Which is the correct expression that relates changes of entropy with the change of pressure for an ideal gas at constant temperature, among the following?

Options

A. `DeltaS = nRT ln P_(2)`
B. `DeltaS =T(P_(2)-P_(1))`
C. `DeltaS = nRln P_(1)/P_(2)`
D. `DeltaS = 2.303 nR ln P_(1)/P_(2)`

Explanation

To solve the problem of determining the correct expression that relates changes of entropy with the change of pressure for an ideal gas at constant temperature, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Entropy Change**:
The general formula for the change in entropy (ΔS) for an ideal gas can be expressed as:
\[ \Delta S = nC_p \ln\left(\frac{T_2}{T_1}\right) + nR \ln\left(\frac{V_2}{V_1}\right) \]
where \( n \) is the number of moles, \( C_p \) is the molar heat capacity at constant pressure, \( R \) is the universal gas constant, \( T_1 \) and \( T_2 \) are the initial and final temperatures, and \( V_1 \) and \( V_2 \) are the initial and final volumes.

2. **Consider Constant Temperature**:
Since we are dealing with a constant temperature process (isothermal), we have \( T_1 = T_2 \). Therefore, the term involving temperature becomes zero:
\[ \Delta S = 0 + nR \ln\left(\frac{V_2}{V_1}\right) = nR \ln\left(\frac{V_2}{V_1}\right) \]

3. **Relate Volume to Pressure**:
Using the ideal gas law, we know that:
\[ PV = nRT \]
Rearranging gives us:
\[ V = \frac{nRT}{P} \]
Therefore, we can express the volumes in terms of pressures:
\[ V_1 = \frac{nRT}{P_1} \quad \text{and} \quad V_2 = \frac{nRT}{P_2} \]

4. **Substituting Volumes into the Entropy Change Formula**:
Now substituting \( V_1 \) and \( V_2 \) into the entropy change expression:
\[ \Delta S = nR \ln\left(\frac{\frac{nRT}{P_2}}{\frac{nRT}{P_1}}\right) = nR \ln\left(\frac{P_1}{P_2}\right) \]

5. **Final Expression**:
Thus, the final expression for the change in entropy in terms of pressure is:
\[ \Delta S = nR \ln\left(\frac{P_1}{P_2}\right) \]

### Conclusion:
The correct expression that relates changes of entropy with the change of pressure for an ideal gas at constant temperature is:
\[ \Delta S = nR \ln\left(\frac{P_1}{P_2}\right) \]