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#### Ideal Gases and Thermal Expansion

##### IDEAL GASES
• 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