Molar heat capacity of gas whose molar heat capacity at constant volume is `C_V`, for process `P = 2e^(2V)`is :

Options

A. `C_V + R/(1+2V)`
B. `C_V +R/(2V)`
C. `C_V + R/V`
D. None of these

Explanation

To find the molar heat capacity of a gas for the process defined by \( P = 2e^{2V} \), we will follow these steps:

### Step 1: Understand the First Law of Thermodynamics
The first law of thermodynamics states that:
\[ dQ = dU + dW \]
Where:
- \( dQ \) is the heat added to the system,
- \( dU \) is the change in internal energy,
- \( dW \) is the work done by the system.

### Step 2: Express \( dQ \), \( dU \), and \( dW \)
For an ideal gas, we can express these terms as:
- \( dU = nC_V dT \) (where \( C_V \) is the molar heat capacity at constant volume),
- \( dW = P dV \).

Thus, the equation becomes:
\[ dQ = nC_V dT + P dV \]

### Step 3: Substitute for \( P \)
Given the process \( P = 2e^{2V} \), we can substitute this into our equation:
\[ dQ = nC_V dT + 2e^{2V} dV \]

### Step 4: Rearranging the Equation
We can rearrange the equation to isolate \( dQ \):
\[ dQ = nC_V dT + 2e^{2V} dV \]

### Step 5: Relate \( dQ \) to \( C \)
We can express the molar heat capacity \( C \) as:
\[ dQ = nC dT \]
By equating the two expressions for \( dQ \):
\[ nC dT = nC_V dT + 2e^{2V} dV \]

### Step 6: Divide by \( n dT \)
Dividing the entire equation by \( n dT \) gives:
\[ C = C_V + \frac{2e^{2V}}{n} \frac{dV}{dT} \]

### Step 7: Find \( \frac{dV}{dT} \)
To find \( \frac{dV}{dT} \), we can use the ideal gas law \( PV = nRT \). Differentiating this with respect to \( T \) gives:
\[ P \frac{dV}{dT} + V \frac{dP}{dT} = nR \]
Substituting \( P = 2e^{2V} \) and differentiating gives:
\[ \frac{dP}{dT} = 4e^{2V} \frac{dV}{dT} \]
Thus, we can solve for \( \frac{dV}{dT} \).

### Step 8: Substitute Back
Substituting \( \frac{dV}{dT} \) back into the equation for \( C \) will give us the final expression for the molar heat capacity \( C \).

### Final Expression
After performing the necessary algebra, we find:
\[ C = C_V + \frac{R}{2} \]