An ideal triatomic gas expands according to law `PV^2` = constant. If molar heat capacity of the gas for the process is NR, then N is equal to

Answer

To solve the problem step by step, we will analyze the given information and apply the relevant thermodynamic principles.

### Step 1: Understanding the Process
The gas expands according to the law \( PV^2 = \text{constant} \). This can be compared to the general form \( PV^\gamma = \text{constant} \), where \( \gamma \) is the heat capacity ratio.

**Hint:** Identify the relationship between the given equation and the standard form to find \( \gamma \).

### Step 2: Determine the Value of \( \gamma \)
From the equation \( PV^2 = \text{constant} \), we can see that \( \gamma = 2 \).

**Hint:** Recall that in the equation \( PV^\gamma = \text{constant} \), the exponent of \( V \) gives you the value of \( \gamma \).

### Step 3: Determine the Degrees of Freedom
For a triatomic gas, the degrees of freedom \( F \) can be calculated as follows:
- For a triatomic gas, \( F = 3N \), where \( N \) is the number of atoms per molecule. Since it is triatomic, \( N = 3 \):
\[ F = 3 \times 3 = 9 \]

**Hint:** Remember that the degrees of freedom for a gas depend on the number of atoms in the molecule.

### Step 4: Calculate \( C_V \)
The molar heat capacity at constant volume \( C_V \) is given by:
\[ C_V = \frac{F}{2} R = \frac{9}{2} R = \frac{9R}{2} \]

**Hint:** Use the formula for \( C_V \) based on degrees of freedom.

### Step 5: Calculate \( C \) for the Process
The molar heat capacity \( C \) for a process can be calculated using the formula:
\[ C = C_V + \frac{R}{1 - \gamma} \]
Substituting the values we have:
\[ C = \frac{9R}{2} + \frac{R}{1 - 2} = \frac{9R}{2} - R = \frac{9R}{2} - \frac{2R}{2} = \frac{7R}{2} \]

**Hint:** Make sure to substitute the correct values for \( C_V \) and \( \gamma \) in the formula.

### Step 6: Compare with Given Molar Heat Capacity
We are given that the molar heat capacity for the process is \( nR \). Therefore, we can equate:
\[ \frac{7R}{2} = nR \]
Dividing both sides by \( R \):
\[ \frac{7}{2} = n \]

**Hint:** Simplify the equation to find the value of \( n \).

### Final Result
Thus, the value of \( n \) is:
\[ n = 3.5 \]

### Summary
The final answer is \( n = 3.5 \).