Q137 / 160 · TS EAMCET 2024 (11 May Shift 1) · Physics

The ratio of the specific heat capacities of a gas is 1.5 . When the gas undergoes an adiabatic process, its volume is doubled and pressure becomes $P_1$. When the gas undergoes isothermal process, its volume is doubled and pressure becomes $P_2$. If $P_1=P_2$, the ratio of the initial pressures of the gas when it undergoes adiabatic and isothermal processes is

  • A
    $\sqrt{3}: \sqrt{2}$
  • B
    $1: 1$
  • C
    $\sqrt{3}: 1$
  • Correct answer
    D
    $\sqrt{2}: 1$

Solution

For a gas, $\gamma=1.5$ For adiabatic Process, $\mathrm{V}_{f}=2 \mathrm{V}_{i}, \mathrm{P}_{f}=\mathrm{P} 1$ $\begin{aligned} & \therefore \mathrm{P}_{\mathrm{i}} \mathrm{~V}_{\mathrm{i}}^\gamma=\mathrm{P}_{\mathrm{f}} \mathrm{~V}_{\mathrm{f}}^\gamma \Rightarrow \mathrm{piVi}^{1.5}=\mathrm{P}_1\left(2 \mathrm{~V}_{\mathrm{i}}\right)^{1.5} \\ & \therefore \quad \mathrm{Pi}_{\text {,ad }}=(2)^{1.5} \mathrm{P}_1 \end{aligned}$
For isothermal process, $\mathrm{V}_{\mathrm{f}}=2 \mathrm{~V}_{\mathrm{i}}, \mathrm{P}_{\mathrm{f}}=\mathrm{P}_2$ $\begin{aligned} & \mathrm{P}_{\mathrm{i}} \mathrm{~V}_{\mathrm{i}}=\mathrm{P}_{\mathrm{f}} \mathrm{~V}_{\mathrm{f}} \Rightarrow \mathrm{P}_{\mathrm{i}} \mathrm{~V}_{\mathrm{i}}=\left(\mathrm{P}_2\right)\left(2 \mathrm{~V}_{\mathrm{i}}\right) \\ & \therefore \quad \mathrm{Pi}_{\text {iso }}=2 \mathrm{P}_2 \\ & \therefore \quad \frac{\mathrm{Pi}_{\mathrm{i}_{\mathrm{ad}}}}{\mathrm{Pi}_{\text {iso }}}=\frac{(2)^{1.5} \mathrm{P}_1}{2 \mathrm{P}_2}=\sqrt{2}: 1 \end{aligned}$

Hence, the correct answer is option: 'D'.

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