Kinetic Theory

The idea that matter is composed of indivisible particles goes back to ancient atomic hypotheses, but it was John Dalton's atomic theory that gave it a scientific basis: every element is made of identical atoms, and compounds form when atoms of different elements combine in fixed ratios.

Avogadro's Law states that equal volumes of all gases, under the same conditions of temperature and pressure, contain an equal number of molecules. This number is called the Avogadro number, N_A = 6.023 × 10²³ per mole, and it is the same for every gas, liquid, or solid.

A mole of a substance is the amount that contains N_A elementary entities (atoms or molecules). If M is the mass of a sample and M₀ is its molar mass (mass of one mole), the number of moles is μ = M/M₀, and the total number of molecules is N = μN_A.

In gases, molecules are far apart (average separation much larger than molecular size) and move about freely, whereas in liquids and solids they are closely packed and strongly interacting. This difference in intermolecular spacing and force explains why gases are easily compressed while solids and liquids are nearly incompressible.

Worked Example

Given: Molar mass of oxygen gas, O₂, M₀ = 32 g/mol; Avogadro number N_A = 6.023 × 10²³ mol^-1. Find the mass of a single oxygen molecule.

Formula: mass of one molecule, m = M₀/N_A

Substitution: m = 32 g mol^-1 ÷ 6.023 × 10²³ mol^-1 = 5.313 × 10^-23 g

Result: m = 5.313 × 10^-23 g = 5.313 × 10^-26 kg per O₂ molecule.

Experiments on gases at low pressure show that they all obey similar simple laws, regardless of their chemical identity — this is why they are called ideal or perfect gases in this limit.

Boyle's Law: For a fixed mass of gas at constant temperature, the pressure is inversely proportional to the volume, i.e. PV = constant. On a P–V graph, this gives a rectangular hyperbola called an isotherm; a higher-temperature isotherm lies further from the origin.

Charles' Law: For a fixed mass of gas at constant pressure, the volume is directly proportional to the absolute (Kelvin) temperature, i.e. V/T = constant. This law is the basis for defining the Kelvin scale, since extrapolated V–T lines for different gases all meet at V = 0, T = 0.

Combining Boyle's law, Charles' law and Avogadro's law gives the ideal gas equation (equation of state of a perfect gas):

PV = μRT, where μ is the number of moles and R is the universal gas constant, R = 8.314 J mol^-1 K^-1. Since μ = N/N_A, this can also be written as PV = NkT, where k = R/N_A = 1.38 × 10^-23 J/K is the Boltzmann constant.

Worked Example

Given: A gas at pressure P₁ = 1.0 × 10⁵ Pa occupies volume V₁ = 4 L. It is compressed isothermally to V₂ = 2 L. Find the new pressure P₂.

Formula: Boyle's law: P₁V₁ = P₂V₂

Substitution: (1.0 × 10⁵ Pa)(4 L) = P₂(2 L)

Result: P₂ = 2.0 × 10⁵ Pa.

The kinetic theory of gases explains macroscopic properties like pressure and temperature in terms of the microscopic motion of molecules. It rests on the following assumptions:

1. A gas consists of a very large number of identical molecules in continuous, random motion.
2. The size of a molecule is negligible compared to the average distance between molecules.
3. Molecules exert no force on one another except during a collision (no intermolecular forces otherwise).
4. Collisions between molecules, and between molecules and the walls, are perfectly elastic and of negligible duration.
5. Between collisions, molecules move in straight lines obeying Newton's laws of motion.
6. The molecular density (number of molecules per unit volume) is uniform throughout the gas.

Pressure of an ideal gas: Molecules colliding elastically with the walls of the container transfer momentum, and this rate of momentum transfer per unit area is the pressure. A careful calculation (summing the effect of molecules moving along all directions inside a cubical container) gives:

P = (1/3) (N/V) m⟨v²⟩ = (1/3) ρ⟨v²⟩

where N is the number of molecules, V is the volume, m is the mass of one molecule, ρ = Nm/V is the mass density of the gas, and ⟨v²⟩ is the mean square speed of the molecules. Equivalently, PV = (1/3) M⟨v²⟩, where M = Nm is the total mass of gas. The square root of ⟨v²⟩ is called the root mean square (rms) speed, v_rms.

Worked Example

Given: Density of nitrogen gas at a certain condition, ρ = 1.25 kg/m³; pressure P = 1.01 × 10⁵ Pa. Find the rms speed of the molecules.

Formula: From P = (1/3)ρv²_rms, we get v_rms = √(3P/ρ)

Substitution: v_rms = √[3 × (1.01 × 10⁵ Pa) / (1.25 kg/m³)] = √(2.424 × 10⁵)

Result: v_rms ≈ 492 m/s.

