1. For spontaneity at constant temperature and pressure, the criterion is:

$\Delta G = \Delta H - T\Delta S$

A process is spontaneous when:

$\Delta G < 0$

2. Given:

$\Delta H = 200\, \text{J mol}^{-1}$ $\Delta S = 40\, \text{J K}^{-1}\text{mol}^{-1}$

So,

$\Delta G = 200 - 40T$

3. For the minimum temperature above which the process becomes spontaneous, set:

$\Delta G = 0$

Thus,

$200 - 40T = 0$

$40T = 200$

$T = \frac{200}{40} = 5\, \text{K}$

4. Therefore, the process will be spontaneous for:

$T > 5\, \text{K}$

So the threshold temperature is $5\,\text{K}$, and among the given options, the correct choice is:

Option C: $5\,\text{K}$.

5. Checking options briefly:

• A: $4\,\text{K}$ — below threshold, not correct
• B: $20\,\text{K}$ — spontaneous above this too, but not the minimum
• C: $5\,\text{K}$ — correct threshold value
• D: $12\,\text{K}$ — not minimum