Kinetic Theory of Gases

A balloon is pushed outward by nothing you can see. Kinetic theory explains pressure, temperature and the gas laws from one picture: a huge number of tiny molecules flying about and colliding.

From molecules to gas laws#

Real gases at low density obey pV = nRT closely; an ideal gas obeys it exactly. A gas is made of molecules, and its measurable behaviour comes from what those molecules do. Kinetic theory builds a simple model of molecular motion and shows it leads to the ideal gas laws.

The assumptions#

1. A gas is made of molecules moving randomly in all directions.
2. A molecule is much smaller than the average spacing between molecules.
3. Molecules exert no forces on each other or on the walls except during collisions.
4. All collisions (molecule–molecule and molecule–wall) are perfectly elastic and take negligible time.
5. Molecules obey Newton’s laws.
6. Left alone, the gas reaches a steady state: the density, and the way molecules are spread over different velocities, are the same everywhere, in every direction, at all times.

Why we can ignore molecule–molecule collisions#

Assumption 6 has a handy consequence. Suppose a collision changes the velocity of some molecule in a small region. Since the number of molecules with each velocity in that region never changes, another collision must, on average, give some nearby molecule the velocity the first one lost. So in calculations we can pretend each molecule simply keeps its velocity.

How realistic is it?#

• At 0.1 atm and room temperature, molecules are about 100 times smaller than their spacing.
• Real molecules do attract each other, but weakly when they are far apart.
• Collisions are elastic unless the temperature is so high that molecules are damaged; collisions with walls are elastic once the walls are at the gas temperature.
• With about 6 × 10²³ molecules in a mole, a steady, uniform distribution is almost guaranteed.

Count the knocks of molecules on a wall, add up their momentum, and out comes the pressure of a gas.

One molecule, one wall#

A gas fills a cubical box of side L. Take one molecule with velocity components vₓ, v_y, v_z. When it strikes the face A₁ perpendicular to x, vₓ reverses and the other components are unchanged (elastic collision). Its momentum changes by −2mvₓ, so the wall receives

It then crosses to the opposite face and back, a distance 2L in x, before hitting A₁ again. So it strikes A₁ every 2L/vₓ seconds, that is vₓ/2L times per second. The average force it exerts on A₁ is

All the molecules#

Adding up, F = (m/L)Σvₓ². Since no direction is special, Σvₓ² = Σv_y² = Σv_z² = ⅓Σv². With N molecules, total mass M = mN and density ρ = M/L³, the pressure F/L² is

The rms speed#

The square root of the mean square speed is the root-mean-square (rms) speed:

Translational kinetic energy

The total translational kinetic energy of the molecules is

What a thermometer really measures, according to kinetic theory, is how fast the molecules are flying about.

Temperature is kinetic energy#

For an ideal gas the internal energy is just the translational kinetic energy of its molecules, ½Mv²_rms. At fixed volume, pV = ⅓Mv² gives p/p_tr = v²/v²_tr; and on the absolute scale p/p_tr = T/273.16 K. So

Mixtures

In a mixture, collisions pass energy from faster to slower molecules until, in equilibrium, the average kinetic energy is the same for every kind:

Boltzmann’s constant and the gas equation#

Putting v² = (v²_tr/273.16 K)T into pV = ⅓Nmv² gives pV = N(⅓mv²_tr/273.16 K)T. The bracket contains ½mv²_tr, the same for every gas at 273.16 K, so it is a universal constant, k:

With N = nN_A (N_A = 6.02 × 10²³ mol⁻¹):

Speeds and energies in terms of T#

Comparing pV = ⅓Nmv²_rms with pV = NkT:

The average speed is a little smaller than the rms speed:

Boyle, Charles, Avogadro, Graham and Dalton found their gas laws by experiment. Kinetic theory derives every one of them from pV = ⅓Nmv²_rms and v²_rms ∝ T.

Boyle’s law#

At a fixed temperature, v²_rms is fixed; for a given mass of gas, N and m are fixed. So pV = ⅓Nmv²_rms is constant:

Charles’s laws#

At fixed pressure, V ∝ v²_rms ∝ T. At fixed volume, p ∝ v²_rms ∝ T (which is really the definition of the absolute scale).

Avogadro’s law#

Take equal volumes of two gases at the same pressure and temperature. From pV = ⅓Nmv² for each,

Same temperature means equal average kinetic energies, ½m₁v₁² = ½m₂v₂². Dividing:

Graham’s law of diffusion#

When two gases at the same pressure and temperature diffuse into each other, it is reasonable that each diffuses at a rate proportional to its rms speed. Since v_rms = √(3p/ρ), at equal pressures:

Dalton’s law of partial pressures#

Pressure comes from molecules hitting the walls, and the total force is just the sum of the forces from each gas’s molecules. Each part, F₁/A, F₂/A…, is the pressure that gas would exert if it filled the container alone:

The rms speed is an average. Inside a gas, some molecules dawdle and a few race. Maxwell worked out the whole spread, and Brown saw its effects through a microscope.

