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Thermal physics (A Level)

15. Ideal Gases

5 min read 9 parts Lesson 19 of 20 Unit quiz

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15. Ideal Gases — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Use the mole, the gas laws and pV = nRT, and derive and apply the kinetic theory result pV = ⅓Nm⟨c²⟩ and the average molecular kinetic energy.Mole = the amount of s…

  • Avogadro constant NA
  • Number of moles
  • Equal volumes
  • Molar mass Mr
  • Number of particles
  • Proportionality

Section 1 of 9The mole and Avogadro's constant

  • Mole = the amount of substance containing as many particles as there are atoms in 12 g of carbon-12.
    • Amount of substance is an SI base quantity with the unit mol.
  • Avogadro constant NA = the number of atoms in 12 g of carbon-12, 6.02 × 1023 mol−1.
    • A carbon-12 atom has mass 12 u = 1.99 × 10−26 kg, so 0.012 kg contains 6.02 × 1023 atoms.
  • Number of moles
    • n = N/NA, where N is the number of particles.
  • Equal volumes
    • gases at the same temperature and pressure with equal volumes contain equal numbers of molecules.

Section 2 of 9Molar mass

  • Molar mass Mr = the mass in grams of one mole of a substance, in g mol−1.
    • One mole of an element has a mass equal to its relative atomic mass in grams.
    • For a compound, add the relative atomic masses; water is 2 × 1 + 16 = 18 g mol−1.
  • Equation
    • n = m/Mr, with mass m in grams.
      What the symbols mean: n = amount of substance (mol) · m = mass (g) · M or Mr = molar mass (g mol−1, numerically equal to Mr)
  • Number of particles
    • multiply the moles by NA.

Section 3 of 9The gas laws

  • Proportionality
    • for a fixed mass of ideal gas, pV ∝ T, with T the thermodynamic temperature in kelvin.
  • Kelvin
    • every gas calculation needs T in K.
ConstantRelationshipExplanation
Volumep ∝ T (p1/T1 = p2/T2)Hotter molecules have more kinetic energy and hit the walls harder and more often
PressureV ∝ T (V1/T1 = V2/T2)Faster molecules push further apart, so the gas expands
Temperaturep ∝ 1/V (p1V1 = p2V2)A smaller volume means more frequent collisions with the walls

Section 4 of 9The ideal gas equation

  • Equation of state
    • pV = nRT, with p in Pa, V in m3, n in moles, R = 8.31 J K−1 mol−1 and T in K.
      What the symbols mean: p = pressure (Pa) · V = volume (m3) · n = amount of gas (mol) · R = 8.31 J mol−1 K−1 · T = temperature (K = °C + 273)
  • Molecular form
    • pV = NkT, where N is the number of molecules and k is the Boltzmann constant, 1.38 × 10−23 J K−1.
  • Ideal gas = a gas that obeys pV = nRT at all pressures, volumes and temperatures.
  • Units
    • convert cm3 to m3 (× 10−6) and °C to K; R and k are given on the data sheet.

Section 5 of 9The Boltzmann constant

  • Definition
    • k = R/NA = 8.31 ÷ (6.02 × 1023) = 1.38 × 10−23 J K−1.
  • Link
    • it connects a microscopic property to a macroscopic one (temperature).
      • (kinetic energy of a molecule)
  • Size
    • it is tiny, because each degree of temperature rise adds only a very small amount of kinetic energy to one molecule.

Section 6 of 9Assumptions of the kinetic theory

  • Molecules
    • identical, hard, perfectly elastic spheres in continuous random motion, in very large numbers.
  • Forces
    • there are no attractive or repulsive forces between molecules, except during collisions.
  • Volume
    • the volume of the molecules is negligible compared with the container.
  • Collisions
    • their duration is negligible compared with the time between them, and they are perfectly elastic with Newton's laws applying.
  • External forces
    • gravity and other external forces are ignored.

Section 7 of 9Deriving pV = ⅓Nm⟨c²⟩

  • Single molecule
    • a molecule of mass m with speed c along the x-axis rebounds elastically from a wall of a cube of side l.
    • Its momentum change is −mc − mc = −2mc, so the wall receives +2mc.
  • Time between collisions
    • the molecule crosses the cube and back, a distance 2l, so the time is 2l/c.
  • Force
    • F = Δp/Δt = 2mc ÷ (2l/c) = mc2/l.
      What the symbols mean: F = force (N) · Δt = time the force acts (s) · Δp = change in momentum (kg m s−1); FΔt is the impulse (N s)
  • Pressure from one molecule
    • p = F/l2 = mc2/l3.
  • N molecules
    • p = Nm⟨c2⟩/l3, using the mean square speed ⟨c2⟩, since each molecule has a different velocity.
  • Three dimensions
    • ⟨c2⟩ = ⟨cx2⟩ + ⟨cy2⟩ + ⟨cz2⟩, and random motion makes the three equal, so only ⅓⟨c2⟩ acts on a given wall.
  • Result
    • with V = l3, pV = ⅓Nm⟨c2⟩.
      What the symbols mean: p = pressure (Pa) · V = volume (m3) · N = number of molecules · m = mass of one molecule (kg) · <c2> = mean square speed (m2 s−2)

Section 8 of 9Density form and r.m.s. speed

  • Density form
    • since ρ = Nm/V, the equation becomes p = ⅓ρ⟨c2⟩.
  • Mean square speed
    • ⟨c2⟩ is the average of the squares of the molecular speeds, in m2 s−2.
  • Root-mean-square speed
    • cr.m.s. = √⟨c2⟩, which is not the same as the mean speed.
  • Mean velocity
    • the average velocity of all molecules is zero, because directions cancel, while squaring keeps ⟨c2⟩ non-zero.

Section 9 of 9Average kinetic energy of a molecule

  • Combining
    • equating pV = NkT with pV = ⅓Nm⟨c2⟩ gives m⟨c2⟩ = 3kT.
  • Average translational kinetic energy of one molecule
    • Ek = ½m⟨c2⟩ = (3/2)kT.
  • All N molecules
    • total Ek = (3/2)NkT.
  • Temperature link
    • the mean kinetic energy is proportional to the thermodynamic temperature, so ⟨c2⟩ ∝ T and cr.m.s. ∝ √T for a fixed gas.

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