Ideal Gases and Thermal Expansion
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
- 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