An ideal gas under goes a quasi static, reversible process in which its molar heat capacity C remains constant. If during this process the relation of pressure P and volume V is given by `PV^n=constant`, then n is given by (Here `C_P and C_V` are molar specific heat at constant pressure and constant volume, respectively):

Options

A. `n=(C-C_(p))/(C-C_(v))`
B. `n=(C_(p)-C)/(C-C_(v))`
C. `n=(C-C_(v))/(C-C_(p))`
D. `n=(C_(p))/(C_(v))`

Answer

To solve the problem, we need to find the value of \( n \) in terms of the molar heat capacities \( C_P \) and \( C_V \) given that the gas follows the relation \( PV^n = \text{constant} \) during a quasi-static, reversible process where the molar heat capacity \( C \) remains constant.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The process described is a polytropic process, where the relationship between pressure \( P \) and volume \( V \) is given by \( PV^n = \text{constant} \).

2. **Molar Heat Capacity Relation**:
For a polytropic process, the molar heat capacity \( C \) can be expressed as:
\[ C = C_V + \frac{R}{1 - n} \]
where \( R \) is the universal gas constant.

3. **Rearranging the Equation**:
We can rearrange the above equation to isolate \( n \):
\[ C - C_V = \frac{R}{1 - n} \]

4. **Cross-Multiplying**:
Cross-multiplying gives:
\[ (C - C_V)(1 - n) = R \]

5. **Expanding the Equation**:
Expanding the left-hand side:
\[ C - C_V - n(C - C_V) = R \]

6. **Isolating \( n \)**:
Rearranging this equation to isolate \( n \):
\[ n(C - C_V) = C - C_V - R \]
\[ n = \frac{C - C_V - R}{C - C_V} \]

7. **Substituting \( R \)**:
We know that \( R = C_P - C_V \). Substituting this into the equation gives:
\[ n = \frac{C - C_V - (C_P - C_V)}{C - C_V} \]
Simplifying this:
\[ n = \frac{C - C_P}{C - C_V} \]

8. **Final Result**:
Thus, the final expression for \( n \) is:
\[ n = \frac{C - C_P}{C - C_V} \]

### Conclusion:
The value of \( n \) in terms of the molar heat capacities \( C_P \) and \( C_V \) is given by:
\[ n = \frac{C - C_P}{C - C_V} \]