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

Explanation

To find the ratio \( \frac{C_P}{C_V} \) for the 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 1: Establish the relationship between pressure and temperature
Given that \( P \propto T^3 \), we can express this relationship mathematically as:
\[ P = k T^3 \]
where \( k \) is a constant.

### Step 2: Use the adiabatic condition
For an adiabatic process, we know that:
\[ P T^{\gamma} = \text{constant} \]
where \( \gamma = \frac{C_P}{C_V} \).

### Step 3: Substitute the expression for pressure
Substituting the expression for \( P \) into the adiabatic condition gives:
\[ (k T^3) T^{\gamma} = \text{constant} \]
This simplifies to:
\[ k T^{3 + \gamma} = \text{constant} \]

### Step 4: Analyze the constant
Since \( k \) is constant, we can say that \( T^{3 + \gamma} \) must also be constant. This implies that:
\[ 3 + \gamma = 0 \]
Thus, we can solve for \( \gamma \):
\[ \gamma = -3 \]

### Step 5: Relate \( \gamma \) to \( C_P \) and \( C_V \)
From the definition of \( \gamma \):
\[ \gamma = \frac{C_P}{C_V} \]
We have found \( \gamma = -3 \), which is not physically meaningful for specific heat ratios. We need to re-evaluate our assumptions.

### Step 6: Correct the approach
Instead of assuming \( P T^{\gamma} \) is constant, we should relate the exponents directly. From the proportionality \( P \propto T^3 \), we can set up the equation:
\[ \frac{P}{T^{3}} = \text{constant} \]
Now, relating this to the adiabatic condition:
\[ P T^{\gamma} = \text{constant} \]
We can equate the two constants:
\[ \frac{P}{T^{3}} = \frac{P}{T^{\gamma}} \]
This leads to:
\[ 3 = \gamma \]

### Step 7: Finalize the ratio
Thus, we conclude:
\[ \frac{C_P}{C_V} = \gamma = 3 \]

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

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