Obtaining the Equilibrium Constant

In order to apply the van I Hoff method of obtaining thermodynamic parameters, some means of measuring the association or dissociation constant of the protein-hgand interaction must be used. The basic principles and many of the experimental methodologies available for obtaining these constants have recently been summarized [32, 33] and are the subject of more extensive coverage in recent reviews (e.g., [34]) and monographs (e.g., [12, 35]). The methods include (extracted from [32] and [33])  

Equihbrium dialysis - Two compartments of a dialysis ceU are divided by a semi-permeable membrane. The protein-hgand complex is aUowed to associate or dissociate across the membrane until equihbrium is attained. By measuring the constituents of the interaction, the binding constant can be obtained from standard formulas. 

Steady-state dialysis - The equihbrium dialysis technique is accelerated by having buffer flow at a constant rate on one side of the semi-permeable membrane and by stirring both sides in order to minimize the concentration gradients [36]. Diafiltration - A type of dialysis equihbrium in which pressure is used to force the ligand-containing solution from one chamber into the protein-containing chamber [37]. 

Ultrafiltration - Pressure or centrifugation is used to force a mixture of known total concentrations of protein and ligand through a semi-permeable membrane [38]. 

Partition equilibrium - Separation occurs between two phases rather than across a semi-permeable membrane. Examples include partition between aqueous and lipid phases or partition between a liquid and a sohd phase (e.g., where the binding sites are embedded on a sohd matrix). 

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From these numbers, a large number of calculations of technical interest can be made. Further, if we divide the equilibrium constant of carbon dioxide by that of steam we obtain the equilibrium constant of the water-gas equilibrium ... 

We obtain the equilibrium constant expression from the reaction ... 

Applying collision theory to the forward and the reverse reactions, and taking the ratio, we obtain the equilibrium constant ... 

CH3COCH3 + HCN . (CH3)2C0.CN.0H, A + B C was studied in aqueous solution (Svirbely Roth, JACS 75 3109, 19531. In one run with initial concentrations of 0.0758 normal for HCN and 0.1164 normal for acetone, the tabulated data were obtained. The equilibrium constant was Ke = 13.87. Find the specific rate. 

The rate of decomposition of an optically active compound was followed by a polarimeter when the tabulated results were obtained. The equilibrium constant is 3.B9. Find the specific rate. 

An interesting but still unexplained case refers to nitrobenzene. The reversible electron exchange between nitrobenzeneand sodium salt of the nitrobenzene- N anion-radical is characterized by the usual constant of 0.40. Stevenson et al. (1987b) used NH3(liq) as a solvent for these measurements at -75°C. Under the same conditions, they obtained the equilibrium constant of 2.1( ) for the electron exchange between nitrobenzene- N and the potassium salt of nitrobenzene- " N anion-radical. Perhaps, the difference between ion radii of sodium and potassium cations is crucial for the stability of the corresponding ion pair with nitrobenzene anion-radical. Such diversity can be pivotal when the electron prefers the heavy or light nitrobenzene. 

Beer s law constants in the NO + 02 system which yields A/Z g = 14 + 2 kcal/mole and S298 = 68 3 eu for the ONOO radical. On the basis of these (albeit approximate) values, one obtains the equilibrium constant... 

Equations 6-8 have been used to obtain Equation 9 Equation 9 is a general equation which can be applied to all mixed associations. The final form of Equation 9 indicates that there are only two solute components (independent variables) involved in the mixed association. When no association occurs, then (dc/dCi)T>c — 1. The superscript eq is used to indicate a mixed association is present. Now that the quantity cMweq has been defined, how do we obtain Mweq, and how do we use it (or its analogs) to obtain the equilibrium constant or constants and the nonideal terms if they are present ... 

A number of studies of H-atom transfer from hydrogen halides to free radicals, R + HX - RH + X, have been done by FPTRMS in which R was detected by photoionization, and its decay was monitored as a function of [HX] under pseudo-first-order conditions. When the rate coefficient is combined with determinations of the rate coefficient of the reverse reaction to obtain the equilibrium constant, the enthalpy of formation of the radical can be deduced. If the kinetics are accurately measured in isolation, this is a direct kinetic method which can be used to confirm (or otherwise) thermodynamic data obtained by classical, indirect kinetic methods which depend on correct mechanistic interpretation. In a number of instances free radical enthalpies of formation by these two different approaches have not been in good agreement. It is not the purpose of this short survey to discuss the differences, but rather to briefly indicate the extent to which the FPTRMS method has contributed to the kinetics of these reactions and to free radical thermochemistry. 

To obtain the equilibrium constant, Kn, we multiply known equilibrium constants for reactions that add to give the net ionic equation for the neutralization reaction. Because CH3C02H is on the left side of the equation and CH3C02- is on the right side, one of the reactions needed is the dissociation of CH3C02H. Since H20 is on the right side of the equation and OH - is on the left side, the other reaction needed is the reverse of the dissociation of H20. Note that H30+ and one H20 molecule cancel when the two equations are added ... 

