A gas is compressed adiabatically till its temperature is doubled. The ratio of its final volume to initial volume will be

Options

A. `1//2`
B. more than `1//2`
C. less than `1//2`
D. between 1 and 2

Explanation

To solve the problem of finding the ratio of the final volume to the initial volume of a gas that is compressed adiabatically until its temperature is doubled, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. The relationship between temperature (T), volume (V), and the heat capacity ratio (γ) is given by the equation:
\[ T V^{\gamma - 1} = \text{constant} \]

### Step 2: Set Up the Initial and Final Conditions
Let:
- Initial temperature = \( T_1 \)
- Final temperature = \( T_2 = 2T_1 \) (since the temperature is doubled)
- Initial volume = \( V_1 \)
- Final volume = \( V_2 \)

### Step 3: Apply the Adiabatic Condition
Using the adiabatic condition, we can write:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]

### Step 4: Substitute the Known Values
Substituting \( T_2 = 2T_1 \) into the equation gives:
\[ T_1 V_1^{\gamma - 1} = (2T_1) V_2^{\gamma - 1} \]

### Step 5: Simplify the Equation
Dividing both sides by \( T_1 \) (assuming \( T_1 \neq 0 \)):
\[ V_1^{\gamma - 1} = 2 V_2^{\gamma - 1} \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ \frac{V_1^{\gamma - 1}}{V_2^{\gamma - 1}} = 2 \]

### Step 7: Take the Reciprocal
Taking the reciprocal of both sides:
\[ \frac{V_2^{\gamma - 1}}{V_1^{\gamma - 1}} = \frac{1}{2} \]

### Step 8: Solve for the Volume Ratio
Taking the \((\gamma - 1)\)th root:
\[ \frac{V_2}{V_1} = \left(\frac{1}{2}\right)^{\frac{1}{\gamma - 1}} \]

### Step 9: Analyze the Result
Since \( \gamma > 1 \), it follows that \( \frac{1}{\gamma - 1} < 1 \). Therefore, \( \left(\frac{1}{2}\right)^{\frac{1}{\gamma - 1}} < \frac{1}{2} \).

This means:
\[ V_2 < \frac{1}{2} V_1 \]

### Conclusion
Thus, the ratio of the final volume to the initial volume is:
\[ \frac{V_2}{V_1} < \frac{1}{2} \]

### Final Answer
The ratio of the final volume to the initial volume is less than \( \frac{1}{2} \).

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