Evaluation of Equilibrium Constants at Different Temperatures

Chemists have determined equilibrium constants for thousands of reactions. It would be an impossibly huge task to catalog such constants for each reaction at every temperature of interest. Fortunately, there is no need to do this. If we determine the equilibrium constant,, for a reaction at one temperature, Ti, and also its we can then estimate the equilibrium constant at a second temperature, T2, using the van t Hoff equation. 

if we know AH° for a reaction and if at a given temperature (say 298 K), we can use the van t Hoff equation to calculate the value of K at any other temperature. 

We found in Example 17-18 that Xp= 4.4 X 10 at25°C(298 K) for the following reac- 

We are given ifp at one temperature, 25°C, and the value of AH°. We are given the second temperature, 2400. K. These data allow us to evaluate the right side of the van t Hoff equation, which gives us In Kj-JK-j-). Because we know, we can find the value for.  

The positive value of AH ° tells us that the reaction is endothermic.Thus, it is reasonable that the equilibrium constant is greater (2.2 x 10 compared to 4.4 X 10 ) at the higher temperature of 2400. K. 

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N-methy lacetamide is known to exist almost exclusively in the irons form whereas N-phenylurethane is found to be 95 % cis and 5 % irons (Russell and Thompson, 1956). Bhaskar and Rao (1967) studied the dimerization equilibria of N-methylaceta-mide and A -phenylurethane by measuring the concentration dependence of the first N—H overtone band. The procedure for determining the dimerization equilibrium constant was similar to that of Liddel and Becker (1957). The enthalpy of dimerization was evaluated from values of at different temperatures. The details of the procedure and uncertainties in the determination of Kj have been described in the paper by Singh and Rao (1967). The values of (liters/mole) at 26 C were 5.4 for N-methylacetamide in CCI4. and 1.5 for N-phenylurethane. The values of —AH were 4.7 and <3.0 kcal/mole, respectively. The values of and — A// for each of these compounds with various bases, e.g., pyridine and benzophenone, were also determined by a method similar to Becker s (1961). The and A// values did not show a clear-cut dependence on the configuration of the N—H bond of the secondary amide. 

At each temperature, the equilibrium concentration of the product was obtained as the difference between the initial and the equilibrium concentrations of phenol, respectively, [PhOH]0 and [PhOH], The latter was determined from the maximum absorbance of the free O-H stretching band. A similar method was used to evaluate the equilibrium concentration of the acceptor acetonitrile, [CH3CN], which is given by the difference between the initial concentration of this acceptor, [CH3CN]0, and the concentration of the product, obtained before. In summary, the equilibrium constant can be represented by equation 14.7 ... 

Instead, a wide variety of spectroscopic and electrochemical titration methods are often employed to determine the equilibrium constants for a molecular recognition process at several different temperatures, which are then analyzed by the van t Hoff equation to give the thermodynamic parameters for the process. However, there is a critical tradeoff between the accuracy of the value obtained and the convenience of the measurement since the thermodynamic parameters, evaluated through the van t Hoff treatment, do not take into account the possible temperature dependence of the enthalpy change, i.e. heat capacity, and are less accurate in principle. In fact, it has been demonstrated with some supramolecular systems that the van t Hoff treatment leads to a curved plot and therefore the thermodynamic parameters deviated considerably from those determined by calorimetry.3132 Hence one should be cautious in handling thermodynamic parameters determined by spectroscopic titration and particularly in comparing the values for distinct systems determined by different methods. 

An adsorption kinetic model was developed to evaluate the adsorption rates of five pure gases (Nj, O2, Ar, CO, and CH4) on a Takeda-3A CMS over a wide range of pressures up to ISatm. The kinetic characteristics of adsorption on the CMS were studied by using the adsorption equilibrium of five pure gases measured at three different temperatures and their physical properties. Since the diffiisional time constants of all the components showed much stronger dependence of pressure than those expected by the traditional Darken relation, a structural diffusion model was applied to predict the strong pressure dependence. The proposed model successfully predicted the dif ional time constant up to high pressure on the CMS. 

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