As in the weak acid-strong base case, we can obtain the equilibrium constant for the neutralization reaction by multiplying known equilibrium constants for reactions that add to give the net ionic equation ... 

To obtain the equilibrium constant in these systems, one can use electrochemical cells such as those described in Section 3.4.8. For example, measurements that involve... 

H-transfer is always ca. 10 faster than 1 4 H-transfer at 600 °K (see Table 3), so it will predominate when the molecular structure of the fuel permits. Simple estimation of the relative concentrations of the hydroperoxyalkyl radicals derived from propane, n-butane and n-pentane illustrates this. Thus, if the relative frequency of attack by OH at primary, secondary and tertiary C—H bond is taken as 2 3 5 [102], then the relative concentrations of propyl, butyl and pentyl radicals may be obtained. The equilibrium constant for reaction (3)... 

Combining equilibrium iii with equilibrium ii, and [CH2(0H)2] (CH20(aq)l one obtains the equilibrium constant... 

A kinetic study of living radical polymerizations of acrylates initiated by the (tetramesitylporphyronato)-cobalt(III) organo complexes (TMP)Co—CH(CH3)C02-Me and (Br8TMP)Co—CH(CH3)C02Me has been reported by Wayland et al.122 They applied an initial excess of the free cobalt complex and obtained the equilibrium constant for the reversible dissociation of the complex—poly(methyl acrylate) bond as K = 4.2 x 10 10 M for (TMP)Co and K= 1.3 x 10 8 M for (BrgTMP)Co from the rate of monomer consumption at 50 °C. The temperature dependence led to a bond... 

When we add two adjacent stepwise equilibria, we multiply the two equilibrium constants to obtain the equilibrium constant for the resulting overall reaction. Thus, for the first two dissociation equilibria for H3PO4, we write... 

By subtracting the last two equations and rearranging, we obtain the equilibrium constant as... 

The electrical method of measuring A is thus a very convenient way of obtaining the equilibrium constant Working out the above example, we see that—... 

The activity towards a target enzyme for a large number of compounds is measured by obtaining the equilibrium constant for the reaction ... 

The recently reported correlation of reactivities, and one-electron oxidation potentials of nucleophiles is examined with new data for hydrazine in aqueous solution and several nucleophiles in (CH3)2SO solution. The correlation fails to apply to these reactions. A thermodynamic cycle is utilized to estimate the free energies of ionization of pyronin-nucleophile adducts both in solution and in the solid state. A satisfying rationalization of the dichotomy of ionic and covalent crystals of these and similar compounds is obtained. The equilibrium constants for reactions of nucleophiles with several types of cations are examined as indicators of specific bonding effects such as steric and gem interactions. 

It may be surprising that all the rate parameters, together with their temperature dependencies, are available from just one or at most two runs. This is so in this case because of the simple and known expression for the kinetics of adsorption/desorption and because we have contrived to obtain the equilibrium constant, together with its temperature dependence, in one run. A full TSR experiment, with several runs, might still be necessary if the rate expression is not known a priori or if other issues, such as diffusion, need to be addressed. The methods used in such cases are described in the preceding discussion of the TS-PF-SSR. 

Equation (29.75) is an important link between quantum mechanics and chemistry. Knowing the energy levels of the molecules, we can calculate the molecular partition functions. Then we use Eq. (29.75) to obtain the equilibrium constant for the chemical reaction. 

Since there are two singly charged ions both in the numerator and in the denominator of this equilibrium concentration quotient and thi ionic concentration never exceeds 0.1 mole 1 , we assume that th. value of is equal, within about 10%, to that of the (thermodynamic < equilibrium constant for the reaction. Dmding this constant by thf solubility product of silver iodide, we obtain the equilibrium constant K - Q s ... 

Finally, multiply the two equilibrium constants to obtain the equilibrium constant for the sum of... 

For reversible reactions the problem of interpretation becomes more difficult since we have basically the same number of data, conversion versus time, but two rate constants to determine. If eonversion data are available up to the point of equilibrium, we may use the equilibrium eomposition together with conversion-time data to solve for the two constants directly. Generally, however, we must assume that the experimenter does not have the time or patience to obtain true equilibrium information (bearing in mind that rates of reaetion are very slow at this point) so that a simple trial procedure is probably most eonvenient. For the first-order forward and reverse case of equation (1-61) one would choose a value of a, which is a function of both kf and k, and test for linearity by plotting In [a/ a — x)] versus time. The correct choice of a will result in a straight line of slope kf + k ), and then the values for a and the slope may be used to solve for the individual constants. If equilibrium data are available, the rate of reaction is zero and the rate equation used directly to obtain the equilibrium constant ... 

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