A gas expands adiabatically at constant pressure such that `T propto 1/V^(3)`, the value of `gamma` of the gas will be

Options

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

Answer

To find the value of gamma (γ) for the gas that expands adiabatically at constant pressure with the relationship \( T \propto \frac{1}{V^3} \), we can follow these steps:

### Step 1: Understand the relationship given
We are given that the temperature \( T \) is inversely proportional to the cube of the volume \( V \):
\[ T \propto \frac{1}{V^3} \]
This can be expressed mathematically as:
\[ T = k \cdot \frac{1}{V^3} \]
where \( k \) is a constant.

### Step 2: Use the adiabatic condition
For an adiabatic process, the relationship between temperature, volume, and gamma is given by:
\[ T V^{\gamma - 1} = \text{constant} \]
We can rewrite this as:
\[ T V^{\gamma - 1} = C \]
where \( C \) is a constant.

### Step 3: Substitute the expression for T
Substituting our expression for \( T \) into the adiabatic condition:
\[ \left(k \cdot \frac{1}{V^3}\right) V^{\gamma - 1} = C \]
This simplifies to:
\[ k \cdot V^{\gamma - 1 - 3} = C \]
or
\[ k \cdot V^{\gamma - 4} = C \]

### Step 4: Analyze the equation
Since \( C \) is a constant, for the equation to hold for all volumes \( V \), the exponent of \( V \) must be zero:
\[ \gamma - 4 = 0 \]

### Step 5: Solve for gamma
From the equation \( \gamma - 4 = 0 \), we can solve for \( \gamma \):
\[ \gamma = 4 \]

### Conclusion
Thus, the value of \( \gamma \) for the gas is:
\[ \boxed{4} \]