Combining the kinetic-theory pressure relation PV = (1/3)Nm⟨v²⟩ with the ideal gas equation PV = NkT gives a direct link between temperature and molecular motion:

(1/3)Nm⟨v²⟩ = NkT ⟹ (1/2)m⟨v²⟩ = (3/2)kT

The left side, (1/2)m⟨v²⟩, is the average translational kinetic energy per molecule. So this equation shows that the average kinetic energy of a gas molecule depends only on its absolute temperature T, and not on the pressure, volume, or the nature (identity) of the gas. This is the kinetic interpretation of temperature.

A useful consequence: at the same temperature, all ideal gases have the same average molecular kinetic energy, but since KE = (1/2)m⟨v²⟩, a lighter molecule must move faster than a heavier one to have the same average KE. Hence v_rms = √(3kT/m) = √(3RT/M₀), where M₀ is the molar mass.

This interpretation also gives a physical meaning to absolute zero: as T → 0 K, the average kinetic energy of molecules tends to zero, i.e. molecular motion (in the ideal gas limit) would cease entirely.

Worked Example

Given: Hydrogen gas (H₂) at T = 300 K; molar mass M₀ = 2 × 10^-3 kg/mol; R = 8.314 J mol^-1K^-1. Find v_rms.

Formula: v_rms = √(3RT/M₀)

Substitution: v_rms = √[3 × 8.314 × 300 / (2 × 10^-3)] = √(3.7413 × 10⁶)

Result: v_rms ≈ 1934 m/s ≈ 1.93 × 10³ m/s.

A molecule can store energy in several independent ways, called degrees of freedom (dof) — translational motion along the three axes, rotational motion about axes, and vibrational motion (kinetic plus potential energy of the vibrating bond).

The law of equipartition of energy states that, for a system in thermal equilibrium, the total energy is distributed equally among all its degrees of freedom, and each quadratic degree of freedom contributes an average energy of (1/2)kT per molecule (equivalently (1/2)RT per mole).

Applying this to different types of molecules:

• Monatomic gas (e.g. He, Ar): only 3 translational dof. Average energy per molecule = 3 × (1/2)kT = (3/2)kT.
• Diatomic gas, rigid rotor (e.g. N₂, O₂ at moderate temperatures): 3 translational + 2 rotational dof = 5 dof. Average energy per molecule = (5/2)kT.
• Diatomic gas with vibration (at high temperature): 5 dof plus 1 vibrational mode, which contributes 2 dof (kinetic + potential energy), giving 7 dof in total. Average energy per molecule = (7/2)kT.
• Polyatomic (non-linear) gas: 3 translational + 3 rotational dof, plus 2 dof for every vibrational mode present.

For one mole of gas, the total internal energy is U = (f/2)RT, where f is the number of degrees of freedom per molecule.

Worked Example

Given: 2 mol of a diatomic gas (treated as a rigid rotor, no vibration) at T = 300 K. Find the total internal energy.

Formula: U = μ(5/2)RT (f = 5 for a rigid diatomic molecule)

Substitution: U = 2 × (5/2) × 8.314 × 300

Result: U = 12471 J ≈ 1.247 × 10⁴ J.

The molar specific heat at constant volume, C_v, is the heat needed to raise the temperature of one mole of gas by 1 K while keeping the volume fixed. Since no work is done at constant volume, all the heat goes into internal energy: C_v = dU/dT.

The molar specific heat at constant pressure, C_p, is the heat needed to raise the temperature of one mole of gas by 1 K at constant pressure. Here, some heat also does work in expanding the gas, so C_p > C_v. Using the ideal gas equation, one can show Mayer's relation: C_p − C_v = R, valid for any ideal gas.

Using U = (f/2)RT from the law of equipartition, C_v = (f/2)R and C_p = (f/2 + 1)R for one mole. The ratio γ = C_p/C_v is an important quantity:

• Monatomic gas (f = 3): C_v = (3/2)R, C_p = (5/2)R, γ = 5/3 ≈ 1.67.
• Diatomic gas, rigid (f = 5): C_v = (5/2)R, C_p = (7/2)R, γ = 7/5 = 1.4.
• Diatomic gas with vibration (f = 7): C_v = (7/2)R, C_p = (9/2)R, γ = 9/7 ≈ 1.29.

These predicted values agree well with experimental specific heats of gases at ordinary temperatures, which is a major success of the kinetic theory and the equipartition law.

Worked Example

Given: 2 mol of a diatomic gas (rigid rotor) is heated at constant pressure so that its temperature rises by ΔT = 50 K. R = 8.314 J mol^-1 K^-1. Find the heat supplied.

Formula: Q = μC_pΔT, with C_p = (7/2)R for a rigid diatomic gas

Substitution: C_p = (7/2)(8.314) = 29.099 J mol^-1K^-1; Q = 2 × 29.099 × 50

Result: Q = 2909.9 J ≈ 2.91 × 10³ J.