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UPSC Civil Services (Mains)

Chemistry optional

  • 2 papers
  • 250 marks each
  • 3 h per paper
  • 5 of 8 questions, Q1 and Q5 compulsory

Paper I covers physical and inorganic chemistry.

UPSC Civil Services (Mains), Chemistry optional · 2020 · Paper I

Q6(a). cₘₚ, cₐᵥ, cᵣₘₛ on the Maxwell distribution curve

  • 10 marks
  • Answered
Question

What does most probable speed (Cₘₚ), average speed (Cₐᵥ) and root mean square speed (Cᵣₘₛ) mean? Indicate these speeds on a Maxwell distribution curve. What does the area under the Maxwell distribution curve represent?

Answer in brief

cₘₚ=√(2RT/M) is the speed at the peak, cₐᵥ=√(8RT/π M) the mean speed and cᵣₘₛ=√(3RT/M) the square root of the mean square speed; cₘₚ<cₐᵥ<cᵣₘₛ in the ratio 1: 1.128: 1.225. The total area under the curve is 1 (all the molecules); the area between two speeds is the fraction of molecules with speeds in that range.

Key points

  • f(c) dc is the fraction of molecules with speeds between c and c+dc. It rises as c² (more ways of having a given speed in 3D), passes through a maximum and falls as e^-Mc²/2RT (Boltzmann factor); it is not symmetric, with a…
  • Most probable speed (speed of the largest fraction of molecules; from df/dc=0): cₘₚ=√((2RT)/(M))
  • Average speed (arithmetic mean of all speeds, ∫₀^∞ c f(c) dc): cₐᵥ=√((8RT)/(π M))
  • Root mean square speed (square root of the mean of c²; it fixes the kinetic energy, 12Mcᵣₘₛ²=32RT, and the pressure, pV=13Nmcᵣₘₛ²): cᵣₘₛ=√((3RT)/(M))

The full model answer

Written out in full, as it should go in the answer booklet. The solution page with its diagrams follows it.

  1. The curve: f(c) dc is the fraction of molecules with speeds between c and c+dc. It rises as c² (more ways of having a given speed in 3D), passes through a maximum and falls as e^-Mc²/2RT (Boltzmann factor); it is not symmetric, with a long tail at high speed.
  2. Most probable speed (speed of the largest fraction of molecules; from df/dc=0): cₘₚ=√((2RT)/(M))
  3. Average speed (arithmetic mean of all speeds, ∫₀^∞ c f(c) dc): cₐᵥ=√((8RT)/(π M))
  4. Root mean square speed (square root of the mean of c²; it fixes the kinetic energy, 12Mcᵣₘₛ²=32RT, and the pressure, pV=13Nmcᵣₘₛ²): cᵣₘₛ=√((3RT)/(M))
  5. Order and ratio: the tail pulls the averages above the peak: cₘₚ:cₐᵥ:cᵣₘₛ=√2:√(8/π):√3=1:1.128:1.225 For N₂ at 300 K: 422, 476 and 517 m s⁻¹.
  6. Area under the curve: ∫₀^∞ f(c) dc=1: the whole area is the total fraction of molecules (all N if dN/dc is plotted). The area between c₁ and c₂ (hatched) is the fraction of molecules with speeds in that range. On heating the curve flattens and moves to higher speeds, but its area stays 1.

Application. the high-speed tail explains why a small rise in temperature greatly increases the fraction of molecules with energy above Eₐ (faster reactions), and why light gases (H₂, He) escape from planetary atmospheres; effusion separation of ²³⁵UF₆ uses the 1/√M dependence.

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Model answer to Q6(a), 2020 Paper I: solution page 29
Solution page 29 of 45
Read the whole 2020 solution2020 question paper

All parts of Question 6