Q2(a). Maxwell distribution of speeds: effect of T and of M

Features of every curve: the plotted quantity is the fraction of molecules per unit speed,

f(c)=1/N(dN_c)/(dc)=4π((M)/(2π RT))^(3/2)c² e^-Mc²/2RT

It starts at zero (the c² factor, few molecules nearly at rest), rises to one maximum and falls in a long exponential tail. The curve is not symmetric: it is skewed to high speeds.

(i) Two temperatures, T₂>T₁ [8 marks]

1. Position of the peak: (df)/(dc)=0 gives 2c-Mc³RT=0, so the most probable speed is cₘₚ=√((2RT)/(M)). It grows as √T, so the peak at T₂ lies to the right.
2. Height of the peak: putting cₘₚ into f, fₘₐₓ=(4)/(√(π))((M)/(2RT))^(1/2)e⁻¹∝√(M/T). The peak at T₂ is lower.
3. Constant area: f is normalised, ∫₀^∞ f dc=1, at every temperature. A curve whose peak moves right and falls must therefore spread out: the distribution at T₂ is broader.
4. The tail: the exponent Mc²/2RT is smaller at T₂, so e^-Mc²/2RT dies away more slowly. A larger fraction of molecules has high speeds, and fewer are slow.
5. Numbers for the plot (N₂, M=28 g mol⁻¹): at 300 K, cₘₚ=422 m s⁻¹ and fₘₐₓ=1.97×10⁻³ s m⁻¹; at 1000 K, cₘₚ=771 m s⁻¹ and fₘₐₓ=1.08×10⁻³ s m⁻¹. The peak moves by √(1000/300)=1.83 and its height falls by the same factor (to 0.548).
6. Three speeds on one curve: cₘₚ:c:cᵣₘₛ=√2:√(8/π):√3=1:1.128:1.225. For N₂ at 300 K they are 422, 476 and 517 m s⁻¹. All three grow as √T.

(ii) Two molar masses, M₂>M₁, at one temperature [7 marks]

SAME Tsame mean kinetic energy 32RT per moleHEAVIER GAScₘₚ∝1/√M

1. M and T enter f only through the ratio M/T. Raising M has the same effect as lowering T, so plot (ii) is the reverse of plot (i).
2. cₘₚ=√(2RT/M) falls as 1/√M: the peak of the heavier gas M₂ lies at a lower speed.
3. fₘₐₓ∝√M, and the area is still 1, so the curve of M₂ is taller and narrower.
4. At the same temperature both gases have the same mean kinetic energy, 12Mcᵣₘₛ²=32RT per mole. A heavier molecule carries this energy at a lower speed, and the large factor M/2RT in the exponent makes high speeds rare.
5. Numbers for the plot (300 K): N₂ (M₁=28 g mol⁻¹) has cₘₚ=422 m s⁻¹ and fₘₐₓ=1.97×10⁻³ s m⁻¹; CO₂ (M₂=44 g mol⁻¹) has cₘₚ=337 m s⁻¹ and fₘₐₓ=2.47×10⁻³ s m⁻¹. The ratio of the peak speeds is √(28/44)=0.798.

Application. the high-speed tail explains why reaction rates rise steeply with temperature: only molecules in the tail have enough energy to react on collision, and the tail fills quickly as T rises. The 1/√M dependence of molecular speed is the basis of Graham's law of effusion and of the separation of ²³⁵UF₆ from ²³⁸UF₆.