In an adiabatic change, the pressure p and temperature T of a diatomic gas are related by the relation `ppropT^alpha`, where `alpha` equals

Options

A. (a) 1.67
B. (b) 0.4
C. (c) 0.6
D. (d) 3.5

Explanation

To solve the problem, we need to find the value of alpha in the relation \( P \propto T^\alpha \) for a diatomic gas undergoing an adiabatic change.

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process, there is no heat exchange with the surroundings. For an ideal gas, the relationship between pressure (P), volume (V), and temperature (T) can be described using the adiabatic condition.

2. **Use the Adiabatic Relation**:
The general relation in an adiabatic process for an ideal gas is given by:
\[ P \cdot V^\gamma = \text{constant} \]
where \( \gamma = \frac{C_p}{C_v} \) is the heat capacity ratio.

3. **Relate Pressure and Temperature**:
For an ideal gas, we can also express the relationship between pressure and temperature during an adiabatic process as:
\[ P \cdot T^{\frac{\gamma}{\gamma - 1}} = \text{constant} \]
This implies that:
\[ P \propto T^{\frac{\gamma}{\gamma - 1}} \]

4. **Identify Gamma for Diatomic Gas**:
For a diatomic gas, the value of \( \gamma \) is typically \( 1.4 \).

5. **Calculate Alpha**:
From the relation derived, we can equate:
\[ \alpha = \frac{\gamma}{\gamma - 1} \]
Substituting \( \gamma = 1.4 \):
\[ \alpha = \frac{1.4}{1.4 - 1} = \frac{1.4}{0.4} = 3.5 \]

6. **Conclusion**:
Therefore, the value of \( \alpha \) is \( 3.5 \).

### Final Answer:
\[ \alpha = 3.5 \]