Question

The \(\Delta H\) and \(\Delta S\) values of a reaction are \(400\,\mathrm{kJ\,mol^{-1}}\) and \(200\,\mathrm{J\,K^{-1}\,mol^{-1}}\) respectively which are constant over a wide range of temperature. The temperature above which the reaction will be spontaneous is

• A. \(2\,K\)
• B. \(400\,K\)
• C. \(2000\,K\)
• D. \(800\,K\)

Hint:

For reactions with \[ \Delta H\gt 0,\qquad \Delta S\gt 0 \] spontaneity is achieved at high temperatures. Use \[ T=\frac{\Delta H}{\Delta S} \] to find the minimum temperature for spontaneity.

Answer 1

The Correct Option is C

Concept: A reaction becomes spontaneous when \[\begin{aligned} \Delta G \lt 0 \end{aligned}\] Using Gibbs free energy relation, \[\begin{aligned} \Delta G=\Delta H-T\Delta S \end{aligned}\] The limiting temperature is obtained by putting \[\begin{aligned} \Delta G=0 \end{aligned}\]

Step 1: Convert all quantities into consistent units. \[\begin{aligned} \Delta H &= 400\,\mathrm{kJ\,mol^{-1}} =400000\,\mathrm{J\,mol^{-1}} \\ \Delta S &= 200\,\mathrm{J\,K^{-1}\,mol^{-1}} \end{aligned}\]

Step 2: Apply the Gibbs free energy equation. At the threshold temperature, \[\begin{aligned} 0=\Delta H-T\Delta S \end{aligned}\] Therefore, \[\begin{aligned} T=\frac{\Delta H}{\Delta S} \end{aligned}\]

Step 3: Calculate the temperature. \[\begin{aligned} T &=\frac{400000}{200} \\ &=2000\,K \end{aligned}\] Hence the reaction becomes spontaneous above \[\begin{aligned} \boxed{2000\,K} \end{aligned}\] Therefore, option \(\mathbf{(C)}\) is correct.