#### Ideal Gases and Thermal Expansion

**IDEAL GASES**

__The Avogadro Constant__

__The Avogadro Constant__

- Avogadro constant (N
_{A}) 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__

__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

: P ∝**Boyle’s Law**^{1}/_{V}, hence pV = constant: V ∝ T, hence**Charles’ Law**^{V}/_{T}= constant- ∴ Ideal gas equation =
^{P1V1}/_{T1}=^{P2V2}/_{T2}

__Kinetic Theory of Gases__

__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__

__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}

- P =
- Rearrange to pV = mc
^{2} - 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}

- pV =
- Mean square velocity, <c>
^{2}is the mean value of the square of the velocities of the molecules.

__Kinetic Energy of a Molecule__

__Kinetic Energy of a Molecule__

- By equating the two formulae in pV, finding a relationship between E
_{k}and T:- nRT =
^{1}/_{3}Nm<c>^{2} ^{3nRT}/_{N}= m<c>^{2}

- nRT =
- Avogadro’s constant, N
_{A}=^{N}/_{n} ^{3RT}/_{2NA}=^{1}/_{2}m<c>^{2}- Boltzmann’s constant, k =
^{R}/_{NA} ^{3}/_{2}kT = E_{k}

__Examples__

__Examples__

- A balloon is filled with helium gas at a pressure of 1.1 × 10
^{5}Pa and temperature of 25 °C. The balloon has a volume of 6.5 × 10^{4}cm^{3}. 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. m
^{3}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 N
_{A}and the number of moles, n, to find the number of particles, N:- N = N
_{A}× n - N = 6.02 × 10
^{23}× 2.89 = 1.75 × 10^{24}

- N = N

- n =

- Firstly, calculate number of moles: pV = nRT; n =