During an adiabatic process, the pressure of a gas is found to be proportional to the square of its absolute temperature. The `γ` for the gas is

Options

A. `3/2`
B. `5/4`
C. `7/5`
D. 2

Answer

To solve the problem, we need to find the value of \( \gamma \) (gamma) for a gas during an adiabatic process, given that the pressure \( P \) is proportional to the square of its absolute temperature \( T \).

### Step-by-Step Solution:

1. **Understanding the Adiabatic Process**:
In an adiabatic process, there is no heat exchange with the surroundings. The relationship for an adiabatic process is given by:
\[ PV^\gamma = \text{constant} \]
where \( P \) is the pressure, \( V \) is the volume, and \( \gamma \) is the heat capacity ratio (specific heat at constant pressure \( C_p \) to specific heat at constant volume \( C_v \)).

2. **Given Relation**:
We are given that:
\[ P \propto T^2 \]
This implies:
\[ P = kT^2 \]
where \( k \) is a constant.

3. **Using the Ideal Gas Law**:
The ideal gas law states:
\[ PV = nRT \]
Rearranging this gives:
\[ V = \frac{nRT}{P} \]

4. **Substituting Volume in the Adiabatic Relation**:
Substitute \( V \) from the ideal gas law into the adiabatic relation:
\[ P \left(\frac{nRT}{P}\right)^\gamma = \text{constant} \]
Simplifying this gives:
\[ P \cdot \frac{(nRT)^\gamma}{P^\gamma} = \text{constant} \]
\[ n^\gamma R^\gamma T^\gamma P^{1 - \gamma} = \text{constant} \]

5. **Rearranging the Expression**:
From the above equation, we can express it as:
\[ PT^{\gamma} = \text{constant} \cdot P^{\gamma - 1} \]

6. **Using the Given Proportionality**:
Since \( P \propto T^2 \), we can write:
\[ PT^2 = \text{constant} \]
Now, comparing this with our derived expression \( PT^\gamma = \text{constant} \), we can equate the powers of \( T \):
\[ \gamma = 2 \]

7. **Conclusion**:
Thus, we find that:
\[ \gamma = 2 \]

### Final Answer:
The value of \( \gamma \) for the gas is \( 2 \).

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