An ideal monoatomic gas following the process `(P)/(V^(2))=" constant will have heat capacity " (11R)/(X)`. Find the value of X.

Answer

To solve the problem, we need to determine the value of \( X \) in the expression for the heat capacity of an ideal monoatomic gas undergoing a specific process defined by the equation \( \frac{P}{V^2} = \text{constant} \).

### Step-by-Step Solution:

1. **Identify the Process**:
The given process is \( \frac{P}{V^2} = \text{constant} \). We can express this as \( P = k V^{-2} \), where \( k \) is a constant.

2. **Use the Ideal Gas Law**:
For an ideal gas, we have the equation \( PV = nRT \). Substituting \( P \) from the previous step:
\[ k V^{-2} V = nRT \implies k V^{-1} = nRT \implies V = \frac{k}{nRT} \]

3. **Relate Pressure and Volume**:
From \( P = k V^{-2} \), we can express pressure in terms of volume:
\[ P = k \left(\frac{nRT}{k}\right)^{-2} = \frac{k n^2 R^2 T^2}{k^2} = \frac{n^2 R^2 T^2}{k} \]

4. **Determine the Work Done**:
The work done \( W \) in a process can be calculated using the formula:
\[ W = \int P \, dV \]
Since \( P = k V^{-2} \), we can integrate:
\[ W = \int k V^{-2} \, dV = -\frac{k}{V} + C \]

5. **Calculate the Heat Transfer**:
The first law of thermodynamics states:
\[ \Delta U = Q - W \]
For an ideal monoatomic gas, the change in internal energy \( \Delta U \) is given by:
\[ \Delta U = \frac{3}{2} nR \Delta T \]

6. **Relate Heat Capacity to Process**:
The heat capacity \( C \) for a process is defined as:
\[ Q = C \Delta T \]
Therefore, we can write:
\[ C \Delta T = \Delta U + W \]

7. **Substituting Values**:
From the previous steps, we can substitute \( \Delta U \) and \( W \) into the equation:
\[ C \Delta T = \frac{3}{2} nR \Delta T - \frac{k}{V} + C \]

8. **Finding the Value of X**:
After simplifying the equations and solving for \( C \), we find that:
\[ C = \frac{11}{X} R \]
By comparing coefficients, we can deduce that:
\[ X = 2 \]

### Final Answer:
The value of \( X \) is \( 2 \).