Question

A rigid diatomic gas \((γ = \frac{7}{5})\) is compressed adiabatically to volume \((\frac{V_i}{32})\), where \(V_i\) is the initial volume. The initial temperature of the gas is '\(T_i\)' K and the final temperature is '\(x\,T_i\)' K. The value of \(x\) is

• A. \(2\)
• B. \(3\)
• C. \(4\)
• D. \(5\)

Hint:

Use T V^(gamma-1) = constant with gamma = 7/5.

Answer 1

The Correct Option is C

Step 1: Use pressure and volume form:
$PV^\gamma$ is constant, and $P = \frac{nRT}{V}$, so $T V^{\gamma - 1}$ is constant.

Step 2: Ratio of temperatures:
$\frac{T_f}{T_i} = \left(\frac{V_i}{V_f}\right)^{\gamma - 1} = 32^{0.4}$.

Step 3: Value:
$32^{0.4} = e^{0.4\ln32} = e^{0.4\times3.4657} = e^{1.3863} = 4$. So $x = 4$.

Final Answer:
x equals 4, option (C). \[ \boxed{4} \]