Question

A sample of gas at temperature \(T\) is adiabatically expanded to double its volume. The work done by the gas in the process is (given, \(γ = \frac{3}{2}\)):

• A. \(W = TR[\sqrt{2}-2]\)
• B. \(W = \frac{T}{R}[\sqrt{2}-2]\)
• C. \(W = \frac{R}{T}[2-\sqrt{2}]\)
• D. \(W = RT[2-\sqrt{2}]\)

Hint:

Adiabatic: \(TV^{\gamma-1}\) is constant and \(W=\frac{nR(T_1-T_2)}{\gamma-1}\) for one mole.

Answer 1

The Correct Option is D

Step 1: Plan:
Use $W = \frac{P_1V_1 - P_2V_2}{\gamma-1}$ with $PV = RT$.

Step 2: Steps:
$P_1V_1 = RT$. For the final state, $P_2V_2 = RT_2 = \frac{RT}{\sqrt2}$, since $T_2 = T(1/2)^{\gamma-1}$ with $\gamma - 1 = \frac12$.
$W = \frac{RT - RT/\sqrt2}{1/2} = 2RT - \sqrt2RT = RT(2-\sqrt2)$.

Final Answer:
The work done is $RT(2-\sqrt2)$, option (D). \[ \boxed{RT(2-\sqrt2)} \]