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Ideal Gases and Thermal Expansion

IDEAL GASES
The Avogadro Constant
  • Avogadro constant (NA) is the number of atoms present in 12 g of Carbon – 12
  • A mole is the amount of substance containing the same number of particles as in 12 g of Carbon – 12.
Equation of State
  • Ideal gas is a gas which obeys the ideal gas equation for all values of P, V, and T.
  • pV = nRT; where n = amount of substance (number of moles)
  • Conditions for the equation to be valid:
    • A fixed amount of gas
    • It must be an ideal gas
  • Boyle’s Law: P ∝ 1/V, hence pV = constant
  • Charles’ Law: V ∝ T, hence V/T = constant
  • ∴ Ideal gas equation = P1V1/T1 = P2V2/T2
Kinetic Theory of Gases
  • Molecular movement causing pressure:
    • Molecules hit and rebound off the walls of the container.
    • The change in momentum gives rise to force.
    • Many impulses averaged to give a constant force, and hence pressure.
  • From the observation of a smoke cell under a microscope, the Brownian (haphazard, random) motion of particles provides evidence of movement of gas molecules.

Basic Assumptions of the Kinetic Theory of Gases

  • Gas contains a large number of particles.
  • They possess negligible inter-molecular forces of attraction.
  • The volume of particles are negligible compared to that of the container.
  • The collisions between the particles are perfectly elastic.
  • There is no time spent in collisions.
  • The average K.E. is directly proportional to the absolute temperature.
Molecular Movement and Pressure

  • Consider a cube of space, with length L, and a particle moving with velocity, c.
  • When the particle collides with a wall, the velocity is reversed and the change is Δp = m(c – (-c)) = 2mc
  • The distance moved by the particle is L + L = 2L
  • Using the speed-distance formula, time between collisions, t = 2L/c
  • Rate of change of momentum (i.e., force),
    F = Δp/t = 2mc/2L/c = mc2/L
  • Using the above quantities to find pressure:
    • P = F/A = mc2/L2 = mc2/L3 = mc2/v
  • Rearrange to pV = mc2
  • Considering N particles in 3D (hence the 1/3) with average speed <c>:
    • pV = 1/3 Nm<c>2 or p = 1/3 ρ<c>2
  • Mean square velocity, <c>2 is the mean value of the square of the velocities of the molecules.
Kinetic Energy of a Molecule
  • By equating the two formulae in pV, finding a relationship between Ek and T:
    • nRT = 1/3 Nm<c>2
    • 3nRT/N =  m<c>2
  • Avogadro’s constant, NA = N/n
  • 3RT/2NA = 1/2 m<c>2
  • Boltzmann’s constant, k = R/NA
  • 3/2 kT = Ek
Examples
  1. A balloon is filled with helium gas at a pressure of 1.1 × 105 Pa and temperature of 25 °C. The balloon has a volume of 6.5 × 104 cm3. Helium may be assumed to be an ideal gas. Determine the number of gas atoms in the balloon.
    • Firstly, calculate number of moles: pV = nRT; n = pV/RT
    • Substitute the information given, converting to standard units, i.e. m3 and Kelvin:
      • n = 1.1 × 105 × 6.5 × 104 × (10-2)3/8.31 × (25 + 273) = 2.89
      • Use the relationship between Avogadro’s constant NA and the number of moles, n, to find the number of particles, N:
        • N = NA × n
        • N = 6.02 × 1023 × 2.89 = 1.75 × 1024
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