The molar specific heat of the process ` V alpha T^(4)` for `CH_(4)` gas at room temperature is :-

Options

A. `4R`
B. `7R`
C. `3R`
D. `8R`

Explanation

To find the molar specific heat of the process where the volume \( V \) is proportional to \( T^4 \) for methane gas (\( CH_4 \)), we can follow these steps:

### Step 1: Understand the relationship between \( V \) and \( T \)
Given that \( V \propto T^4 \), we can express this relationship mathematically as:
\[ V = k T^4 \]
where \( k \) is a constant.

### Step 2: Use the Ideal Gas Law
From the Ideal Gas Law, we know:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature.

### Step 3: Substitute \( V \) into the Ideal Gas Law
Substituting our expression for \( V \) into the Ideal Gas Law gives:
\[ P(k T^4) = nRT \]
This simplifies to:
\[ P = \frac{nR}{k} T^{-3} \]

### Step 4: Establish the relationship between \( P \) and \( V \)
Rearranging the equation gives us:
\[ PV^{-3/4} = \text{constant} \]
This indicates that the process can be compared to an adiabatic process, where:
\[ PV^\gamma = \text{constant} \]
Here, we can identify \( \gamma = \frac{3}{4} \).

### Step 5: Find the specific heat capacity
The specific heat capacity \( C \) for the process can be calculated using the formula:
\[ C = C_v + \frac{R}{1 - \gamma} \]
For methane (\( CH_4 \)), the specific heat capacity at constant volume \( C_v \) is typically \( \frac{3R}{2} \) (for a diatomic gas, it is usually \( \frac{5R}{2} \), but methane can be approximated as a simple gas).

### Step 6: Substitute \( \gamma \) into the specific heat capacity formula
Substituting \( C_v \) and \( \gamma \) into the equation:
\[ C = \frac{3R}{2} + \frac{R}{1 - \frac{3}{4}} = \frac{3R}{2} + \frac{R}{\frac{1}{4}} = \frac{3R}{2} + 4R \]
Combining these gives:
\[ C = \frac{3R}{2} + \frac{8R}{2} = \frac{11R}{2} \]

### Conclusion
Thus, the molar specific heat capacity of the process \( V \propto T^4 \) for methane gas at room temperature is:
\[ C = 7R \]