Which of the following gives entropy change of an ideal gas for isobaric process ?

Question

Accepted Answer

The correct answer for the entropy change of an ideal gas during an isobaric process is given by Option 2: $C_{p} \ln\left(\frac{T_{2}}{T_{1}}\right)$. Let's explore why this is the correct choice and why the other options are incorrect.

### Explanation for Option 2:

1. **Isobaric Process**: An isobaric process is one in which the pressure remains constant. For an ideal gas undergoing an isobaric process, the change in entropy ($\Delta S$) can be calculated using the formula:
   $$
   \Delta S = C_{p} \ln\left(\frac{T_{2}}{T_{1}}\right)
   $$
   where $C_{p}$ is the specific heat capacity at constant pressure, $T_{1}$ is the initial temperature, and $T_{2}$ is the final temperature.

2. **Justification**: This formula arises because, in an isobaric process, the heat added to the system is related to the change in temperature at constant pressure. The specific heat capacity at constant pressure, $C_{p}$, is used because the process is isobaric. The natural logarithm of the ratio of the final to initial temperatures reflects the proportional change in entropy due to temperature change.

### Explanation for Incorrect Options:

1. **Option 1: $R \ln\left(\frac{P_{1}}{P_{2}}\right)$**:
   - This formula is not applicable for an isobaric process because it involves a change in pressure ($P_{1}$ and $P_{2}$). In an isobaric process, the pressure remains constant, so this expression does not describe the entropy change correctly.

2. **Option 3: $C_{V} \ln\left(\frac{T_{2}}{T_{1}}\right)$**:
   - This formula is used for calculating the entropy change in an isochoric process (constant volume), not an isobaric process. In an isobaric process, the volume changes, and the specific heat capacity at constant volume, $C_{V}$, is not applicable.

3. **Option 4: $R \ln\left(\frac{V_{2}}{V_{1}}\right)$**:
   - This expression is related to the entropy change in an isothermal process (constant temperature) for an ideal gas, where the volume changes. It does not apply to an isobaric process where the temperature changes and pressure remains constant.

In conclusion, Option 2 is the correct choice because it accurately describes the entropy change for an ideal gas during an isobaric process, using the specific heat capacity at constant pressure and the change in temperature. The other options do not correctly apply to the conditions of an isobaric process.