH, and related terminology" 5291: 4361: 2968: 1017: 2951: 2107:

etc. of the reactants A, B etc. This means solving a set of nonlinear equations of mass-balance

7531: 3906:{\displaystyle \mathbf {\left(J^{\mathrm {T} }WJ+\lambda I\right)\delta p=J^{\mathrm {T} }Wr} } 8196: 8076: 8004: 7994: 7916: 7879: 7851: 7828: 7791: 7648: 7596: 6893: 6665: 5299: 4502: 3142: 3138: 470: 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

8135: 8021: 7908: 7820: 7746: 7709: 7675: 7638: 7588: 7546: 7131: 3824: 3541: 3251: 2081: 1325: 1291: 873: 150:. Where this is not possible, consideration should be given to possible activity variation. 7460: 4215:

using the refined values of the equilibrium constants to obtain the speciation. The matrix

8097: 7477: 7326: 5025: 4956: 4771: 1360:

It is assumed that the collected experimental data comprise a set of data points. At each

500: 295: 202: 198: 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. 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

8208: 7769: 6892:

When the weights have been correctly derived from estimates of experimental error, the

4284: 1429:

The chemical model consists of a set of chemical species present in solution, both the

147: 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,

8225: 8061: 7912: 7824: 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}\,} 5295: 3262: 2976: 781: 154: 7608: 7558: 3261:. Firstly the objective function is linearised by approximating it as a first-order 7450: 7572:

Silva, Andre M. N.; Kong, Xiaole; Hider, Robert C. (2009). "Determination of the p

7470: 7405:

it is common practice the fix the values for the corresponding binary complexes M

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}} 201:. If the electrode is calibrated using activity standards it is assumed that the 5323: 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}}}} 777: 459: 193:

A free concentration or activity {A} of a species A is measured by means of an

6346:{\displaystyle \beta _{13}=K\beta _{12};K={\frac {\beta _{13}}{\beta _{12}}}\,} 3141:

of the observations. It is rare for this to be known. However, when it is, the

8071: 7592: 7156: 7135: 6999: 6661: 1406:

Calculation of the concentrations of all the chemical species in each solution

4999:) it is implied that the experimental errors are uncorrelated and all equal: 7979: 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

1013: 755:{\displaystyle {\bar {\delta }}={\frac {\sum x_{i}\delta _{i}}{\sum x_{i}}}} 496: 467: 162: 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 7496: 7456: 5242: 832: 773: 495:

values between about 2 and 11 can be measured directly by potentiometric

299: 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} } 7643: 7626: 7365: 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

7345: 1418: 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.

5294:

is the square root of variance. These error estimates reflect only

7876:

Computational Methods for the Determination of Formation Constants

463: 372:{\displaystyle \mathrm {pH} ={\frac {nF}{RT}}\left(E^{0}-E\right)} 1691:

When using general-purpose computer programs, it is usual to use

864: 863:

in the determination of the stability constants of complexes of

7933: 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

1549: 1534: 1477: 1463: 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 6998: 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\}} } 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}} } 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

