The specific heat of a gas in a polytropic process is given by-

Options

A. `(R )/(gamma - 1) + (R )/(N - 1)`
B. `(R )/(1 - gamma) + (R )/(1 - N)`
C. `(R )/(gamma - 1) - (R )/(N - 1)`
D. `(R )/(1 - gamma) - (R )/(1 - N)`

Answer

To find the specific heat of a gas in a polytropic process, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Polytropic Process**:
The polytropic process for a gas can be represented by the equation \( pV^n = \text{constant} \), where \( n \) is the polytropic index.

2. **Relate Specific Heats**:
The specific heat \( C \) in a polytropic process can be expressed in terms of the specific heats at constant pressure \( C_p \) and constant volume \( C_v \):
\[ n = \frac{C - C_p}{C - C_v} \]

3. **Use Known Formulas**:
We need to use the following known relationships:
- \( C_v = \frac{R}{\gamma - 1} \)
- \( C_p - C_v = R \)

4. **Rearranging the Equation**:
From the equation \( n = \frac{C - C_p}{C - C_v} \), we can cross-multiply to get:
\[ n(C - C_v) = C - C_p \]

5. **Isolate \( C \)**:
Rearranging gives us:
\[ nC - nC_v = C - C_p \]
This can be rearranged to:
\[ nC - C = nC_v - C_p \]
Factoring out \( C \) from the left side:
\[ C(n - 1) = nC_v - C_p \]

6. **Substituting \( C_p \) and \( C_v \)**:
We know that:
- \( C_p = C_v + R \)
Substituting this into the equation gives:
\[ C(n - 1) = nC_v - (C_v + R) \]
Simplifying this:
\[ C(n - 1) = (n - 1)C_v - R \]

7. **Expressing \( C_v \)**:
Substitute \( C_v = \frac{R}{\gamma - 1} \):
\[ C(n - 1) = (n - 1)\frac{R}{\gamma - 1} - R \]

8. **Factor Out \( R \)**:
\[ C(n - 1) = \frac{(n - 1)R}{\gamma - 1} - R \]
\[ C(n - 1) = R\left(\frac{(n - 1)}{\gamma - 1} - 1\right) \]

9. **Solve for \( C \)**:
Now, divide both sides by \( (n - 1) \):
\[ C = \frac{R\left(\frac{(n - 1)}{\gamma - 1} - 1\right)}{(n - 1)} \]
Simplifying gives:
\[ C = R\left(\frac{1}{\gamma - 1} - \frac{1}{n - 1}\right) \]

10. **Final Expression**:
Rearranging gives the final expression for the specific heat \( C \):
\[ C = R\left(\frac{n - \gamma}{(n - 1)(\gamma - 1)}\right) \]