One mole of an ideal gas undergoes a process whose molar heat capacity is 4R and in which work done by gas for small change in temperature is given by the relation `dW=2RdT`, then find the degree of freedom of gas

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the problem, we need to find the degree of freedom of the gas given the molar heat capacity and the work done during a small change in temperature. Let's break down the solution step by step.

### Step 1: Understand the given information
We have:
- Molar heat capacity \( C = 4R \)
- Work done by the gas for a small change in temperature \( dW = 2RdT \)
- Number of moles \( n = 1 \)

### Step 2: Use the first law of thermodynamics
The first law of thermodynamics states:
\[ dQ = dU + dW \]
where \( dQ \) is the heat added to the system, \( dU \) is the change in internal energy, and \( dW \) is the work done by the system.

### Step 3: Express \( dU \) in terms of \( C_v \)
For one mole of gas, the change in internal energy can be expressed as:
\[ dU = nC_v dT \]
Since \( n = 1 \), we have:
\[ dU = C_v dT \]

### Step 4: Substitute \( dQ \) and \( dW \) into the first law
From the given information, we can express \( dQ \) as:
\[ dQ = C dT = 4R dT \]
And from the work done:
\[ dW = 2R dT \]

Substituting these into the first law equation:
\[ 4R dT = C_v dT + 2R dT \]

### Step 5: Simplify the equation
Now we can simplify the equation:
\[ 4R dT = C_v dT + 2R dT \]
Subtract \( 2R dT \) from both sides:
\[ (4R - 2R) dT = C_v dT \]
This simplifies to:
\[ 2R dT = C_v dT \]

### Step 6: Solve for \( C_v \)
Since \( dT \) is not zero, we can divide both sides by \( dT \):
\[ C_v = 2R \]

### Step 7: Relate \( C_v \) to degrees of freedom
The molar heat capacity at constant volume \( C_v \) is related to the degrees of freedom \( f \) of the gas by the equation:
\[ C_v = \frac{f}{2} R \]
Substituting \( C_v = 2R \) into this equation gives:
\[ 2R = \frac{f}{2} R \]

### Step 8: Solve for \( f \)
Dividing both sides by \( R \) (assuming \( R \neq 0 \)):
\[ 2 = \frac{f}{2} \]
Multiplying both sides by 2:
\[ f = 4 \]

### Conclusion
The degree of freedom of the gas is \( f = 4 \).