One mole of an ideal monatomic gas undergoes a process described by the equation `PV^(3)`= constant. The heat capacity of the gas during this process is

Options

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

Explanation

To find the heat capacity of one mole of an ideal monatomic gas undergoing a process described by the equation \( PV^3 = \text{constant} \), we can follow these steps:

### Step 1: Identify the process and its characteristics
The given process is defined by the equation \( PV^3 = \text{constant} \). This indicates that the gas is undergoing a polytropic process where the relationship between pressure (P) and volume (V) is not linear.

### Step 2: Determine the value of \( X \)
In the context of heat capacities, we can express the heat capacity \( C \) during a process as:
\[ C = C_v + \frac{R}{1 - X} \]
where \( C_v \) is the molar heat capacity at constant volume, \( R \) is the universal gas constant, and \( X \) is the exponent in the polytropic process equation. Here, \( X = 3 \).

### Step 3: Calculate \( C_v \) for a monatomic ideal gas
For a monatomic ideal gas, the degrees of freedom \( F \) is 3. Therefore, the molar heat capacity at constant volume \( C_v \) is given by:
\[ C_v = \frac{F}{2} R = \frac{3}{2} R \]

### Step 4: Substitute values into the heat capacity equation
Now, substituting \( C_v \) and \( X \) into the heat capacity equation:
\[ C = C_v + \frac{R}{1 - X} = \frac{3}{2} R + \frac{R}{1 - 3} \]
\[ C = \frac{3}{2} R + \frac{R}{-2} \]

### Step 5: Simplify the equation
Now, we simplify the equation:
\[ C = \frac{3}{2} R - \frac{1}{2} R \]
\[ C = \frac{3R - R}{2} = \frac{2R}{2} = R \]

### Conclusion
Thus, the heat capacity \( C \) of the gas during this process is:
\[ C = R \]