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 the ratio `(C_(P))/(C_(V))` is

Options

A. `7//5`
B. `5//3`
C. `3//2`
D. `2`

Explanation

To solve the problem, we need to find the ratio \( \frac{C_P}{C_V} \) for an ideal gas undergoing a specific process. We are given that the molar heat capacity \( C \) for the process is \( 4R \) and the work done by the gas for a small change in temperature is given by \( dW = 2R dT \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Molar heat capacity for the process: \( C = 4R \)
- Work done by the gas: \( dW = 2R dT \)
- Number of moles \( n = 1 \) (since it is given as one mole of gas).

2. **Apply the First Law of Thermodynamics**:
The first law 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.

3. **Express \( dQ \) in Terms of Molar Heat Capacity**:
Since we have one mole of gas, we can express \( dQ \) as:
\[ dQ = nC dT = 1 \cdot C dT = C dT \]
Substituting the value of \( C \):
\[ dQ = 4R dT \]

4. **Substitute \( dW \) into the First Law Equation**:
Now substituting \( dQ \) and \( dW \) into the first law equation:
\[ 4R dT = dU + 2R dT \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ dU = 4R dT - 2R dT = 2R dT \]

6. **Relate Change in Internal Energy to Heat Capacities**:
The change in internal energy \( dU \) can also be expressed in terms of \( C_V \):
\[ dU = nC_V dT = 1 \cdot C_V dT = C_V dT \]
Thus, we have:
\[ C_V dT = 2R dT \]
This implies:
\[ C_V = 2R \]

7. **Calculate \( C_P \)**:
We know the relation between \( C_P \) and \( C_V \) for an ideal gas:
\[ C_P - C_V = R \]
Substituting \( C_V = 2R \):
\[ C_P - 2R = R \]
Therefore:
\[ C_P = 3R \]

8. **Find the Ratio \( \frac{C_P}{C_V} \)**:
Now we can find the ratio:
\[ \frac{C_P}{C_V} = \frac{3R}{2R} = \frac{3}{2} \]

### Final Answer:
The ratio \( \frac{C_P}{C_V} \) is \( \frac{3}{2} \).