MHTCET 15 Apr 2026 Shift 2 Paper

Concept:In an adiabatic process, temperature and volume are related by $T V^{\gamma-1} = \text{constant}$, and r.m.s. speed depends on temperature as $v_{\text{rms}} \propto \sqrt{T}$.Explanation:The r.m.s. speed of an ideal gas is $v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$, so $v_{\text{rms}} \propto \sqrt{T}$.If the r.m.s. speed becomes one-third, then $\frac{v_2}{v_1} = \frac{1}{3}$.Since $v \propto \sqrt{T}$, we get $\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \frac{1}{3}$.Squaring both sides gives $\frac{T_2}{T_1} = \frac{1}{9}$.For an adiabatic process, $T V^{\gamma-1} = \text{constant}$, so $\frac{T_2}{T_1} = \left(\frac{V_1}{V_2}\right)^{\gamma-1}$.Given $\gamma = 1.5$, we have $\gamma - 1 = 0.5 = \frac{1}{2}$.Thus, $\frac{1}{9} = \left(\frac{V_1}{V_2}\right)^{1/2}$.Squaring: $\frac{1}{81} = \frac{V_1}{V_2}$.Therefore, $\frac{V_2}{V_1} = 81$, meaning the gas must be expanded 81 times.Answer:Option D: 81 times.