For a certain process, pressure of diatomic gas varies according to the relation `P = aV^2`, where a is constant. What is the molar heat capacity of the gas for this process ?

Options

A. `(17R)/6`
B. `(6R)/(17)`
C. `(13R)/6`
D. `(16R)/7`

Answer

To find the molar heat capacity of a diatomic gas for the given process where the pressure varies as \( P = aV^2 \), we can follow these steps:

### Step 1: Identify the type of process
The given relation \( P = aV^2 \) indicates that the process can be compared to a polytropic process, which is generally expressed as \( PV^m = \text{constant} \).

### Step 2: Determine the value of \( m \)
From the equation \( P = aV^2 \), we can rewrite it as:
\[ PV^{-2} = a \]
This shows that \( m = -2 \).

### Step 3: Use the formula for molar heat capacity in a polytropic process
The molar heat capacity \( C \) for a polytropic process is given by the formula:
\[ C = C_V + \frac{R}{1 - m} \]
where \( C_V \) is the molar heat capacity at constant volume, \( R \) is the universal gas constant, and \( m \) is the polytropic index.

### Step 4: Calculate \( C_V \) for a diatomic gas
For a diatomic gas, the degrees of freedom are 5 (3 translational and 2 rotational). Therefore, the molar heat capacity at constant volume is:
\[ C_V = \frac{5R}{2} \]

### Step 5: Substitute values into the heat capacity formula
Now substituting \( C_V \) and \( m \) into the heat capacity formula:
\[ C = \frac{5R}{2} + \frac{R}{1 - (-2)} \]
This simplifies to:
\[ C = \frac{5R}{2} + \frac{R}{1 + 2} = \frac{5R}{2} + \frac{R}{3} \]

### Step 6: Find a common denominator and combine the terms
To combine these fractions, we need a common denominator, which is 6:
\[ C = \frac{15R}{6} + \frac{2R}{6} = \frac{17R}{6} \]

### Conclusion
Thus, the molar heat capacity of the gas for this process is:
\[ C = \frac{17R}{6} \]