Question

For an isomerization reaction \( A \rightleftharpoons B \), the temperature dependence of the equilibrium constant is given by: \[ \log_e K = 4.0 - \frac{2000}{T} \] {The value of \( \Delta S^\circ \) at Hook is, therefore:}

• A. \( 4R \)
• B. \( 5R \)
• C. \( 400R \)
• D. \( 2000R \)

Hint:

The temperature dependence of the equilibrium constant can be related to entropy change using the van't Hoff equation. The equation provides a way to determine \( \Delta S^\circ \) if the temperature dependence of \( K \) is known.

Answer 1

The Correct Option is A

We are given the equation for the temperature dependence of the equilibrium constant \( K \): \[ \log_e K = 4.0 - \frac{2000}{T} \] This equation is a form of the van't Hoff equation, which relates the change in the equilibrium constant to the change in temperature. The general form of the van't Hoff equation is: \[ \frac{d \ln K}{dT} = \frac{\Delta H^\circ}{RT^2} \] However, we can also use the relation: \[ \frac{d (\log_e K)}{dT} = -\frac{\Delta H^\circ}{2.303RT^2} \] By differentiating \( \log_e K = 4.0 - \frac{2000}{T} \) with respect to \( T \), we obtain: \[ \frac{d (\log_e K)}{dT} = \frac{2000}{T^2} \] Equating this to the expression from the van't Hoff equation: \[ \frac{2000}{T^2} = -\frac{\Delta H^\circ}{2.303R T^2} \] From this, we can solve for \( \Delta H^\circ \): \[ \Delta H^\circ = -2000 \times 2.303R = -4606R \] Now, to find \( \Delta S^\circ \), we use the relation: \[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \] At equilibrium, \( \Delta G^\circ = -RT \ln K \), so: \[ \Delta S^\circ = \frac{\Delta H^\circ}{T} = \frac{2000}{T} \, {which gives} \, \Delta S^\circ = 4R. \] Thus, the value of \( \Delta S^\circ \) is \( 4R \).