P–V diagram of a monatomic gas is a straight line passing through origin. The molar heat capacity of the gas in the process will be:

Options

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

Explanation

To find the molar heat capacity of a monatomic gas in a process represented by a straight line on a P-V diagram that passes through the origin, we can follow these steps:

### Step 1: Understand the P-V Relationship
Since the P-V diagram is a straight line passing through the origin, we can express the relationship between pressure (P) and volume (V) as:
\[ P \propto V \]
This implies that:
\[ P \cdot V^{-1} = \text{constant} \]
This indicates that the process can be described as a polytropic process.

### Step 2: Identify the Polytropic Index
In a general polytropic process, the relationship can be expressed as:
\[ P \cdot V^n = \text{constant} \]
where \( n \) is the polytropic index. From our earlier relationship, we can see that:
\[ n = -1 \]

### Step 3: Use the Molar Heat Capacity Formula
The molar heat capacity \( C \) for a polytropic process is given by:
\[ C = C_v + \frac{R}{1 - n} \]
where \( C_v \) is the molar specific heat at constant volume, and \( R \) is the gas constant.

### Step 4: Determine \( C_v \) for a Monatomic Gas
For a monatomic gas, the degrees of freedom \( f \) is 3. The molar specific heat at constant volume is given by:
\[ C_v = \frac{f}{2} R = \frac{3}{2} R \]

### Step 5: Substitute Values into the Heat Capacity Formula
Now, substituting \( C_v \) and \( n \) into the heat capacity formula:
\[ C = \frac{3}{2} R + \frac{R}{1 - (-1)} \]
\[ C = \frac{3}{2} R + \frac{R}{2} \]

### Step 6: Simplify the Expression
Now, simplifying the expression:
\[ C = \frac{3}{2} R + \frac{1}{2} R = \frac{4}{2} R = 2R \]

### Conclusion
Thus, the molar heat capacity of the gas in this process is:
\[ C = 2R \]