The temperature of a gas is $-68^{\circ} \mathrm{C}$. To what temperature should it be heated, so that the r…

The temperature of a gas is $-68^{\circ} \mathrm{C}$. To what temperature should it be heated, so that the r.m.s. velocity of the molecules be doubled?
  1. $357^{\circ} \mathrm{C}$
  2. $457^{\circ} \mathrm{C}$
  3. $547^{\circ} \mathrm{C}$
  4. $820^{\circ} \mathrm{C}$

Solution

$\mathrm{T}_1=-68^{\circ} \mathrm{C}=-68+273 \mathrm{~K}=205 \mathrm{~K}$ R.M.S. velocity, $\mathrm{v}_{\mathrm{rms}} \propto \sqrt{\mathrm{T}}$ $\begin{array}{ll} & \frac{\left(\mathrm{v}_{\mathrm{rms}}\right)_2}{\left(\mathrm{v}_{\mathrm{rms}}\right)_1}=\sqrt{\frac{\mathrm{T}_2}{\mathrm{~T}_1}}=2 \\ \therefore \quad & \frac{\mathrm{T}_2}{\mathrm{~T}_1}=4 \\ \therefore \quad & \mathrm{T}_2=4 \times 205=820 \mathrm{~K} \\ \therefore \quad & \mathrm{T}_2=547^{\circ} \mathrm{C} \end{array}$

Asked in: MHT CET 2023 (14 May Shift 1)

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