A-LevelPhysicsIdeal gases9702/42 · May/June 2021 · Q29 Marks

An ideal gas has a volume of 3.1 × 10⁻³m³ at a pressure of 8.5 × 10⁵ Pa and a temperature of 290 K, as shown in [Figure 2.1]. The gas suddenly expands to a volume of 6.3 × 10⁻³m³. During the expansion, no thermal energy is transferred. The final pressure of the gas is 2.7 × 10⁵ Pa at temperature T_F, as shown in [Figure 2.1].

(a)Show that the number of gas molecules is 6.6 × 10²³.[3]
(b)(i)Show that the final temperature T_F of the gas is 190 K.[1]
(b)(ii)The average translational kinetic energy E_K of a molecule of an ideal gas is given by E_K = (3/2)kT where T is the thermodynamic temperature and k is the Boltzmann constant. Calculate the increase in internal energy ΔU of the gas.[3]
(c)Use the first law of thermodynamics to explain why the external work W done on the gas during the expansion is equal to the increase in internal energy in (b)(ii).[2]

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Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

[1] pV = nRT
[1] pV = nRT and N = N_A OR pV = NkT
[1] 3.1 × 10⁻³ × 8.5 × 10⁵ = (N × 290 × 8.31) / (6.02 × 10²³) so N = 6.6 × 10²³ OR 3.1 × 10⁻³ × 8.5 × 10⁵ = N × 1.38 × 10⁻²³ × 290 so N = 6.6 × 10²³
[1] (3.1 × 10⁻³ × 8.5 × 10⁵) / 290 = (6.3 × 10⁻³ × 2.7 × 10⁵) / T so T = 190 K OR 6.3 × 10⁻³ × 2.7 × 10⁵ = 6.6 × 10²³ × 1.38 × 10⁻²³ × T so T = 190 K
[1] ΔU = 3/2 x k x ΔT x N
[1] = 3/2 × 1.38 × 10⁻²³ × (190 – 290) × 6.6 × 10²³
[1] = -1400 J
[1] ΔU = q + W
[1] q = 0 so ΔU = W

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