Given that the equilibrium constant k for the reaction a(g) ⇌ b(g) is 3.2 × 10^{-3} at 0°C and 8.1 × 10^{-3} at 50°C, what is the enthalpy of vaporization (ΔH_vap) for the process? (Assume R = 8.314 J·mol^{-1}·K^{-1})

• A. 8.2 kJ·mol^{-1}
• B. 2.3 kJ·mol^{-1}
• C. 5.1 kJ·mol^{-1}
• D. 4.6 kJ·mol^{-1}

Answer: (D)

Verified step by step guidance

1. Identify that the problem involves finding the enthalpy change (\( \Delta H_{vap} \)) from equilibrium constants at two different temperatures using the Van't Hoff equation.
2. Write down the Van't Hoff equation in its linear form relating equilibrium constants and temperature: \[ \ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \] where \( K_1 \) and \( K_2 \) are the equilibrium constants at temperatures \( T_1 \) and \( T_2 \) respectively, and \( R \) is the gas constant.
3. Convert the given temperatures from Celsius to Kelvin by adding 273.15: \[ T_1 = 0 + 273.15 = 273.15 \text{ K} \] \[ T_2 = 50 + 273.15 = 323.15 \text{ K} \]
4. Calculate the natural logarithm of the ratio of the equilibrium constants: \[ \ln\left(\frac{K_2}{K_1}\right) = \ln\left(\frac{8.1 \times 10^{-3}}{3.2 \times 10^{-3}}\right) \]
5. Rearrange the Van't Hoff equation to solve for \( \Delta H_{vap} \): \[ \Delta H_{vap} = -R \times \frac{\ln\left(\frac{K_2}{K_1}\right)}{\left( \frac{1}{T_2} - \frac{1}{T_1} \right)} \] Substitute the values of \( R \), \( \ln\left(\frac{K_2}{K_1}\right) \), \( T_1 \), and \( T_2 \) into this equation to find \( \Delta H_{vap} \).