Maxwell’s speed distribution#

Oxygen at 300 K has an rms speed of about 480 m/s, but plenty of molecules are slower and plenty faster. If dN molecules have speeds between v and v + dv:

The peak of the curve is the most probable speed:

The long tail matters: a small fraction of molecules move many times faster than average. In nuclear fusion experiments, it is these fast particles that react.

Thermodynamic state and equation of state#

The relation between p, V and T is the equation of state: pV = nRT for an ideal gas. For a real gas the molecules’ size and attraction matter. Van der Waals proposed

Brownian motion#

In 1827 the botanist Robert Brown watched pollen grains in water under a microscope. Though the water was still, the grains jiggled and wandered randomly. Water molecules, too small to see, hit each grain from all sides; at any instant the kicks don’t quite balance, so the grain lurches one way, then another.

• The particles must be light: a floating log is too massive to respond to molecular kicks.
• Brownian motion grows with temperature, and is livelier in liquids of low viscosity and density.
• Einstein’s 1905 theory of Brownian motion let the size of molecules be worked out, strong evidence that atoms are real.

Kinetic theory ignores molecular attraction. Squeeze or cool a gas enough and attraction takes over: the gas condenses. This is the physics of vapour, evaporation and boiling.

Vapour and the critical temperature#

At low pressure and high temperature molecules are far apart and their attraction is weak compared with their kinetic energy. Compress the gas or cool it and the attraction grows in importance, until the gas liquefies. But above a certain temperature the molecules are so energetic that no pressure can liquefy the gas.

On a pressure–volume graph each curve at fixed temperature is an isotherm. Water vapour at 350°C begins to liquefy at about 163 atm; at 370°C, about 194 atm. Along the flat part vapour and liquid coexist; once all is liquid, the curve rises steeply because liquids barely compress. Above 374.1°C the isotherm never flattens.

Evaporation#

Leave ether in three dishes, one with a lid, one under a large jar, one open. After a few hours the lidded dish is nearly full, the jar has lost some, the open dish is almost empty. Molecules near the surface occasionally escape (evaporation). In a closed space they pile up until as many return to the liquid each second as leave it; in the open they drift away, and wind carries them off even faster, which is why clothes dry quicker on a windy day.

Saturated vapour#

In a closed space above a liquid, equilibrium is reached when evaporation and return balance. Inject more vapour and the extra condenses; take some away and more evaporates. The space can hold only a certain maximum amount of vapour at a given temperature.

Higher temperature means faster molecules and more escapes, so the SVP rises steeply with temperature:

Saturation doesn’t need a liquid present: keep injecting vapour into a closed vessel and condensation starts once the SVP is reached. Air holding the maximum vapour is “saturated”.

Boiling#

Heat a liquid and its molecules gain energy everywhere, not just at the surface. Eventually bubbles of vapour form throughout and rise out: boiling. Bubbles can survive only when the vapour pressure inside them can match the pressure on the liquid:

Evaporation versus boiling

Evaporation: only fast molecules at the surface escape, at any temperature. Boiling: molecules throughout the liquid form vapour, at one temperature for a given pressure.

Why grass is wet on a clear morning, why a humid day feels sticky, and why dry ice skips the liquid stage.

Dew point#

Air at 20°C containing vapour at 8.94 mm Hg is unsaturated: it would take 17.5 mm Hg to saturate it. Cool it to 10°C, where the SVP is 8.94 mm Hg, and it becomes saturated.

Humidity#

Absolute humidity is the mass of water vapour per unit volume of air, usually in g/m³. Relative humidity compares what is present with what would saturate the same air:

Measuring humidity: Regnault’s hygrometer#

Two test tubes, A and B, have silvered lower outer surfaces, each holding a thermometer. Tube A contains ether. Water drained slowly from a vessel draws air through the ether, evaporating it and cooling tube A. When A reaches the dew point, dew clouds its silver surface, while B stays shiny. Note A’s temperature; stop the airflow, let A warm, and note when the mist clears. The average is the dew point; B’s thermometer gives the air temperature. Then

Phase diagrams and the triple point#

The SVP curve, plotted against temperature, marks where liquid and vapour coexist. Above it only liquid is stable, below it only vapour. It is also the boiling point as a function of pressure. Similar curves mark where solid and liquid coexist (melting) and where solid and vapour coexist (sublimation). The three curves divide the p–T plane into solid, liquid and vapour regions and meet at the triple point, where all three phases coexist.

• Water: triple point at 4.58 mm Hg and 273.16 K (the fixed point of the kelvin scale).
• Carbon dioxide: 216.55 K and 5.11 atm. At 1 atm it can only be solid or vapour, so solid CO₂ turns straight to gas: dry ice.

Dew and fog#

On winter nights, grass, leaves and windowpanes cool by radiation below the air temperature. The air touching them reaches its dew point and droplets of dew condense. If the whole air mass cools enough, droplets form on dust in the air and float as fog.