Question

A sample of 1 mole gas at temperature T is adiabatically expanded to double its volume. If adiabatic constant for the gas is \(\gamma=\frac{3}{2}\) , then the work done by the gas in the process is:

• A. \(RT[2-\sqrt2]\)
• B. \(\frac{R}{T}[2-\sqrt2]\)
• C. \(RT[2+\sqrt2]\)
• D. \(\frac{T}{R}[2-\sqrt2]\)

Answer 1

The Correct Option is A

Step 1: Adiabatic condition The adiabatic condition is given by:

\(TV^{\gamma-1} = \text{constant.}\)

For the initial and final states:

\(T(V)^{\frac{3}{2}-1} = T_f(2V)^{\frac{3}{2}-1}.\)

Simplify:

\(TV^{\frac{1}{2}} = T_f(2V)^{\frac{1}{2}},\)

\(TV\sqrt{V} = T_f\sqrt{2V}.\)

Cancel \(\sqrt{V}\):

\(T = T_f\sqrt{2}.\)

Solve for \(T_f\):

\(T_f = \frac{T}{\sqrt{2}}.\)

Step 2: Work done in adiabatic expansion The work done in an adiabatic process is given by:

\(W.D. = \frac{nR}{1-\gamma}\left[T_f - T\right].\)

Substitute \(T_f = \frac{T}{\sqrt{2}}, \gamma = \frac{3}{2},\) and \(n = 1\):

\(W.D. = \frac{R}{1-\frac{3}{2}}\left[\frac{T}{\sqrt{2}} - T\right].\)

Simplify:

\(W.D. = \frac{R}{-\frac{1}{2}}\left[\frac{T}{\sqrt{2}} - T\right],\)

\(W.D. = -2R\left[\frac{T}{\sqrt{2}} - T\right].\)

Factorize:

\(W.D. = 2RT\left[1 - \frac{1}{\sqrt{2}}\right].\)

Simplify further:

\(W.D. = RT\left[2 - \sqrt{2}\right].\)

Final Answer: \(RT\left[2 - \sqrt{2}\right].\)