At which temperature would a reaction with [tex]\(\Delta H = -92 \text{ kJ/mol}\)[/tex] and [tex]\(\Delta S = -0.199 \text{ kJ/(mol} \cdot \text{K)}\)[/tex] be spontaneous?

A. [tex]\(500 \text{ K}\)[/tex]

B. [tex]\(400 \text{ K}\)[/tex]

C. [tex]\(700 \text{ K}\)[/tex]

D. [tex]\(600 \text{ K}\)[/tex]



Answer :

To determine at which temperature the reaction is spontaneous, we need to calculate the Gibbs free energy change ([tex]\(\Delta G\)[/tex]) for each temperature given in the choices. The Gibbs free energy change can be calculated using the formula:

[tex]\[ \Delta G = \Delta H - T \Delta S \][/tex]

where:
- [tex]\(\Delta H\)[/tex] is the enthalpy change of the reaction.
- [tex]\(T\)[/tex] is the temperature in Kelvin.
- [tex]\(\Delta S\)[/tex] is the entropy change of the reaction.

Given:
- [tex]\(\Delta H = -92 \)[/tex] kJ/mol
- [tex]\(\Delta S = -0.199 \)[/tex] kJ/(mol·K)

We need to evaluate [tex]\(\Delta G\)[/tex] for each of the following temperatures: [tex]\(500 K\)[/tex], [tex]\(400 K\)[/tex], [tex]\(700 K\)[/tex], and [tex]\(600 K\)[/tex].

1. At [tex]\(500 K\)[/tex]:
[tex]\[ \Delta G = -92 - (500 \times -0.199) \][/tex]
[tex]\[ \Delta G = -92 + 99.5 \][/tex]
[tex]\[ \Delta G = 7.5 \text{ kJ/mol} \][/tex]

2. At [tex]\(400 K\)[/tex]:
[tex]\[ \Delta G = -92 - (400 \times -0.199) \][/tex]
[tex]\[ \Delta G = -92 + 79.6 \][/tex]
[tex]\[ \Delta G = -12.4 \text{ kJ/mol} \][/tex]

3. At [tex]\(700 K\)[/tex]:
[tex]\[ \Delta G = -92 - (700 \times -0.199) \][/tex]
[tex]\[ \Delta G = -92 + 139.3 \][/tex]
[tex]\[ \Delta G = 47.3 \text{ kJ/mol} \][/tex]

4. At [tex]\(600 K\)[/tex]:
[tex]\[ \Delta G = -92 - (600 \times -0.199) \][/tex]
[tex]\[ \Delta G = -92 + 119.4 \][/tex]
[tex]\[ \Delta G = 27.4 \text{ kJ/mol} \][/tex]

A reaction is spontaneous when [tex]\(\Delta G\)[/tex] is less than zero (i.e., [tex]\(\Delta G < 0\)[/tex]).

From our calculations:
- At [tex]\(500 K\)[/tex], [tex]\(\Delta G = 7.5\)[/tex] kJ/mol (not spontaneous).
- At [tex]\(400 K\)[/tex], [tex]\(\Delta G = -12.4\)[/tex] kJ/mol (spontaneous).
- At [tex]\(700 K\)[/tex], [tex]\(\Delta G = 47.3\)[/tex] kJ/mol (not spontaneous).
- At [tex]\(600 K\)[/tex], [tex]\(\Delta G = 27.4\)[/tex] kJ/mol (not spontaneous).

Therefore, the reaction is only spontaneous at [tex]\(400 K\)[/tex].

Thus, the correct answer is:
B. [tex]\[400 K\][/tex]