Find equation of process for which heat capacity is `C = 7/2 R` for a mono-atomic gas.

Options

A. (a)PV = constant
B. (b)`PV^2`= constant
C. (c)`P^2 V`=constant
D. (d)None of these

Explanation

To find the equation of the process for which the heat capacity \( C = \frac{7}{2} R \) for a monoatomic gas, we can follow these steps:

### Step 1: Understand the relationship between heat capacity and polytropic processes
For a polytropic process, the heat capacity \( C \) can be expressed as:
\[ C = C_V + \frac{R}{1 - n} \]
where \( C_V \) is the molar heat capacity at constant volume, \( R \) is the universal gas constant, and \( n \) is the polytropic index.

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

### Step 3: Substitute the values into the equation
We are given that:
\[ C = \frac{7}{2} R \]
Substituting \( C_V \) into the heat capacity equation:
\[ \frac{7}{2} R = \frac{3}{2} R + \frac{R}{1 - n} \]

### Step 4: Simplify the equation
Subtract \( \frac{3}{2} R \) from both sides:
\[ \frac{7}{2} R - \frac{3}{2} R = \frac{R}{1 - n} \]
This simplifies to:
\[ \frac{4}{2} R = \frac{R}{1 - n} \]
\[ 2R = \frac{R}{1 - n} \]

### Step 5: Solve for the polytropic index \( n \)
Dividing both sides by \( R \) (assuming \( R \neq 0 \)):
\[ 2 = \frac{1}{1 - n} \]
Cross-multiplying gives:
\[ 2(1 - n) = 1 \]
Expanding this:
\[ 2 - 2n = 1 \]
Rearranging gives:
\[ 2n = 1 \]
\[ n = \frac{1}{2} \]

### Step 6: Write the equation of the process
The equation for a polytropic process is given by:
\[ PV^n = \text{constant} \]
Substituting \( n = \frac{1}{2} \):
\[ PV^{\frac{1}{2}} = \text{constant} \]
This can be rewritten as:
\[ P V^{\frac{1}{2}} = k \quad \text{(where \( k \) is a constant)} \]
or equivalently:
\[ P^2 V = \text{constant} \]

### Conclusion
Thus, the equation of the process for which the heat capacity is \( C = \frac{7}{2} R \) for a monoatomic gas is:
\[ P^2 V = \text{constant} \]