A one mole of an ideal gas expands adiabatically ato constant pressure such that its temperature `Tpropto (1)/(sqrt(V))` .The value of the adiabatic constant gas is

Options

A. a.1.3
B. b.1.5
C. c.1.67
D. d.2

Explanation

To solve the problem, we need to find the value of the adiabatic constant (γ) for a one mole of an ideal gas that expands adiabatically under the condition that its temperature \( T \) is inversely proportional to \( \frac{1}{\sqrt{V}} \).

### Step-by-Step Solution:

1. **Understand the Relationship**:
The problem states that \( T \propto \frac{1}{\sqrt{V}} \). This can be expressed mathematically as:
\[ T = k \cdot \frac{1}{\sqrt{V}} \]
where \( k \) is a constant.

2. **Rearranging the Equation**:
We can rewrite the equation as:
\[ T \sqrt{V} = k \]
This indicates that the product \( T \sqrt{V} \) remains constant during the process.

3. **Using the Ideal Gas Law**:
The ideal gas law states:
\[ PV = nRT \]
For one mole of gas (\( n = 1 \)):
\[ PV = RT \]
From this, we can express pressure \( P \) as:
\[ P = \frac{RT}{V} \]

4. **Substituting Pressure into the Adiabatic Condition**:
The adiabatic condition for an ideal gas is given by:
\[ PV^\gamma = \text{constant} \]
Substituting for \( P \):
\[ \left(\frac{RT}{V}\right)V^\gamma = \text{constant} \]
Simplifying this gives:
\[ RTV^{\gamma - 1} = \text{constant} \]

5. **Comparing the Two Constants**:
We have two expressions that are constant:
- From the temperature relationship: \( T \sqrt{V} = k \)
- From the adiabatic process: \( RTV^{\gamma - 1} = \text{constant} \)

Since both are constants, we can set them equal:
\[ T \sqrt{V} \propto RTV^{\gamma - 1} \]

6. **Substituting for \( T \)**:
Substitute \( T = \frac{k}{\sqrt{V}} \) into the adiabatic equation:
\[ \frac{k}{\sqrt{V}} \cdot \sqrt{V} \propto R V^{\gamma - 1} \]
This simplifies to:
\[ k \propto R V^{\gamma - 1} \]

7. **Equating Powers of \( V \)**:
From the relationship \( T \sqrt{V} = k \) and \( RTV^{\gamma - 1} = \text{constant} \), we can compare the powers of \( V \):
\[ \frac{1}{2} = \gamma - 1 \]
Solving for \( \gamma \):
\[ \gamma - 1 = \frac{1}{2} \implies \gamma = 1 + \frac{1}{2} = \frac{3}{2} \]

8. **Conclusion**:
The value of the adiabatic constant \( \gamma \) is:
\[ \gamma = \frac{3}{2} \text{ or } 1.5 \]

### Final Answer:
The value of the adiabatic constant \( \gamma \) is \( \frac{3}{2} \) or 1.5.