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 find the coefficient of volume expansion of an ideal gas expanding under the condition \( PT^2 = \text{constant} \), we can follow these steps:

### Step 1: Understand the relationship given
We start with the relationship \( PT^2 = \text{constant} \). This implies that the product of pressure \( P \) and the square of the temperature \( T \) remains constant during the expansion of the gas.

### Step 2: Use the ideal gas law
From the ideal gas law, we know that:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.

### Step 3: Express pressure in terms of volume and temperature
We can express pressure \( P \) in terms of volume \( V \) and temperature \( T \):
\[ P = \frac{nRT}{V} \]

### Step 4: Substitute pressure into the given relationship
Substituting this expression for \( P \) into the equation \( PT^2 = \text{constant} \):
\[ \left(\frac{nRT}{V}\right)T^2 = \text{constant} \]
This simplifies to:
\[ \frac{nRT^3}{V} = \text{constant} \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ V = \frac{nRT^3}{\text{constant}} \]
Let’s denote the constant as \( C \):
\[ V = C nRT^3 \]

### Step 6: Differentiate volume with respect to temperature
To find the coefficient of volume expansion \( \gamma \), we need to differentiate \( V \) with respect to \( T \):
\[ \frac{dV}{dT} = C nR \cdot 3T^2 \]

### Step 7: Calculate the coefficient of volume expansion
The coefficient of volume expansion \( \gamma \) is defined as:
\[ \gamma = \frac{1}{V} \frac{dV}{dT} \]
Substituting \( \frac{dV}{dT} \) and \( V \):
\[ \gamma = \frac{C nR \cdot 3T^2}{C nRT^3} \]
This simplifies to:
\[ \gamma = \frac{3T^2}{T^3} = \frac{3}{T} \]

### Step 8: Conclusion
Thus, the coefficient of volume expansion \( \gamma \) of the gas is:
\[ \gamma = \frac{3}{T} \]

### Final Answer
The correct answer is option 3: \( \frac{3}{T} \).

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