An ideal gas is expanding such that `PT^2=constant.` The coefficient of volume expansion of the gas is-

Options

A. `(1)/(T)`
B. `(2)/(T)`
C. `(3)/(T)`
D. `(4)/(T)`

Explanation

To solve the problem, we need to find the coefficient of volume expansion (γ) of an ideal gas that is expanding according to the relation \( PT^2 = \text{constant} \).

### Step-by-Step Solution:

1. **Understanding the Given Relation**:
We start with the relation given in the problem:
\[ PT^2 = \text{constant} \]
This implies that the product of pressure (P) and the square of temperature (T) remains constant during the expansion of the gas.

2. **Using the Ideal Gas Law**:
The ideal gas law states:
\[ PV = nRT \]
From this, we can express pressure (P) as:
\[ P = \frac{nRT}{V} \]

3. **Substituting P in the Given Relation**:
Substitute the expression for P into the relation \( PT^2 = \text{constant} \):
\[ \left(\frac{nRT}{V}\right)T^2 = \text{constant} \]
This simplifies to:
\[ \frac{nRT^3}{V} = \text{constant} \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ V = \frac{nRT^3}{\text{constant}} \]
Let’s denote the constant as \( K \) (where \( K = \frac{nR}{\text{constant}} \)):
\[ V = K T^3 \]

5. **Differentiating Volume with Respect to Temperature**:
Differentiate both sides of the equation \( V = K T^3 \) with respect to T:
\[ \frac{dV}{dT} = 3K T^2 \]

6. **Expressing Change in Volume**:
We can express the change in volume in terms of the original volume:
\[ dV = 3K T^2 dT \]

7. **Relating Change in Volume to Coefficient of Volume Expansion**:
The coefficient of volume expansion (γ) is defined as:
\[ \gamma = \frac{1}{V} \frac{dV}{dT} \]
Substituting \( dV \) into this equation gives:
\[ \gamma = \frac{1}{V} (3K T^2) \]

8. **Substituting for V**:
From our earlier equation, we know \( V = K T^3 \). Thus:
\[ \gamma = \frac{3K T^2}{K T^3} = \frac{3}{T} \]

9. **Final Result**:
Therefore, the coefficient of volume expansion (γ) is:
\[ \gamma = \frac{3}{T} \]

### Conclusion:
The coefficient of volume expansion of the gas is \( \frac{3}{T} \).