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

Options

A. PV = constant
B. `PV^2`= constant
C. `P^2 V`=constant
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 degrees of freedom
For a monoatomic gas, the molar specific heat at constant volume \( C_V \) is given by:
\[ C_V = \frac{f}{2} R \]
where \( f \) is the degrees of freedom. For a monoatomic gas, \( f = 3 \), so:
\[ C_V = \frac{3}{2} R \]

### Step 2: Use the formula for heat capacity in a polytropic process
The heat capacity \( C \) for a polytropic process is given by:
\[ C = C_V + \frac{R}{1 - X} \]
where \( X \) is the polytropic index.

### Step 3: Substitute the known values into the equation
We know that:
\[ C = \frac{7}{2} R \]
Substituting \( C_V \) into the equation gives:
\[ \frac{7}{2} R = \frac{3}{2} R + \frac{R}{1 - X} \]

### Step 4: Simplify the equation
Now, let's simplify the equation:
1. Subtract \( \frac{3}{2} R \) from both sides:
\[ \frac{7}{2} R - \frac{3}{2} R = \frac{R}{1 - X} \]
2. This simplifies to:
\[ 2 R = \frac{R}{1 - X} \]
3. Dividing both sides by \( R \) (assuming \( R \neq 0 \)):
\[ 2 = \frac{1}{1 - X} \]

### Step 5: Solve for \( X \)
Now, we can solve for \( X \):
1. Cross-multiply:
\[ 2(1 - X) = 1 \]
2. Expanding gives:
\[ 2 - 2X = 1 \]
3. Rearranging gives:
\[ 2X = 2 - 1 \]
4. Thus:
\[ 2X = 1 \quad \Rightarrow \quad X = \frac{1}{2} \]

### Step 6: Write the equation of the process
The general equation for a polytropic process is:
\[ PV^X = \text{constant} \]
Substituting \( X = \frac{1}{2} \):
\[ PV^{\frac{1}{2}} = \text{constant} \]
To express it in a more recognizable form, we can square both sides:
\[ P^2 V = \text{constant} \]

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