2021 A/L Physics Past Paper · Question 05 (English Medium)

Answer: (4) absolute temperature.

What the question asks

Find the quantity that the mean kinetic energy of an ideal gas molecule depends on.

What each letter means:

$m$ ($\text{kg}$): mass of one molecule.

$\overline{c^2}$ ($\text{m}^2\ \text{s}^{-2}$): the mean of the squares of the molecule speeds.

$\tfrac{1}{2} m\overline{c^2}$ ($\text{J}$): mean kinetic energy of one molecule.

$T$ ($\text{K}$): absolute temperature.

$k$ ($\text{J K}^{-1}$): the Boltzmann constant.

Relevant theory

The relation between mean kinetic energy and the temperature of gas molecula

when heat is supplied, if its temperature rises, then its kinetic energy increases.

(as KE ∝ θ in kelvin)

Reading the options

For example, take two balloons in the shade. One is small and tight. The other is big and soft.

Their pressure, volume and density are different. But they are at the same temperature $T$, so their air molecules move just as fast. Each molecule has the same kinetic energy, $\tfrac{1}{2} m\overline{c^2}$.

Now carry one balloon into the hot sun. Its temperature $T$ goes up.

Its molecules move faster, so each one carries more kinetic energy.

$$\tfrac{1}{2} m\overline{c^2} = \tfrac{3}{2} kT$$

In words: $$the\,mean\,kinetic\,energy\,of\,one\,molecule = a\,constant$$ $\times$ the absolute temperature.

The relation between the mean kinetic energy and the temperature is:

Here $k$ is the Boltzmann constant and $T$ is the absolute temperature, in kelvin.

Only $T$ appears. Pressure, volume, density and heat capacity do not.

Why the other options fail

Pressure is not in $\tfrac{3}{2}kT$.

Volume is not in $\tfrac{3}{2}kT$.

Density is not in $\tfrac{3}{2}kT$.

Heat capacity is not in $\tfrac{3}{2}kT$.

Conclusion

The mean kinetic energy is $\tfrac{3}{2}kT$. It depends only on the absolute temperature.

The answer is (4).

For an ideal gas, temperature is a measure of the mean kinetic energy of the molecules.

Double the absolute temperature and the mean kinetic energy doubles.

Exam tip