To determine the temperature at which a given reaction becomes spontaneous, we use the Gibbs free energy change equation:
[tex]\[ \Delta G = \Delta H - T\Delta S \][/tex]
For a reaction to be spontaneous, the Gibbs free energy change ([tex]\(\Delta G\)[/tex]) must be negative:
[tex]\[ \Delta G < 0 \][/tex]
Rewriting the above inequality, we get:
[tex]\[ \Delta H - T\Delta S < 0 \][/tex]
Rearranging the terms to solve for the temperature ([tex]\(T\)[/tex]):
[tex]\[ T\Delta S > \Delta H \][/tex]
[tex]\[ T > \frac{\Delta H}{\Delta S} \][/tex]
We are given the values:
- [tex]\(\Delta H = 176 \, \text{kJ/mol}\)[/tex]
- [tex]\(\Delta S = 0.285 \, \text{kJ/(K} \cdot \text{mol)}\)[/tex]
Plugging in these values into our inequality, we get:
[tex]\[ T > \frac{176 \, \text{kJ/mol}}{0.285 \, \text{kJ/(K} \cdot \text{mol)}} \][/tex]
[tex]\[ T > 617.5438596491229 \, \text{K} \][/tex]
Therefore, the reaction will be spontaneous at temperatures greater than [tex]\(617 \, \text{K}\)[/tex].
So the correct answer is:
B. [tex]\(T > 617 \, \text{K}\)[/tex]