8116: 8090: 8040: 7967: 7516:Rossotti, F. J. C.; Rossotti, H. (1961). 5110:is the minimum value of the objective function and 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: 2907: 2784: 2737: 2515: 2428: 2373: 2060: 1978: 1897: 1794: 1680: 1564: 1344: 1310: 1277: 1219: 1140: 1086: 978: 892: 754: 637: 560: 443: 371: 273: 131: 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: 7848:The Determination and Use of Stability Constants 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 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 {} } 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 8: 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 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: 7930: 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}} 1445:that relate the particular combination of 7788:Excel for Chemists: A Comprehensive Guide 7713: 7642: 7620: 7618: 7280: 7248: 7243: 7241: 7206: 7205: 7175: 7173: 7103: 7102: 7091: 7090: 7077: 7062: 7061: 7032: 7031: 7020: 7018: 6854: 6848: 6831: 6826: 6820: 6798: 6786: 6774: 6769: 6757: 6752: 6736: 6729: 6724: 6711: 6704: 6699: 6686: 6681: 6675: 6646: 6640: 6627: 6613: 6608: 6603: 6602: 6593: 6591: 6574: 6568: 6555: 6542: 6529: 6515: 6510: 6505: 6504: 6495: 6493: 6476: 6470: 6457: 6444: 6431: 6417: 6412: 6407: 6406: 6397: 6395: 6342: 6334: 6324: 6318: 6303: 6287: 6281: 6250: 6245: 6236: 6225: 6220: 6204: 6199: 6190: 6174: 6163: 6158: 6142: 6137: 6128: 6122: 6075: 6070: 6057: 6052: 6047: 6029: 6024: 6016: 6002: 5997: 5983: 5976: 5975: 5973: 5965: 5958: 5956: 5955: 5953: 5947: 5933: 5928: 5923: 5922: 5920: 5881: 5870: 5865: 5849: 5844: 5835: 5820: 5815: 5810: 5792: 5787: 5782: 5768: 5761: 5760: 5758: 5750: 5743: 5741: 5740: 5738: 5732: 5727: 5723: 5710: 5705: 5695: 5684: 5679: 5663: 5658: 5649: 5632: 5627: 5622: 5602: 5597: 5592: 5578: 5571: 5570: 5568: 5560: 5553: 5551: 5550: 5548: 5542: 5537: 5533: 5520: 5515: 5501: 5496: 5480: 5475: 5466: 5446: 5441: 5418: 5413: 5399: 5392: 5391: 5389: 5381: 5374: 5372: 5371: 5369: 5363: 5358: 5345: 5340: 5336: 5334: 5219: 5209: 5202: 5201: 5196: 5179: 5178: 5164: 5163: 5153: 5143: 5138: 5136: 5083: 5082: 5068: 5067: 5057: 5048: 5042: 4882: 4877: 4867: 4862: 4838: 4824: 4819: 4809: 4800: 4794: 4741: 4723: 4722: 4703: 4698: 4696: 4654: 4637: 4636: 4619: 4618: 4608: 4594: 4593: 4580: 4562: 4561: 4542: 4537: 4535: 4468: 4467: 4453: 4452: 4439: 4421: 4420: 4403: 4401: 4365: 4363: 4342: 4337: 4328: 4323: 4318: 4308: 4303: 4300: 4262: 4249: 4235: 4224: 4222: 4190: 4189: 4176: 4161: 4160: 4149: 4141: 4139: 4102: 4097: 4092: 4084: 4082: 4055: 4035: 3998: 3988: 3969: 3947: 3941: 3889: 3888: 3848: 3847: 3837: 3835: 3791: 3779: 3774: 3758: 3753: 3750: 3705: 3693: 3688: 3672: 3667: 3664: 3622: 3621: 3590: 3589: 3579: 3577: 3486: 3461: 3460: 3455: 3445: 3433: 3427: 3344: 3328: 3310: 3304: 3291: 3279: 3197: 3192: 3179: 3169: 3157: 3137:, should be, ideally, the inverse of the 3117: 3111: 3098: 3088: 3078: 3068: 3056: 3029: 3024: 3011: 3006: 2993: 2987: 2896: 2883: 2865: 2864: 2858: 2845: 2840: 2836: 2826: 2825: 2819: 2785:{\displaystyle \beta _{10}=\beta _{01}=1} 2770: 2757: 2751: 2725: 2716: 2707: 2698: 2686: 2664: 2647: 2633: 2624: 2615: 2606: 2594: 2572: 2555: 2547: 2545: 2501: 2467: 2463: 2462: 2461: 2446: 2441: 2416: 2412: 2397: 2392: 2342: 2333: 2324: 2315: 2300: 2278: 2263: 2248: 2243: 2238: 2225: 2216: 2207: 2198: 2183: 2161: 2146: 2131: 2126: 2121: 2117: 2115: 2057: 2044: 2039: 2028: 2027: 2021: 2008: 2003: 1998: 1975: 1965: 1960: 1946: 1938: 1937: 1927: 1926: 1920: 1880: 1862: 1843: 1833: 1820: 1814: 1813: 1807: 1785: 1772: 1758: 1751: 1750: 1748: 1740: 1733: 1731: 1730: 1728: 1721: 1716: 1711: 1709: 1666: 1657: 1648: 1639: 1622: 1617: 1610: 1605: 1598: 1583: 1577: 1554: 1548: 1547: 1539: 1533: 1532: 1518: 1511: 1510: 1508: 1500: 1493: 1491: 1490: 1488: 1476: 1475: 1462: 1461: 1460: 1459: 1457: 1333: 1327: 1299: 1293: 1266: 1244: 1238: 1167: 1159: 1109: 1046: 1034: 1028: 938: 926: 920: 881: 875: 743: 728: 718: 708: 694: 693: 691: 618: 550: 536: 430: 421: 405: 393: 352: 323: 312: 310: 260: 234: 225: 213: 117: 106: 99: 88: 76: 65: 58: 47: 43: 35: 7518:The Determination of Stability Constants 3656:are added iteratively to the parameters 7508: 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: 7625:Hibbert, D.B.; Thordarson, P. (2017). 