Question

A diatomic gas $\left(\gamma = \frac{7}{5}\right)$ is compressed adiabatically to volume $\frac{\text{V}_0}{32}$ , where $\text{V}_0$ is its initial volume. The initial temperature of the gas is $\text{T}_\text{i}$ in kelvin and the final temperature is $\text{x}\text{T}_\text{i}$ in kelvin. The value of $x$ is

• A. 5
• B. 4
• C. 3
• D. 2

Hint:

Adiabatic compression always heats the gas. If volume decreases, temperature must increase.

Answer 1

The Correct Option is B

Step 1: Concept
For an adiabatic process, the relationship between temperature and volume is $TV^{\gamma - 1} = \text{constant}$.

Step 2: Meaning
$\frac{T_2}{T_1} = \left( \frac{V_1}{V_2} \right)^{\gamma - 1}$. Here, $\gamma - 1 = \frac{7}{5} - 1 = \frac{2}{5}$.

Step 3: Analysis
$V_1/V_2 = 32$.
$x = \frac{T_f}{T_i} = (32)^{2/5}$.
$32 = 2^5$, so $x = (2^5)^{2/5} = 2^2 = 4$.

Step 4: Conclusion
The value of $x$ is 4. Final Answer: (B)