During an adiabatic process. The pressure of a gas is to be proportional to the cube of its absolute temperature The value of ` gamma ` for the gas is :

Options

A. ` (5)/(3)`
B. ` (7)/(5)`
C. ` (3)/(2)`
D. ` (11)/(9)`

Explanation

To find the value of gamma (γ) for a gas during an adiabatic process where the pressure (P) is proportional to the cube of its absolute temperature (T), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We start with the given relationship that pressure (P) is proportional to the cube of the absolute temperature (T). This can be expressed mathematically as:
\[ P \propto T^3 \]
This implies:
\[ P = kT^3 \]
where \( k \) is a constant.

2. **Using the Ideal Gas Law**:
From the ideal gas law, we know:
\[ PV = nRT \]
Rearranging this gives us:
\[ T = \frac{PV}{nR} \]

3. **Substituting for T**:
We can substitute this expression for T into our earlier equation:
\[ P = k\left(\frac{PV}{nR}\right)^3 \]
Simplifying this, we get:
\[ P = k\frac{P^3V^3}{(nR)^3} \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ P^3 = \frac{(nR)^3}{k} V^3 P \]
Dividing both sides by P (assuming P ≠ 0):
\[ P^2 = \frac{(nR)^3}{k} V^3 \]

5. **Finding the Relationship**:
We can express this in terms of a constant:
\[ P^2 V^3 = \text{constant} \]

6. **Using the Adiabatic Condition**:
In an adiabatic process, we have:
\[ PV^\gamma = \text{constant} \]
Comparing this with our derived equation \( P^2 V^3 = \text{constant} \), we can equate the powers of P and V:
\[ \gamma = \frac{P^2}{PV^3} = \frac{2}{3} \]

7. **Final Value of Gamma**:
Thus, we find that:
\[ \gamma = \frac{3}{2} \]

### Conclusion:
The value of gamma (γ) for the gas is \( \frac{3}{2} \).

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