For the reaction CO(g)+3 H2(g) ⇌CH4(g)+H2 O(g) K =2.15...

For the reaction $$\begin{aligned}\mathrm{CO}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{CH}_{4}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \\K &=2.15 \times 10^{11} \mathrm{at} 200^{\circ} \mathrm{C} \\K &=4.56 \times 10^{8} \mathrm{at} 260^{\circ} \mathrm{C}\end{aligned}$$determine $\Delta_{\mathrm{r}} H^{\circ}$ by using the van't Hoff equation (13.25) and by using tabulated data in Appendix D. Compare the two results, and comment on how good the assumption is that $\Delta_{\mathrm{r}} H^{\circ}$ is essentially independent of temperature in this case.

For the reaction
$$\begin{aligned}\mathrm{CO}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{CH}_{4}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \\K &=2.15 \times 10^{11} \mathrm{at} 200^{\circ} \mathrm{C} \\K &=4.56 \times 10^{8} \mathrm{at} 260^{\circ} \mathrm{C}\end{aligned}$$determine $\Delta_{\mathrm{r}} H^{\circ}$ by using the van't Hoff equation (13.25) and by using tabulated data in Appendix D. Compare the two results, and comment on how good the assumption is that $\Delta_{\mathrm{r}} H^{\circ}$ is essentially independent of temperature in this case.

Key Concepts

van't Hoff Equation

This equation relates the changes in the equilibrium constant of a reaction to the temperature and the standard reaction enthalpy. By taking the derivative of the natural logarithm of the equilibrium constant with respect to temperature, one can determine the overall enthalpy change of the reaction. It serves as a bridge between thermodynamics and equilibrium, allowing predictions about how the equilibrium composition changes with temperature.

Standard Reaction Enthalpy

Standard reaction enthalpy (?rH°) represents the heat absorbed or released during a chemical reaction under standard conditions. It can be determined either from experimental data and temperature-dependent equilibrium constants using the van't Hoff equation or calculated from tabulated thermodynamic data, such as the heats of formation for the reactants and products.

Temperature Dependence of Equilibrium Constant

The equilibrium constant of a chemical reaction is sensitive to temperature changes. This temperature dependence is a direct consequence of the reaction’s enthalpy change. Analyzing how the equilibrium constant varies with temperature provides insight into the endothermic or exothermic nature of the reaction, providing a thermodynamic perspective on chemical equilibria.

Assumption of Temperature Independence of ?rH°

In many analyses, ?rH° is assumed to remain constant over a limited temperature range to simplify calculations using the van't Hoff equation. This assumption is generally acceptable when the temperature variation is not large; however, any significant deviation in heat capacity can lead to inaccuracies. Comparing results derived from different methods helps assess the validity of this approximation.

Use of Tabulated Thermodynamic Data

Thermodynamic tables provide standard enthalpies of formation and other related data for various substances. By applying Hess’s law, one can calculate the standard reaction enthalpy by taking the weighted sum of the formation enthalpies of the products and subtracting those of the reactants. This method serves as an independent check on values obtained from experimental equilibrium measurements.

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