GATE 2025 PH – Question 30
One mole of an ideal monatomic gas at absolute temperature $T$ undergoes free expansion to double its original volume, so that the entropy change is $\Delta S_1$. An identical amount of the same gas at absolute temperature $2T$ undergoes isothermal expansion to double its original volume, so that the entropy change is $\Delta S_2$. The value of $\Delta S_1/\Delta S_2$ (in integer) is ____
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Correct answer: 1
Explanation
Entropy is a state function, so the entropy change depends only on the initial and final states.
For an **ideal gas** the entropy change when the volume changes from $V_i$ to $V_f$ at constant temperature is
$$\Delta S=nR\ln\frac{V_f}{V_i}.$$
**Free expansion** (to double the volume) of an ideal gas does not change the temperature, so the initial and final states are the same as for an isothermal expansion to double the volume:
$$\Delta S_1=R\ln2 .$$
**Isothermal expansion** at $2T$, with volume doubled:
$$\Delta S_2=R\ln2 .$$
The temperature does not matter, so
$$\frac{\Delta S_1}{\Delta S_2}=\mathbf{1}.$$