IDEAL GASES

N (total number of atoms/ ions/ molecules in a substance) =

...n (number of moles of the substance) x N_A (Avogadro constant) (6.02e23)

m (mass of a substance) =

...n (number of moles of the substance) x M (molar mass of the substance)

A model that describes all substances as made of atoms/ ions/ molecules arranged differently depending on the phase of the substance

Z = atoms/ molecules/ ions

.

An ideal gas contains a very large number of Z moving in random directions with random speeds.

The Z of the ideal gas occupy a negligible volume compared to the volume of the ideal gas.

The collisions of Z with each other and the container walls are perfectly elastic (no E_k is lost).

The time of collisions between the Z is negligible compared to the time between the collisions.

Electrostatic forces between Z are negligible except during collisions.

For a fixed amount of gas at a constant volume, pressure is directly proportional to absolute temperature

p / T = constant

p₁ / T₁ = p₂ / T₂

For a fixed amount of gas at a constant temperature, pressure is inversely proportional to volume

pV = constant

p₁ / V₁ = p₂ / V₂

For a fixed amount of gas at a constant pressure, volume is directly proportional to absolute temperature

V / T = constant

_V1 / T₁ = V₂ / T₂

pV / T = constant

p₁V₁ / T₁ = p₂V₂ / T₂

pV = nRT

.

p = pressure (Pa)

V = volume (m³)

n = number of moles (mol)

R = molar gas constant (8.31 JK^-1mol^-1)

T = temperature (K) (^oC + 273)

• Straight line

• Through the origin

• pV ∝ T

• Gradient = nR

pressure exerted by the gas x volume of the gas =

...1/3 x number of particles in the gas x mass of each particle x mean square speed of the particles

.

p V = 1/3 N m c²

The distribution of the speeds of particles in a gas at a given temperature

.

↑ temperature of gas

= ↑ range of speeds of particles

= ↑ most probable speed and root mean square speed

= distribution more spread out

k = R / N_A

= 1.38e-23 JK^-1

pV = nRT and k = R / N_A

∴ pV = nkN_AT

.

N = nNA

∴ pV = NkT

E_k = 3/2 k T

= (2.07e-23) T

p V = 1/3 N m c² and pV = NkT

1/3 N m c² = NkT

1/3 m c² = kT

2/3 (1/2 m c²) = kT

1/2 m c² = 3/2 kT

.

E_{k av} = 1/2 m c² 

∴ E_k ∝ T

Internal energy of a gas = kinetic energy + potential energy

...Electrostatic forces between particles in an ideal gas are negligible except during collisions

......No electrical potential energy

.........Internal energy = kinetic energy

............2 x temperature = 2 x average kinetic energy = 2 x internal energy