To determine the temperature at which the reaction [tex]\( HCl(g) + Ni_3(g) \rightarrow NH_4Cl(s) \)[/tex] is feasible, we need to apply the Gibbs free energy equation:
[tex]\[ \Delta G = \Delta H - T \Delta S \][/tex]
For the reaction to be feasible (spontaneous), the Gibbs free energy change ([tex]\(\Delta G\)[/tex]) should be less than or equal to zero:
[tex]\[ \Delta G \leq 0 \][/tex]
Setting [tex]\(\Delta G\)[/tex] to zero for the threshold temperature:
[tex]\[ 0 = \Delta H - T \Delta S \][/tex]
We can rearrange this equation to solve for the temperature [tex]\( T \)[/tex] at which the reaction becomes feasible:
[tex]\[ T = \frac{\Delta H}{\Delta S} \][/tex]
Given:
- Enthalpy change ([tex]\(\Delta H\)[/tex]) = +946 kJ/mol
- Entropy change ([tex]\(\Delta S\)[/tex]) = +127 J/K·mol
Since the enthalpy change is given in kilojoules and the entropy change is in joules, we need to convert [tex]\(\Delta H\)[/tex] from kJ to J.
[tex]\[ \Delta H = 946 \text{ kJ/mol} \times 1000 \text{ J/kJ} = 946000 \text{ J/mol} \][/tex]
Now, substitute [tex]\(\Delta H\)[/tex] and [tex]\(\Delta S\)[/tex] into the equation:
[tex]\[ T = \frac{946000 \text{ J/mol}}{127 \text{ J/K·mol}} \][/tex]
[tex]\[ T = 7448.818897637795 \text{ K} \][/tex]
Therefore, the temperature at which the reaction becomes feasible is approximately [tex]\( 7448.8 \text{ K} \)[/tex], making option B the correct answer.