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. 2979:. First the residuals are defined as 2084:is introduced into the calculations. 1278:{\displaystyle K_{AB}=K\times K_{AC}} 606:Fluorescence (luminescence) intensity 7: 4511:, using the linear formula. Let the 2429:{\displaystyle =10^{-\mathrm {pH} }} 19:are determined in order to quantify 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. 7213: 7210: 7207: 7110: 7107: 7104: 7092: 7063: 7039: 7036: 7033: 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: 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} 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 7771:Statistics in Physical Science 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). 3821:Levenberg–Marquardt algorithm 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 7790:(3rd ed.). Wiley-VCH. 7696:Potvin, P. G. (1990). 7497:standard formation enthalpy 1443:stoichiometric coefficients 189:Potentiometric measurements 8253: 7370:liquid junction potentials 6982:significantly larger than 4513:variance-covariance matrix 3139:variance-covariance matrix 509:liquid junction potentials 8032:Thermodynamic equilibrium 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: 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: 7898: 7897: 7893: 7886: 7873: 7872: 7865: 7858: 7845: 7844: 7840: 7810: 7809: 7805: 7798: 7785: 7784: 7780: 7766: 7765: 7758: 7751:10.1137/0710036 7728: 7727: 7723: 7715:10.1139/v90-337 7695: 7694: 7687: 7680:10.1139/v82-347 7665: 7664: 7660: 7624: 7623: 7616: 7578: 7571: 7570: 7566: 7539:Pure Appl. Chem 7534: 7530: 7529: 7525: 7515: 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: 7276: 7261: 7257: 7244: 7238: 7237: 7201: 7186: 7182: 7170: 7169: 7160: 7139: 7098: 7086: 7057: 7056: 7052: 7051: 7027: 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 6850: 6827: 6822: 6817: 6816: 6782: 6770: 6765: 6753: 6748: 6725: 6700: 6672: 6671: 6636: 6623: 6598: 6588: 6587: 6564: 6551: 6538: 6525: 6500: 6490: 6489: 6466: 6453: 6440: 6427: 6402: 6392: 6391: 6378: 6375: 6372: 6371: 6369: 6368: 6364: 6357: 6330: 6320: 6299: 6283: 6278: 6277: 6246: 6232: 6221: 6200: 6186: 6170: 6159: 6138: 6124: 6119: 6118: 6110: 6107: 6106: 6105: 6103: 6101: 6071: 5943: 5917: 5916: 5911: 5888: 5887: 5877: 5866: 5845: 5831: 5797: 5728: 5706: 5702: 5701: 5691: 5680: 5659: 5645: 5609: 5538: 5516: 5512: 5511: 5497: 5476: 5462: 5442: 5428: 5414: 5359: 5341: 5331: 5330: 5321: 5315: 5286: 5283: 5282: 5281: 5279: 5276: 5272: 5269: 5265: 5260: 5257: 5256: 5255: 5253: 5250: 5246: 5195: 5194: 5190: 5189: 5174: 5159: 5158: 5139: 5133: 5132: 5126: 5120: 5117: 5111: 5107: 5078: 5063: 5062: 5044: 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: 4848: 4840: 4834: 4814: 4796: 4791: 4790: 4783: 4765: 4718: 4717: 4713: 4712: 4699: 4693: 4692: 4678: 4650: 4632: 4614: 4604: 4589: 4557: 4556: 4552: 4551: 4538: 4532: 4531: 4522: 4516: 4506: 4496: 4463: 4448: 4416: 4415: 4411: 4410: 4398: 4397: 4390: 4387: 4360: 4359: 4302: 4297: 4296: 4258: 4231: 4230: 4226: 4225: 4219: 4218: 4185: 4156: 4155: 4151: 4150: 4136: 4135: 4127: 4121: 4079: 4078: 4071: 4052: 4051: 4032: 4031: 4027: 3994: 3984: 3965: 3943: 3938: 3937: 3930: 3921: 3917: 3884: 3843: 3842: 3838: 3832: 3831: 3813: 3773: 3752: 3747: 3746: 3739: 3734: 3728: 3724: 3687: 3666: 3661: 3660: 3650: 3649:The increments 3617: 3585: 3584: 3580: 3574: 3573: 