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: Understanding the Question:
The gas is taken as one mole, since the options contain \(R\) and \(T\) without \(n\). In an adiabatic expansion \(TV^{\gamma-1} = \text{constant}\).

Step 2: Final temperature:
\(T_2 = T\left(\frac{V_1}{V_2}\right)^{\gamma-1} = T\left(\frac12\right)^{1/2} = \frac{T}{\sqrt2}\).

Step 3: Work done:
\[ W = \frac{R(T_1 - T_2)}{\gamma - 1} = \frac{R\left(T - \frac{T}{\sqrt2}\right)}{1/2} = 2RT\left(1 - \frac{1}{\sqrt2}\right) = RT(2 - \sqrt2) \]

Step 4: Other options:
Options A and B have \(\sqrt2 - 2\), which is negative. Work done by an expanding gas is positive, so these are wrong. Option C has \(R/T\), which has the wrong units.

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