An ideal has undergoes a polytropic given by equation `PV^(n)` = constant. If molar heat capacity of gas during this process is arithmetic mean of its molar heat capacity at constant pressure and constant volume then value of n is

Options

A. Zero
B. `-1`
C. `+1`
D. `gamma`

Answer

To solve the problem, we need to find the value of \( n \) for an ideal gas undergoing a polytropic process defined by the equation \( PV^n = \text{constant} \). We are given that the molar heat capacity \( C \) during this process is the arithmetic mean of the molar heat capacities at constant pressure \( C_p \) and constant volume \( C_v \).

### Step-by-Step Solution:

1. **Understanding Molar Heat Capacity**:
The molar heat capacity \( C \) during the polytropic process is given as:
\[ C = \frac{C_p + C_v}{2} \]

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

3. **Setting the Two Expressions Equal**:
Since both expressions represent the molar heat capacity during the process, we can set them equal to each other:
\[ \frac{C_p + C_v}{2} = C_v + \frac{R}{1 - n} \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ \frac{C_p + C_v}{2} - C_v = \frac{R}{1 - n} \]
Simplifying the left side:
\[ \frac{C_p - C_v}{2} = \frac{R}{1 - n} \]

5. **Cross-Multiplying**:
Cross-multiplying gives:
\[ (C_p - C_v)(1 - n) = 2R \]

6. **Solving for \( n \)**:
Expanding the left side:
\[ C_p - C_v - n(C_p - C_v) = 2R \]
Rearranging to isolate \( n \):
\[ -n(C_p - C_v) = 2R - (C_p - C_v) \]
Thus,
\[ n = \frac{C_p - C_v - 2R}{C_p - C_v} \]

7. **Using the Relation Between \( C_p \) and \( C_v \)**:
We know that \( R = C_p - C_v \). Substituting this into the equation:
\[ n = \frac{C_p - C_v - 2(C_p - C_v)}{C_p - C_v} \]
This simplifies to:
\[ n = \frac{-C_p + Cv}{C_p - C_v} = -1 \]

8. **Final Result**:
Therefore, the value of \( n \) is:
\[ n = -1 \]

### Conclusion:
The value of \( n \) for the given polytropic process is \( -1 \).