An ideal gas is expanding such that `PT^(2)= `constant. The coefficient of volume expansion of lthe gas is:

Options

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

Answer

To find the coefficient of volume expansion of an ideal gas expanding such that \( P T^2 = \) constant, we can follow these steps:

### Step 1: Understand the relationship given
We have the equation \( P T^2 = C \) where \( C \) is a constant. This means that the pressure \( P \) and the temperature \( T \) are related in such a way that their product with \( T^2 \) remains constant.

### 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,
- \( T \) is the temperature.

### Step 3: Express Pressure in terms of Volume and Temperature
From the ideal gas law, we can express pressure \( P \) as:
\[ P = \frac{nRT}{V} \]

### Step 4: Substitute Pressure in the given relationship
Substituting \( P \) into the equation \( P T^2 = C \):
\[ \frac{nRT}{V} T^2 = C \]
This simplifies to:
\[ \frac{nRT^3}{V} = C \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ V = \frac{nRT^3}{C} \]

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

### Step 7: Calculate the Coefficient of Volume Expansion
The coefficient of volume expansion \( \beta \) is given by:
\[ \beta = \frac{1}{V} \frac{dV}{dT} \]
Substituting \( V \) and \( \frac{dV}{dT} \):
\[ \beta = \frac{1}{\frac{nRT^3}{C}} \cdot \left(\frac{nR}{C} \cdot 3T^2\right) \]
This simplifies to:
\[ \beta = \frac{3}{T} \]

### Final Answer
Thus, the coefficient of volume expansion of the gas is:
\[ \beta = \frac{3}{T} \]

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