The change in entropy when the pressure of perfect gas is changed isothermally from `P_1` to `P_2` is

Options

A. `/_\S=nRln(P_1+P_2)`
B. `/_\S=nRln(P_2//P_1)`
C. `/_\S=nRln(P_1//P_2)`
D. `/_\S=nRln((P_1+P_2)/(P_2))`

Answer

To find the change in entropy (ΔS) when the pressure of a perfect gas is changed isothermally from \( P_1 \) to \( P_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Conditions**: The process is isothermal, meaning the temperature (T) remains constant. For a perfect gas, we can use the ideal gas law, which states that \( PV = nRT \).

2. **Formula for Change in Entropy**: The change in entropy for a reversible process can be expressed as:
\[ \Delta S = \frac{dQ}{T} \]
where \( dQ \) is the heat transfer.

3. **Relate Heat Transfer to Work Done**: For an ideal gas undergoing an isothermal process, the heat transfer \( dQ \) can be related to the work done \( dW \):
\[ dQ = dW \]
In terms of pressure and volume, this becomes:
\[ dW = -P dV \]
Therefore, we can write:
\[ dQ = P dV \]

4. **Substituting Ideal Gas Law**: Using the ideal gas law \( P = \frac{nRT}{V} \), we substitute for \( P \):
\[ dQ = \frac{nRT}{V} dV \]

5. **Integrate to Find Change in Entropy**: Now we can substitute \( dQ \) into the entropy formula:
\[ \Delta S = \int_{V_1}^{V_2} \frac{dQ}{T} = \int_{V_1}^{V_2} \frac{nRT}{V} \frac{dV}{T} \]
Since T is constant, we can take it out of the integral:
\[ \Delta S = \frac{nR}{T} \int_{V_1}^{V_2} \frac{dV}{V} \]
The integral \( \int \frac{dV}{V} \) evaluates to \( \ln \frac{V_2}{V_1} \):
\[ \Delta S = \frac{nR}{T} \ln \frac{V_2}{V_1} \]

6. **Relate Volume to Pressure**: From the ideal gas law, we know that \( P_1 V_1 = P_2 V_2 \). Rearranging gives:
\[ \frac{V_2}{V_1} = \frac{P_1}{P_2} \]
Substituting this back into the entropy change equation:
\[ \Delta S = \frac{nR}{T} \ln \left(\frac{P_1}{P_2}\right) \]

7. **Final Result**: Thus, the change in entropy when the pressure of a perfect gas is changed isothermally from \( P_1 \) to \( P_2 \) is:
\[ \Delta S = nR \ln \left(\frac{P_1}{P_2}\right) \]