The vant Hoff Equation

El Shahawi concluded that polyurethanes were preferred over carbon because of, among other properties, ease of regeneration. To support this, they determined the H values of a number of pesticides by determining partition coefficient as a function of temperature. The partitioning of a solute is in part defined by the vant Hoff equation. The integrated form of the equation is ... 

The magnitude of the equilibrium displacement depends on AT and AH for the equilibrium. The sign of AH1 determines the direction of the shift in accordance with the Vant Hoff equation. A typical equipment might discharge 10-25 kV in 1 cm3 of solution. Such a voltage discharge may produce about 45 J in 10-6 s corresponding to 4.5 x 107 W and temperature rise of 5-10°C. 

K represents the real equilibrium constant for an exchange between Z = 0 and Z = Zmax. The standard free energy of exchange is obtained as AG = — RT In K. The enthalpy of exchange is derived from the Vant Hoff equation, and the entropy change is calculated from the AG and AH values ... 

Figure 6.6 shows the effect of temperature on the experimental efficiency of anthracene [5] and acetophenone [22] eluted with pure micellar solutions. The efficiency of acetophenone on a C18 column was multiplied by 12 when the temperature was raised from 25°C to 73°C [22]. A threefold enhancement was obtained for anthracene [5]. The retention factors were decreased by temperature. The magnitude of the k decrease is far less important than what was observed with alcohol additions [34]. The k decrease with temperature are predicted by the Vant Hoff equation (eq. 6.4). The equilibrium enthalpies and entropies of the retention process were measured in MLC for various surfactants [36]. These studies showed that the micellar retention process is essentially governed by entropic effects [3 6]. 

The temperature dependence of the Henry constant obeys the vant Hoff equation ... 

A mathematically equivalent form of the vant Hoff equation is... 

FIGURE 5.5 Plot of the vant Hoff equation as given in equation 5.19. Plots like this are one graphical way of determining... 

To analyze sorption of penetrants, including hydrocarbons, in glassy polymers, the dual-mode sorption model is most frequently used. For a number of glassy polymers, correlations between the constants of the dual-mode sorption model and the condensabUity of hydrocarbons have been established (see, e.g.. Figure 9.4a through 9.4c and data presented in Refs. [18-20]). Temperature dependence of model constants is described by Vant-Hoff equation, where the exponent contains heat of penetrant sorption A//s. This quantity is essentially dependent on the heat of penetrant condensation, AHcond- A//s = AH ond + A//i, where AHi is partial molar enthalpy of penetrant dissolution in the polymer, AHi = [d( AGi/T)/d( l/T)]c, AGi is the partial molar free... 

Osmotic Pressure—The pressure exerted due to difference in solute concentration between two distinct zones and can be calculated using Vant Hoff equation ( rr = CRT/M, C = concentration of the solution, R = gas constant, T = temperature, M= molecular weight). 

Ihe same work in the absence of equihbrium under standard conditions maybe expressed through Vant Hoffs equation (equation (1.109))... 

The temperature dependence of K follows the usual vant Hoff equation [Eq. (2.19)]. 

The equilibrium constant which appears in the denominator of the time constant will vary with temperature according to a vant Hoff equation (K- whereas the temperature dependence of the pore diffusivity... 

Thermal swing is probably the most common system. The temperature dependence of the adsorption equilibrium constant is governed by a vant Hoff equation [Eq. (2.20)]. Since adsorption is an exothermic process and since the more strongly adsorbed species generally have the higher heats of adsorption, a large decrease in the adsorbed phase concentration in equilibrium with a specified fluid concentration may be achieved by a comparatively modest increase in temperature. This means that at elevated temperatures the iso-... 

Figure A4. Application of the vant Hoff test in the case of (a) RNase A and (b) RNase BS. The large divergences at the limits of the denaturation temperature range are artefacts due to the form of the analytical expression of equation A3 3.

The Hood s equation was based on the experimental results. Some theoretical significance to this equation was given by Vant Hoff (1884) on the basis of the effect of temperature on equilibrium constants. This idea was extended by Arrhenius in his attempt to obtain the relation between rate constant and temperature. The relation obtained was successfully applied by him to the effect of temperature data for a number of reactions and the equation is usually called the Arrhenius equation. 

The temperature dependence of ki ) is given by two Arrhenius equations and one vant Hoff relation ... 

Such data are summarized in the form of vant Hoff plots in which a series of isocompositional lines are plotted as a function of reciprocal temperature and log partial pressure. The results for the above system are presented in Figure 8.8. The dashed lines in the right-hand portion represent the metastable (single phase) extensions into the two-phase region. The enthalpy for the oxidation process, Equation 8.1, can be obtained from the slope of the lines in the single-ph ase region. 

This approach assumes the constancy of AH over the temperature range that is, its accuracy depends largely on the linearity of flie vant Hoff regression. Estimation of AS from R X intercept of the regression, as implied by equation [13.21.2.15], is invalid due to the large propagation of measurement errors encountered in extrapolating the 1/T (x-... 

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