Q137 / 160 · TS EAMCET 2024 (10 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, $\mathrm{Vf}=2 \mathrm{Vi}, \mathrm{Pf}=\mathrm{P} 1$ $\begin{aligned} & \therefore \mathrm{P}_{\mathrm{i}} \mathrm{~V}_{\mathrm{i}}^\gamma=\mathrm{P}_{\mathrm{f}} \mathrm{~V}_{\mathrm{f}}^\gamma \Rightarrow \mathrm{PiVi}^{1.5}=\mathrm{P}_1\left(2 \mathrm{~V}_{\mathrm{i}}\right)^{1.5} \\ & \therefore \quad \mathrm{Pi}_{\mathrm{g}_{\mathrm{ad}}}=(2)^{1.5} \mathrm{P}_1 \end{aligned}$
For isothermal process, $\mathrm{V}_{\mathrm{f}}=2 \mathrm{~V}_{\mathrm{i}}, \mathrm{P}_{\mathrm{f}}=\mathrm{P}_2$ $\begin{aligned} & P_i V_i=P_f V_f \Rightarrow P_i V_i=\left(P_2\right)\left(2 V_i\right) \\ & \therefore \quad \mathrm{Pi}_{\text {,iso }}=2 P_2 \\ & \therefore \quad \frac{P_{i_{\text {ad }}}}{P_{\text {i riso }}}=\frac{(2)^{1.5} \mathrm{P}_1}{2 P_2}=\sqrt{2}: 1 \end{aligned}$

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

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