A monoatomic ideal gas initially at temperature ' $\mathrm{T}_1$ ' is enclosed in a cylinder fitted with…

A monoatomic ideal gas initially at temperature ' $\mathrm{T}_1$ ' is enclosed in a cylinder fitted with massless, frictionless piston. By releasing the piston suddenly the gas is allowed to expand to adiabatically to a temperature ' $\mathrm{T}_2$ '. If ' $\mathrm{L}_1$ ' and ' $\mathrm{L}_2$ ' are the lengths of the gas columns before and after expansion respectively, then $\frac{T_2}{T_1}$ is
  1. $\frac{\mathrm{L}_1}{\mathrm{~L}_2}$
  2. $\frac{\mathrm{L}_2}{\mathrm{~L}_1}$
  3. $\left(\frac{L_1}{L_2}\right)^{2 / 3}$
  4. $\left(\frac{\mathrm{L}_2}{\mathrm{~L}_1}\right)^{2 / 3}$

Solution

For an adiabatic process $\begin{aligned} & \mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1} \\ \therefore \quad & \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\gamma-1} \end{aligned}$ For a monoatomic gas, $r=\frac{5}{3}$ $\begin{aligned} & \Rightarrow \gamma-1=\frac{5}{3}-1=\frac{2}{3} \\ & \mathrm{~V}_1=\mathrm{AL}_1 \text { and } \mathrm{V}_2=\mathrm{AL}_2 \\ \therefore \quad & \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left[\frac{\mathrm{AL}_1}{\mathrm{AL}_2}\right]^{2 / 3}=\left[\frac{\mathrm{L}_1}{\mathrm{~L}_2}\right]^{2 / 3} \\ & \gamma-1=\frac{2}{3} \end{aligned}$ For an adiabatic process, $\begin{aligned} & \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\gamma-1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{\frac{2}{3}} \\ \mathrm{~V} & \propto \mathrm{L} \\ \therefore \quad \frac{\mathrm{T}_2}{\mathrm{~T}_1} & =\left(\frac{\mathrm{L}_1}{\mathrm{~L}_2}\right)^{\frac{2}{3}} \end{aligned}$

Asked in: MHT CET 2023 (12 May Shift 2)

Practice more Kinetic Theory of Gases and Radiation questions on Aicharya