Thermal physics (A Level)

15. Ideal Gases

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15. Ideal Gases — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Use the mole, the gas laws and pV = nRT, and derive and apply the kinetic theory result pV = ⅓Nm⟨c²⟩ and the average molecular kinetic energy.Mole = the amount of s…

Avogadro constant NA
Number of moles
Equal volumes
Molar mass Mr
Number of particles
Proportionality

Section 1 of 11The mole and Avogadro's constant

Mole

Mole the amount of substance containing as many particles as there are atoms in 12 g of carbon-12.

Amount of substance is an SI base quantity with the unit mol.

Avogadro constant NA

Avogadro constant NA the number of atoms in 12 g of carbon-12, 6.02 × 10²³ mol⁻¹.

A carbon-12 atom has mass 12 u = 1.99 × 10⁻²⁶ kg, so 0.012 kg contains 6.02 × 10²³ atoms.

Number of moles

Number of moles n = N/N_A, where N is the number of particles.

Equal volumes

Equal volumes gases at the same temperature and pressure with equal volumes contain equal numbers of molecules.

Section 2 of 11Molar mass

Molar mass

Molar mass Mr

Molar mass Mr the mass in grams of one mole of a substance, in g mol⁻¹.

One mole of an element has a mass equal to its relative atomic mass in grams.
For a compound, add the relative atomic masses; water is 2 × 1 + 16 = 18 g mol⁻¹.

Equation

Equation n = m/M_r, with mass m in grams.
What the symbols mean: n = amount of substance (mol) · m = mass (g) · M or M_r = molar mass (g mol⁻¹, numerically equal to M_r)

Number of particles

Number of particles multiply the moles by N_A.

Section 3 of 11The gas laws

Proportionality

Proportionality for a fixed mass of ideal gas, pV ∝ T, with T the thermodynamic temperature in kelvin.

Kelvin

Kelvin every gas calculation needs T in K.

Constant	Relationship	Explanation
Volume	p ∝ T (p₁/T₁ = p₂/T₂)	Hotter molecules have more kinetic energy and hit the walls harder and more often
Pressure	V ∝ T (V₁/T₁ = V₂/T₂)	Faster molecules push further apart, so the gas expands
Temperature	p ∝ 1/V (p₁V₁ = p₂V₂)	A smaller volume means more frequent collisions with the walls

Section 4 of 11The ideal gas equation

Equation of state

Equation of state pV = nRT, with p in Pa, V in m³, n in moles, R = 8.31 J K⁻¹ mol⁻¹ and T in K.
What the symbols mean: p = pressure (Pa) · V = volume (m³) · n = amount of gas (mol) · R = 8.31 J mol⁻¹ K⁻¹ · T = temperature (K = °C + 273)

Molecular form

Molecular form pV = NkT, where N is the number of molecules and k is the Boltzmann constant, 1.38 × 10⁻²³ J K⁻¹.

Ideal gas

Ideal gas a gas that obeys pV = nRT at all pressures, volumes and temperatures.

Units

Units convert cm³ to m³ (× 10⁻⁶) and °C to K; R and k are given on the data sheet.

Section 5 of 11The Boltzmann constant

Definition

Definition k = R/N_A = 8.31 ÷ (6.02 × 10²³) = 1.38 × 10⁻²³ J K⁻¹.

Link

Link it connects a microscopic property to a macroscopic one (temperature).
- (kinetic energy of a molecule)

Size

Size it is tiny, because each degree of temperature rise adds only a very small amount of kinetic energy to one molecule.

Section 6 of 11Assumptions of the kinetic theory

Molecules

Molecules identical, hard, perfectly elastic spheres in continuous random motion, in very large numbers.

Forces

Forces there are no attractive or repulsive forces between molecules, except during collisions.

Volume

Volume the volume of the molecules is negligible compared with the container.

Collisions

Collisions their duration is negligible compared with the time between them, and they are perfectly elastic with Newton's laws applying.

External forces

External forces gravity and other external forces are ignored.

Section 7 of 11Deriving pV = ⅓Nm⟨c²⟩

Deriving pV = ⅓Nm⟨c²⟩

Single molecule

Single molecule a molecule of mass m with speed c along the x-axis rebounds elastically from a wall of a cube of side l.
Its momentum change is −mc − mc = −2mc, so the wall receives +2mc.

Time between collisions

Time between collisions the molecule crosses the cube and back, a distance 2l, so the time is 2l/c.

Force

F = Δp/Δt = 2mc ÷ (2l/c) = mc²/l.
What the symbols mean: F = force (N) · Δt = time the force acts (s) · Δp = change in momentum (kg m s⁻¹); FΔt is the impulse (N s)

Pressure from one molecule

Pressure from one molecule p = F/l² = mc²/l³.

N molecules

N molecules p = Nm⟨c²⟩/l³, using the mean square speed ⟨c²⟩, since each molecule has a different velocity.

Three dimensions

⟨c²⟩ = ⟨c_x²⟩ + ⟨c_y²⟩ + ⟨c_z²⟩, and random motion makes the three equal, so only ⅓⟨c²⟩ acts on a given wall.

Result

Result with V = l³, pV = ⅓Nm⟨c²⟩.
What the symbols mean: p = pressure (Pa) · V = volume (m³) · N = number of molecules · m = mass of one molecule (kg) · <c²> = mean square speed (m² s⁻²)

Section 8 of 11Density form and r.m.s. speed

Density form

Density form since ρ = Nm/V, the equation becomes p = ⅓ρ⟨c²⟩.

Mean square speed

⟨c²⟩ is the average of the squares of the molecular speeds, in m² s⁻².

Root-mean-square speed

Root-mean-square speed c_r.m.s. = √⟨c²⟩, which is not the same as the mean speed.

Mean velocity

Mean velocity the average velocity of all molecules is zero, because directions cancel, while squaring keeps ⟨c²⟩ non-zero.

Section 9 of 11Average kinetic energy of a molecule

1Combining

Combining equating pV = NkT with pV = ⅓Nm⟨c²⟩ gives m⟨c²⟩ = 3kT.

2Average translational kinetic energy of one molecule

E_k = ½m⟨c²⟩ = (3/2)kT.

3All N molecules

All N molecules total E_k = (3/2)NkT.

4Temperature link

Temperature link the mean kinetic energy is proportional to the thermodynamic temperature, so ⟨c²⟩ ∝ T and c_r.m.s. ∝ √T for a fixed gas.

Section 10 of 11Key terms

Term	Meaning
Mole	The amount of substance containing as many particles as there are atoms in 12 g of carbon-12.
Avogadro constant	The number of particles in one mole, 6.02 × 10²³ mol⁻¹.
Molar mass	The mass of one mole of a substance in g mol⁻¹.
Ideal gas	A gas that obeys pV = nRT at all pressures, volumes and temperatures.
Boltzmann constant	k = R/N_A = 1.38 × 10⁻²³ J K⁻¹.
Mean square speed	The average of the squares of the speeds of the molecules.
Root-mean-square speed	The square root of the mean square speed.

Section 11 of 11Exam technique

Using temperature in °C in gas equations — T must be in kelvin.

Forgetting to convert volume to m³ — 1 cm³ = 10⁻⁶ m³.

Confusing mean speed with r.m.s. speed — r.m.s. is the square root of the mean of the squares.

Using (3/2)kT as the total kinetic energy of the gas — it is for one molecule, so multiply by N.

Mixing up n and N — n counts moles (use R) and N counts molecules (use k).

Try it in the virtual labPressure law and absolute zero \n