The temperature of a gas is $-80^{\circ} \mathrm{C}$. To what temperature the gas should be heated so that…

The temperature of a gas is $-80^{\circ} \mathrm{C}$. To what temperature the gas should be heated so that the r.m.s. speed is increased by 2 times?
  1. $499^{\circ} \mathrm{C}$
  2. $772^{\circ} \mathrm{C}$
  3. $1464^{\circ} \mathrm{C}$
  4. $1737^{\circ} \mathrm{C}$

Solution

$T_1=-80^{\circ} \mathrm{C}=-80+273 \mathrm{~K}=193 \mathrm{~K}$ R.M.S. velocity, $\mathrm{v}_{\mathrm{rms}} \propto \sqrt{\mathrm{T}}$
As the speed is increased by two times, final speed becomes $\left(2 \mathrm{~V}_{\mathrm{rms}}\right)_1$ $\begin{array}{ll} & \frac{\left(v_{\mathrm{mss}}\right)_2}{\left(v_{\mathrm{ms}}\right)_1}=\sqrt{\frac{T_2}{T_1}}=2 \\ \therefore \quad & \frac{T_2}{T_1}=4 \\ \therefore \quad & T_2=4 \times 193=772 \mathrm{~K}=499^{\circ} \mathrm{C} \end{array}$

Asked in: MHT CET 2024 (09 May Shift 1)

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