Question:
\text{For } A \rightarrow B, \Delta H = 4 \text{ kcal mol}^{-1}, \Delta S = 10 \text{ cal mol}^{-1} \text{ K}^{-1}, \text{ the reaction is spontaneous when the temperature is:}
\text{1. 400 K}
\text{2. 300 K}
\text{3. 500 K}
\text{4. None of the above}
Solution:
$\text{Hint: The Gibbs Free Energy is simply a method of telling whether a chemical process will take place spontaneously or non-spontaneously.}$
$\text{To determine the temperature at which the reaction becomes spontaneous, we need to use the Gibbs free energy equation:}$
$\Delta\text{G} = \Delta\text{H} - \text{T}\Delta\text{S}$
$\text{For a reaction to be spontaneous, }\Delta\text{G must be negative:}$
$\Delta\text{G} < 0$
$\text{Given:}$
$\Delta\text{H} = 4 \text{ kcal mol}^{-1} = 4000 \text{ cal mol}^{-1}$ $\text{(converting to the same units as }\Delta\text{S)}$
$\Delta\text{S} = 10 \text{ cal mol}^{-1} \text{ K}^{-1}$
$\text{Step 1: Set up the inequality for spontaneity}$
$\Delta\text{H} - \text{T}\Delta\text{S} < 0$
$\text{Step 2: Solve for the temperature T}$
$\Delta\text{H} < \text{T}\Delta\text{S}$
$\frac{\Delta\text{H}}{\Delta\text{S}} < \text{T}$
$\frac{4000 \text{ cal mol}^{-1}}{10 \text{ cal mol}^{-1} \text{ K}^{-1}} < \text{T}$
$400 \text{ K} < \text{T}$
$\text{This means the reaction will be spontaneous at temperatures above 400 K.}$
$\text{At exactly 400 K, }\Delta\text{G} = 0\text{, which means the reaction is at equilibrium.}$
$\text{At temperatures below 400 K, }\Delta\text{G} > 0\text{, so the reaction is non-spontaneous.}$
$\text{At temperatures above 400 K, }\Delta\text{G} < 0\text{, so the reaction is spontaneous.}$
$\text{Looking at the given options:}$
$\text{- 400 K: }\Delta\text{G} = 0\text{ (equilibrium, not spontaneous)}$
$\text{- 300 K: }\Delta\text{G} > 0\text{ (non-spontaneous)}$
$\text{- 500 K: }\Delta\text{G} < 0\text{ (spontaneous)}$
$\text{Therefore, the reaction is spontaneous at 500 K, making option 3 the correct answer.}$