A-Level Physics May/June 2024 Q2(b)(iii): The gas in X consists of molecules that each have a mass that is four times the mass of…

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  The temperature of the two gases is the same, and the average kinetic energy of the molecules, $\langle E_k \rangle$, is directly proportional to the absolute temperature ($T$). Therefore, the average kinetic energy of the molecules is the same in both cylinders X and Y.

  The average kinetic energy is given by the equation $\langle E_k \rangle = \frac{1}{2}m\langle c^2 \rangle$, where $m$ is the mass of a molecule and $\langle c^2 \rangle$ is the mean-square speed. The root-mean-square (r.m.s.) speed is $c_{rms} = \sqrt{\langle c^2 \rangle}$.

  Since $\langle E_k \rangle$ is the same for both gases: $\frac{1}{2}m_X \langle c_X^2 \rangle = \frac{1}{2}m_Y \langle c_Y^2 \rangle$ $m_X \langle c_X^2 \rangle = m_Y \langle c_Y^2 \rangle$

  This shows that the mean-square speed is inversely proportional to the mass ($\,\langle c^2 \rangle \propto \frac{1}{m}\,$), and therefore the r.m.s. speed is inversely proportional to the square root of the mass ($c_{rms} \propto \frac{1}{\sqrt{m}}$).

  We are given that $m_X = 4m_Y$. To find the ratio of the r.m.s. speeds: $\frac{c_{rms, X}}{c_{rms, Y}} = \sqrt{\frac{m_Y}{m_X}} = \sqrt{\frac{m_Y}{4m_Y}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$

  Therefore, the r.m.s. speed of the molecules in X is half the r.m.s. speed of the molecules in Y.

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  How the marks are awarded

  • B1 — Stating that because the temperature is the same, the average kinetic energy of the molecules is the same in both cylinders.
  • B1 — Correctly relating r.m.s. speed to mass, for example by stating $c_{rms} \propto 1/\sqrt{m}$ or by showing the derivation from the kinetic energy formula.
  • B1 — Correctly calculating the ratio and concluding that the r.m.s. speed in X is half the r.m.s. speed in Y.

  Common mistakes

  • Stating that r.m.s. speed is inversely proportional to mass ($c \propto 1/m$) instead of the square root of mass, leading to an incorrect ratio of 1/4.
  • Inverting the mass ratio in the calculation, leading to an answer of 2.
  • Forgetting to state the initial link between same temperature and same average kinetic energy, thus losing the first mark.
  • Calculating the ratio of the mean-square speeds (1/4) but forgetting to take the square root to find the ratio of the r.m.s. speeds.

  Examiner tip: For kinetic theory comparisons, always start from the fundamental principle that for gases at the same temperature, their average molecular kinetic energy is identical.

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