During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio `C_P//C_V` for the gas is

Options

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

Explanation

To solve the problem, we need to determine the ratio \( \frac{C_P}{C_V} \) for a gas undergoing an adiabatic process where the pressure \( P \) is proportional to the cube of its absolute temperature \( T \).

### Step-by-Step Solution:

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

2. **Using the Adiabatic Condition**:
For an adiabatic process, we have the relation:
\[ P T^{\gamma - 1} = \text{constant} \]
where \( \gamma = \frac{C_P}{C_V} \).

3. **Equating the Two Relationships**:
From the first step, we can express \( P \) in terms of \( T \):
\[ P = k T^3 \]
Substituting this into the adiabatic condition gives:
\[ (k T^3) T^{\gamma - 1} = \text{constant} \]
Simplifying this, we get:
\[ k T^{3 + \gamma - 1} = \text{constant} \]
or
\[ k T^{\gamma + 2} = \text{constant} \]

4. **Comparing Exponents**:
Since both expressions must be equal to a constant, the exponent of \( T \) must be the same. Therefore, we can set:
\[ \gamma + 2 = 0 \]
Solving for \( \gamma \) gives:
\[ \gamma = -2 \]

5. **Finding the Ratio \( \frac{C_P}{C_V} \)**:
We know that:
\[ \frac{C_P}{C_V} = \gamma \]
Thus:
\[ \frac{C_P}{C_V} = -2 \]

### Conclusion:
The ratio \( \frac{C_P}{C_V} \) for the gas is \( \frac{3}{2} \).