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 (γ) for an ideal gas given that \( P T^{11} = \text{constant} \).

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
We are given that \( P T^{11} = C \), where \( C \) is a constant. This implies a relationship between pressure (P), temperature (T), and volume (V) for the gas.

2. **Using the Ideal Gas Law**:
From the ideal gas law, we know that:
\[ PV = nRT \]
Rearranging gives us:
\[ T = \frac{PV}{nR} \]

3. **Substituting T in the Given Equation**:
Substitute \( T \) from the ideal gas law into the equation \( P T^{11} = C \):
\[ P \left( \frac{PV}{nR} \right)^{11} = C \]
This simplifies to:
\[ P^{12} V^{11} = C (nR)^{11} \]

4. **Differentiating the Equation**:
To find the volume expansion coefficient, we need to differentiate the equation with respect to temperature (T). First, we can express \( V \) in terms of \( T \) and differentiate:
\[ V^{11} = \frac{C (nR)^{11}}{P^{12}} \]
Taking the derivative with respect to \( T \):
\[ 11 V^{10} \frac{dV}{dT} = -\frac{12 C (nR)^{11}}{P^{13}} \frac{dP}{dT} \]

5. **Relating Volume Change to Temperature Change**:
The volume expansion coefficient \( \gamma \) is defined as:
\[ \gamma = \frac{1}{V} \frac{dV}{dT} \]
Rearranging gives:
\[ \frac{dV}{dT} = \gamma V \]

6. **Substituting Back**:
Substitute \( \frac{dV}{dT} \) back into the differentiated equation:
\[ 11 V^{10} \gamma V = -\frac{12 C (nR)^{11}}{P^{13}} \frac{dP}{dT} \]
Simplifying gives:
\[ 11 \gamma V^{11} = -\frac{12 C (nR)^{11}}{P^{13}} \frac{dP}{dT} \]

7. **Solving for γ**:
Rearranging to isolate \( \gamma \):
\[ \gamma = -\frac{12 C (nR)^{11}}{11 P^{13} V^{11}} \frac{dP}{dT} \]

8. **Final Expression**:
Since we are looking for a general expression, we can conclude that the volume expansion coefficient \( \gamma \) is related to the changes in pressure and temperature through the ideal gas law and the given relationship.

### Conclusion:
The volume expansion coefficient \( \gamma \) for the ideal gas under the condition \( P T^{11} = \text{constant} \) can be expressed in terms of the derivatives of pressure with respect to temperature.