Klotz equation

Flow dialysis follows the procedmes of Colowick and Womack (1969) (17) as modified by Haiech et al. (1981) (18). The dialysis cell is described by Feldman (1978) (19). A 2 ml aliquot of protein (1-2 mg/ml) is placed into an upper Teflon chamber, and both the upper and lower chamber (0.1-0.2 ml) are temperature controlled at 25°C. A peristaltic pump that perfuses the lower chamber is set at a flow rate of 1 ml/min. Non-radioactive calciiun is added in steps to yield a final calcium concentration of 0-700 pM. Data are analyzed using the Hill equation and the Adair-Klotz equation. 

Actually, we have already discussed this case in Chapter 4, although the emphasis there was more as an example of an unsteady-state problem (i.e., nature of the problem) than as an example of the class of problems associated with adsorption and ion exchange. The important result of that analysis was the Klotz equation... 

This is the same as the Klotz equation of Chapter 4, equation (4-168), with the exception of the (—1) term. This discrepancy is the result of the fact that in order to use the step-function we had to neglect the material in the band in writing the material balance. 

Note that this is an expansion on the development of Section 9.2a.) Consider the adsorption of two components, 1 and 2, in a bed consisting of an intimate mixture of two adsorbents, each spedlic for one of the components. For each of the two it may be assumed that the Klotz equation apphes, and that there is no interaction between components. 

Experimentally, v are measured by varying [A] and data analysis is performed to evaluate two binding parameters, n and K, often via linear transformation such as the Klotz equation and the Scatchard equation (Table 10.1). 

Approach Direct Klotz equation Scatchard equation... 

Klotz equation v vs A plot Straight line Non-cooperativity... 

Figure 23-5. Dependence of the number Na of complexed A positions on the logarithm of the concentration [B] of uncomplexed B molecules for the binding of L-leucine (B) to the a-isopropyl maleate synthetase enzyme. The broken line corresponds to the Klotz equation with n = 2 and K = 500000 liter/mol ( ) or K = 55000 liter /mol (—). (After data by E. Teng-Leary and G. B. Kohlhaw.)...

The remaining question is how we got from G3MP2 (OK) = —117.672791 to G3MP2 Enthalpy = —117.667683. This is not a textbook of classical thermodynamics (see Klotz and Rosenberg, 2000) or statistical themiodynamics (see McQuarrie, 1997 or Maczek, 1998), so we shall use a few equations from these fields opportunistically, without explanation. The definition of heat capacity of an ideal gas... 

The volume of a solution is sometimes expressed as a function of composition and the partial molar volume is then obtained by differentiation. For example, Klotz and Rosenburg2 have expressed the volume of aqueous sodium chloride solutions at 298.15 K and ambient pressure as a function of the molality m of the solution by the equation ... 

Thus, p can be estimated from the observed pressure and the ideal pressure calculated from the molar volume and the ideal gas equation. Klotz and Rosenburg3 report that the error in using equation (6.23) to calculate p is less than 1% for O2 up to a pressure of 10 MPa. For CO2 (g) the error is 1% at 2.5 MPa and 4% at 5 MPa. 

The fugacity coefficient can be calculated from other equations of state such as the van der Waals, Redlick-Kwong, Peng-Robinson, and Soave,d but the calculation is complicated, since these equations are cubic in volume, and therefore they cannot be solved explicitly for Vm, as is needed to apply equation (6.12). Klotz and Rosenburg4 have shown a way to get around this problem by eliminating p from equation (6.12) and integrating over volume, but the process is not easy. For the van der Waals equation, they end up with the relationship... 

SCATCHARD PLOT KLOTZ PLOT ALLOSTERISM COOPERATIVITY HILL EQUATION AND PLOT WOMACK-COLOWICK DIALYSIS METHOD... 

Klotz (K8) put forward the following equation, based on the law of mass action, to describe reversible binding ... 

The stoichiometry and energetics of alkyl pyrazines binding to soy proteins were determined from the free, [L], and bound, u, ligand concentration and the amount of protein (12). A double reciprocal plot of 1/u vs 1/[L], a Klotz plot, yielded a straight line with the equation ... 

Hunston and co-workers (Klotz and Hunston, 1971 Hunston, 1975 Thakur et al., 1980) have presented an approximation technique for describing the binding of small molecules to proteins. The method exploits the formal similarity between the appropriate chemical equations and the equations that are... 

Substituting this result back into equation (4-164) and rearranging gives [I.M. Klotz, Chem. Rev., 39, 241 (1946)]... 

The equation can be rewritten in Klotz form (double reciprocal) as... 

G. Sposito, On the use of the Langmuir equation in the interpretation of adsorption phenomena. II The two-surface Langmuir equation. Soil Sci. Soc. Am. J. 46 1147 (1982). See also I. M. Klotz, Numbers of receptor sites from Scatchard graphs Facts and fantasies. Science 217 1247 (1982). (In the literature of macromolecular chemistry, Fig. 4.2 is known as a Scatchard plot.)... 

The data points in the plot of r/[D] vs. r for MPVI + MO in 0.01 M NaCl (Figure 4) shows a downward curvature, which is expected from the nonlinear nature of the Klotz plots shown in Figure 3. By taking into account the neighboring bound dye-dye interaction, Equation 2 can be... 

The first part (237 pp.) of Klotz s book, which is issued separately under the title Introduction to Chemical Thermodynamics , gives an account of the laws and basic equations of classical thermodynamics as applied to systems of constant composition. The second part is concerned with systems of variable composition. The book has a good treatment of the Third Law and its chemical applications but reaction equilibrium is covered only to a limited extent. 

---