Helium gas is subjected to a polytropic process in which the heat supplied to the gas is four times the work done by it. The molar heat capacity of the gas for the process is: (R is universal gas constant)

Options

A. `R//2`
B. `R`
C. `2//R`
D. `3//R`

Explanation

To find the molar heat capacity of helium gas subjected to a polytropic process where the heat supplied is four times the work done, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the First Law of Thermodynamics**:
The first law states that the heat added to the system (ΔQ) is equal to the change in internal energy (ΔU) plus the work done by the system (W). Mathematically, this is expressed as:
\[ \Delta Q = \Delta U + W \]

2. **Set Up the Given Relationship**:
According to the problem, the heat supplied to the gas is four times the work done:
\[ \Delta Q = 4W \]

3. **Substitute into the First Law**:
Substitute ΔQ from the previous step into the first law equation:
\[ 4W = \Delta U + W \]

4. **Rearrange the Equation**:
Rearranging gives:
\[ \Delta U = 4W - W = 3W \]

5. **Relate Heat Capacity to Internal Energy**:
The molar heat capacity (C) for a polytropic process can be defined as:
\[ \Delta Q = nC\Delta T \]
where \( n \) is the number of moles and \( \Delta T \) is the change in temperature.

6. **Express ΔU in Terms of Molar Heat Capacity**:
The change in internal energy for an ideal gas can be expressed as:
\[ \Delta U = nC_v\Delta T \]
where \( C_v \) is the molar heat capacity at constant volume.

7. **Combine the Equations**:
Now we can equate the expressions for ΔQ:
\[ nC\Delta T = 4W \]
and for ΔU:
\[ \Delta U = 3W = nC_v\Delta T \]

8. **Express Work in Terms of ΔT**:
From ΔU = 3W, we can express W as:
\[ W = \frac{\Delta U}{3} = \frac{nC_v\Delta T}{3} \]

9. **Substitute W into the Heat Equation**:
Substitute W back into the heat equation:
\[ nC\Delta T = 4\left(\frac{nC_v\Delta T}{3}\right) \]
Simplifying gives:
\[ C = \frac{4C_v}{3} \]

10. **Calculate \( C_v \) for Helium**:
Helium is a monatomic gas, and its molar heat capacity at constant volume is given by:
\[ C_v = \frac{3}{2}R \]

11. **Substitute \( C_v \) into the Heat Capacity Equation**:
Now substitute \( C_v \) into the equation for \( C \):
\[ C = \frac{4}{3} \left(\frac{3}{2}R\right) = 2R \]

### Final Answer:
The molar heat capacity of helium gas for the polytropic process is:
\[ C = 2R \]