The relationship between the temperature of a reaction, its standard enthalpy change, and the equilibrium constant at that temperature can be expressed as the following linear equation: $$ \ln K=\frac{-\Delta H^{\circ}}{R T}+\text { constant } $$ (a) Explain how this equation can be used to determine \(\Delta H^{\circ}\) experimentally from the equilibrium constants at several different temperatures. (b) Derive the preceding equation using relationships given in this chapter. To what is the constant equal?

To determine ΔH° experimentally, we need to have equilibrium constants (K) at various temperatures (T). According to the given equation: \[ \ln K=\frac{-\Delta H^{\circ}}{R T}+\text { constant } \] This is a linear equation where the slope equals \(\frac{-\Delta H^{\circ}}{R}\) and the intercept is equal to the constant. By measuring K values at different temperatures (T) and making a plot of \(\ln K\) against \(\frac{1}{T}\), we can find the slope of the line corresponding to this dataset. Once we have the slope, we can calculate \(\Delta H^{\circ}\) using the formula: \[ \Delta H^{\circ} = -R \times (\text{slope}) \]

To derive the equation, we use the van 't Hoff equation: \[ \frac{\mathrm{d}(\ln K)}{\mathrm{d}T} = \frac{\Delta H^{\circ}}{RT^2} \] Now, we'll integrate this equation. Let's assume that the constant of integration is "C". Integrating both sides with respect to T, we get: \[ \int\frac{\mathrm{d}(\ln K)}{\mathrm{d}T} \mathrm{d}T = \int\frac{\Delta H^{\circ}}{RT^2} \mathrm{d}T \] \[ \ln K = -\frac{\Delta H^{\circ}}{R} \int\frac{1}{T^2} \mathrm{d}T + C \] Now, integrating the left side: \[ \int\frac{1}{T^2} \mathrm{d}T = \int T^{-2} \mathrm{d}T = -\frac{1}{T} \] Hence, the equation becomes: \[ \ln K = -\frac{\Delta H^{\circ}}{R}(-\frac{1}{T}) + C \] Simplifying the equation, we get: \[ \ln K = \frac{\Delta H^{\circ}}{RT} + C \] Comparing to the equation given in the exercise: \[ \ln K = \frac{-\Delta H^{\circ}}{R T}+\text { constant } \] We see that the constant C corresponds to the "constant" term in the given equation.