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. `(3)/(2)R`
B. `(5)/(2) R`
C. `2R`
D. `R`

Answer

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 type
The equation \( PV^n = \text{constant} \) indicates that this is a polytropic process. Here, \( n = 3 \).

### Step 2: Recall the formula for heat capacity in a polytropic process
The heat capacity \( C \) for a polytropic process is given by the formula:
\[ C = C_v + \frac{R}{1 - n} \]
where:
- \( C_v \) is the molar heat capacity at constant volume,
- \( R \) is the universal gas constant,
- \( n \) is the polytropic index.

### Step 3: Determine \( C_v \) for a monatomic gas
For a monatomic ideal gas, the molar heat capacity at constant volume \( C_v \) is:
\[ C_v = \frac{3}{2} R \]

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

### Step 5: Simplify the expression
Calculate \( \frac{R}{1 - 3} \):
\[ \frac{R}{1 - 3} = \frac{R}{-2} = -\frac{R}{2} \]
Now substitute this back into the equation for \( C \):
\[ C = \frac{3}{2} R - \frac{R}{2} \]

### Step 6: Combine the terms
Combine the terms:
\[ C = \frac{3}{2} R - \frac{1}{2} R = \frac{2}{2} R = R \]

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