3558: 3555: 3549: 3548: 3546: 3545: 3534: 3520: 3516: 3513: 3509: 3482: 3478: 3447: 3429: 3424: 3423: 3415: 3403: 3397: 3394: 3388: 3387: 3385: 3384: 3378: 3374: 3370: 3364: 3363:The increments 3340: 3324: 3320: 3312: 3287: 3276: 3275: 3266: 3247: 3242: 3237: 3227: 3222: 3217: 3175: 3154: 3153: 3132: 3107: 3094: 3084: 3053: 3052: 2989: 2984: 2983: 2972: 2965: 2944: 2940: 2933: 2927: 2922:is the initial 2921: 2892: 2879: 2878: 2854: 2839: 2838: 2821: 2816: 2815: 2808: 2766: 2753: 2748: 2747: 2732: 2731: 2721: 2703: 2682: 2653: 2643: 2640: 2639: 2629: 2611: 2590: 2561: 2551: 2542: 2541: 2534: 2497: 2477: 2469: 2457: 2438: 2437: 2408: 2389: 2388: 2368: 2367: 2352: 2351: 2338: 2320: 2296: 2253: 2235: 2234: 2221: 2203: 2179: 2136: 2112: 2111: 2106: 2102: 2097: 2076:in particular. 2040: 2035: 2023: 2004: 1995: 1994: 1961: 1942: 1922: 1917: 1916: 1876: 1872: 1858: 1839: 1835: 1821: 1804: 1803: 1781: 1768: 1706: 1705: 1662: 1644: 1635: 1616: 1604: 1600: 1579: 1574: 1573: 1454: 1453: 1439:complex species 1435:complex species 1427: 1412:Model selection 1397: 1393: 1385: 1379: 1371: 1365: 1361: 1358: 1329: 1324: 1323: 1295: 1290: 1289: 1262: 1240: 1235: 1234: 1192: 1169: 1156: 1155: 1106: 1105: 1062: 1048: 1030: 1025: 1024: 1003: 954: 940: 922: 917: 916: 904:, of a reagent 877: 872: 871: 857: 849: 831: 827: 824: 817: 805: 801: 798: 771: 739: 735: 724: 714: 710: 688: 687: 680: 675: 674: 671: 661: 658: 649: 615: 614: 608: 597: 593: 590: 580: 576: 572: 533: 532: 522: 517: 501:glass electrode 490: 486: 479: 455: 417: 401: 390: 389: 348: 347: 343: 333: 325: 307: 306: 285: 244: 236: 221: 210: 209: 203:Nernst equation 199:glass electrode 191: 175: 166:titrant added. 105: 87: 86: 64: 46: 45: 32: 31: 24: 12: 11: 5: 8250: 8248: 8240: 8239: 8234: 8224: 8223: 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: 7959: 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: 13: 10: 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: 7892: 7887: 7881: 7877: 7870: 7868: 7864: 7859: 7853: 7850:. Wiley-VCH. 7849: 7842: 7839: 7834: 7830: 7826: 7822: 7818: 7814: 7807: 7804: 7799: 7793: 7789: 7782: 7779: 7773: 7772: 7763: 7761: 7757: 7752: 7748: 7744: 7740: 7736: 7732: 7725: 7722: 7716: 7711: 7707: 7703: 7699: 7692: 7690: 7686: 7681: 7677: 7673: 7669: 7662: 7659: 7654: 7650: 7645: 7640: 7636: 7632: 7628: 7621: 7619: 7615: 7610: 7606: 7602: 7598: 7594: 7590: 7586: 7582: 7575: 7568: 7565: 7560: 7556: 7552: 7548: 7544: 7540: 7533: 7527: 7524: 7519: 7512: 7509: 7503: 7498: 7494: 7489: 7485: 7481: 7479: 7475: 7472: 7468: 7464: 7462: 7458: 7454: 7452: 7448: 7444: 7443: 7442: 7439: 7433: 7431: 7424: 7422: 7387: 7385: 7382: 7375: 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: 6994: 6992: 6986: 6979: 6966: 6964: 6954: 6947: 6939: 6929: 6919: 6912: 6904: 6895: 6887: 6885: 6878: 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: 5312: 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: 4384: 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: 2962: 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: 2094: 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: 1424: 1422: 1420: 1411: 1408: 1405: 1402: 1401: 1400: 1389: 1382: 1375: 1368: 1355: 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 854: 852: 846: 844: 842: 837: 836: 821: 819: 813: 808: 795: 793: 791: 787: 786:potentiometry 783: 779: 775: 768: 744: 740: 736: 729: 725: 719: 715: 711: 705: 696: 686: 685: 684: 668: 666: 655: 653: 632: 629: 626: 623: 620: 613: 612: 611: 605: 603: 600: 587: 585: 554: 551: 547: 544: 541: 538: 531: 530: 529: 527: 519: 514: 512: 510: 504: 502: 498: 493: 484: 476: 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: 188: 186: 184: 180: 172: 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

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Knowledge (XXG)

Determination of equilibrium constants

Index