To determine the temperature at which the reaction [tex]\( I_2(s) \rightleftharpoons I_2(g) \)[/tex] is at equilibrium, we start with the concept of Gibbs free energy [tex]\(\Delta G\)[/tex]. At equilibrium, [tex]\(\Delta G\)[/tex] is zero. The relationship between [tex]\(\Delta G\)[/tex], [tex]\(\Delta H\)[/tex], and [tex]\(\Delta S\)[/tex] is given by the equation:
[tex]\[
\Delta G = \Delta H - T\Delta S
\][/tex]
At equilibrium, we set [tex]\(\Delta G\)[/tex] to zero:
[tex]\[
0 = \Delta H - T\Delta S
\][/tex]
We need to solve this equation for [tex]\( T \)[/tex]:
[tex]\[
T = \frac{\Delta H}{\Delta S}
\][/tex]
We are given:
[tex]\[
\Delta H = 62.4 \text{ kJ/mol}
\][/tex]
[tex]\[
\Delta S = 0.145 \text{ kJ/(mol*K)}
\][/tex]
Now we plug these values into the equation:
[tex]\[
T = \frac{62.4 \text{ kJ/mol}}{0.145 \text{ kJ/(mol*K)}}
\][/tex]
Performing the division gives us:
[tex]\[
T = 430.3448275862069 \text{ K}
\][/tex]
Upon examining the options provided:
A. [tex]\(0.002 \text{ K}\)[/tex]
B. [tex]\(62 \text{ K}\)[/tex]
C. [tex]\(157 \text{ K}\)[/tex]
D. [tex]\(430 \text{ K}\)[/tex]
The closest value to our calculated temperature is [tex]\(430 \text{ K}\)[/tex]. Therefore, the correct answer is:
D. [tex]\(430 \text{ K}\)[/tex]
