An ideal gas is taken through a process `PT^(3)=` constant. The coefficient of thermal expansion of the gas in the given process is

Options

A. 1/T
B. 2/T
C. 3/T
D. 4/T

Explanation

To solve the problem of finding the coefficient of thermal expansion of an ideal gas undergoing a process where \( P T^3 = \text{constant} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Process**:
We are given that \( P T^3 = \text{constant} \). This implies that as pressure \( P \) and temperature \( T \) change, their product with \( T^3 \) remains constant.

2. **Use the Ideal Gas Law**:
Recall the ideal gas law, which states:
\[ PV = nRT \]
where \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( V \) is the volume.

3. **Express Pressure in Terms of Volume and Temperature**:
From the ideal gas law, we can express pressure \( P \) as:
\[ P = \frac{nRT}{V} \]
Substituting this into the given process equation \( P T^3 = \text{constant} \):
\[ \frac{nRT}{V} T^3 = \text{constant} \]
This simplifies to:
\[ \frac{nR T^4}{V} = \text{constant} \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ T^4 = \text{constant} \cdot V \]
This indicates that \( T^4 \) is proportional to \( V \).

5. **Differentiate the Relationship**:
To find the coefficient of thermal expansion \( \gamma \), we differentiate both sides with respect to temperature:
\[ \frac{d(T^4)}{dT} = \frac{dV}{dT} \]
Using the chain rule, we have:
\[ 4T^3 \frac{dT}{dT} = \frac{dV}{dT} \Rightarrow 4T^3 = \frac{dV}{dT} \]

6. **Relate Changes in Volume and Temperature**:
We can express the changes in volume and temperature as:
\[ \frac{\Delta V}{V} = \frac{dV}{dT} \Delta T \]
Therefore, we can write:
\[ \frac{\Delta V}{V} = 4T^3 \Delta T \]

7. **Define the Coefficient of Thermal Expansion**:
The coefficient of thermal expansion \( \gamma \) is defined as:
\[ \gamma = \frac{1}{V} \frac{\Delta V}{\Delta T} \]
Substituting our expression for \( \Delta V \):
\[ \gamma = \frac{1}{V} (4T^3 \Delta T) \frac{1}{\Delta T} = \frac{4T^3}{V} \]

8. **Final Expression**:
Since \( V \) is not constant, we can express \( \gamma \) as:
\[ \gamma = \frac{4}{T} \]

### Conclusion:
Thus, the coefficient of thermal expansion of the gas in the given process is:
\[ \gamma = \frac{4}{T} \]