For an ideal gas `PT^(11)` = constant then volume expansion coefficient is equal to :-

Options

A. `(11)/(T)`
B. `(1)/(T)`
C. `(12)/(T)`
D. `(2)/(T)`

Explanation

To solve the problem, we need to find the volume expansion coefficient for an ideal gas given that \( P T^{11} = \text{constant} \).

### Step-by-Step Solution:

1. **Understand the Given Relation**:
We start with the relation \( P T^{11} = \text{constant} \). Let's denote this constant as \( K \). Therefore, we can express pressure \( P \) in terms of temperature \( T \):
\[ P = \frac{K}{T^{11}} \]

2. **Use the Ideal Gas Law**:
The ideal gas law is given by:
\[ PV = nRT \]
Substituting our expression for \( P \) into the ideal gas law gives:
\[ \frac{K}{T^{11}} V = nRT \]

3. **Rearranging the Equation**:
Rearranging the equation to solve for volume \( V \):
\[ V = \frac{nRT^{12}}{K} \]
Here, we see that volume \( V \) is directly proportional to \( T^{12} \):
\[ V \propto T^{12} \]

4. **Finding the Volume Expansion Coefficient**:
The volume expansion coefficient \( \beta_V \) is defined as:
\[ \beta_V = \frac{1}{V} \frac{\Delta V}{\Delta T} \]
From our proportionality \( V \propto T^{12} \), we can express the change in volume \( \Delta V \) as:
\[ \Delta V = 12V \frac{\Delta T}{T} \]

5. **Substituting into the Coefficient Formula**:
Now substituting \( \Delta V \) into the volume expansion coefficient formula:
\[ \beta_V = \frac{1}{V} \cdot 12V \cdot \frac{\Delta T}{T} \]
Simplifying this gives:
\[ \beta_V = 12 \frac{\Delta T}{T} \]

6. **Final Expression for Volume Expansion Coefficient**:
Therefore, the volume expansion coefficient is:
\[ \beta_V = \frac{12}{T} \]

### Conclusion:
The volume expansion coefficient for the ideal gas under the given condition is \( \beta_V = \frac{12}{T} \).