Find the molar heat capacity (in terms of `R`) of a monoatomic ideal gas undergoing the process: `PV^(1//2) = constant`?

Options

A. (a)`(15)/(2)R`
B. (b)`(23)/(2)R`
C. (c)`(7)/(2)R`
D. (d)Zero

Answer

To find the molar heat capacity of a monoatomic ideal gas undergoing the process \( PV^{1/2} = \text{constant} \), we can follow these steps:

### Step 1: Identify the relationship
We start with the given process:
\[ PV^{1/2} = \text{constant} \]
This can be rewritten in the form:
\[ PV^n = \text{constant} \]
where \( n = \frac{1}{2} \).

### Step 2: Use the formula for molar heat capacity
The molar heat capacity \( C \) for a process can be expressed in terms of the heat capacity at constant volume \( C_v \) and the gas constant \( R \) as:
\[ C = C_v + \frac{R}{1 - n} \]
Here, \( n = \frac{1}{2} \).

### Step 3: Substitute the value of \( n \)
Substituting \( n = \frac{1}{2} \) into the formula gives:
\[ C = C_v + \frac{R}{1 - \frac{1}{2}} = C_v + \frac{R}{\frac{1}{2}} = C_v + 2R \]

### Step 4: Determine \( C_v \) for a monoatomic ideal gas
For a monoatomic ideal gas, the molar heat capacity at constant volume \( C_v \) is given by:
\[ C_v = \frac{3}{2} R \]

### Step 5: Substitute \( C_v \) into the equation for \( C \)
Now, substituting \( C_v \) into the equation for \( C \):
\[ C = \frac{3}{2} R + 2R \]

### Step 6: Combine the terms
To combine the terms, we can express \( 2R \) as \( \frac{4}{2} R \):
\[ C = \frac{3}{2} R + \frac{4}{2} R = \frac{7}{2} R \]

### Final Answer
Thus, the molar heat capacity \( C \) of the monoatomic ideal gas undergoing the process \( PV^{1/2} = \text{constant} \) is:
\[ C = \frac{7}{2} R \]