B.3 Gas laws

Pressure

Pressure is perpendicular force distributed over area:

$$P=\frac{F_{\perp}}{A}$$

Its SI unit is the pascal, $$1\,\mathrm{Pa}=1\,\mathrm{N\,m^{-2}}$$.

Use the normal component

Only the component of force perpendicular to the surface contributes to pressure on that surface. A tangential component produces shear rather than normal pressure.

Read the proportionality

At fixed force, doubling area halves pressure. At fixed area, doubling the perpendicular force doubles pressure. Pressure is a scalar even though the force producing it has direction.

Common trap

Do not use total force if the force is angled. Resolve it perpendicular to the surface first, and keep area in square metres.

Amount of substance

The amount of substance $$n$$ counts how many groups of $$N_A$$ particles are present:

$$n=\frac{N}{N_A}$$

where N is the number of particles and $$N_A$$ is the Avogadro constant.

Connect mass to moles

If molar mass is M, then

$$n=\frac{m}{M}$$

Use matching units for m and M. Once n is known, the number of particles is $$N=nN_A$$.

Use ratios efficiently

For equal numbers of particles, the samples contain equal amounts in moles even if their masses differ. For isotope or element comparisons, calculate moles before comparing particle counts.

Common trap

Do not confuse N (number of particles) with $$N_A$$ (particles per mole) or n (amount in moles).

Ideal-gas model

An ideal gas is a kinetic-theory model: particles are in constant random motion, occupy negligible volume compared with the container, and interact negligibly except during collisions. Collisions are treated as elastic.

Connect microscopic and macroscopic quantities

Temperature is related to average translational kinetic energy. Pressure comes from momentum transfer when particles collide with the container walls. More energetic or more frequent collisions can increase pressure.

It is an approximation

Real gases have finite-size particles and intermolecular forces. The ideal model is most reliable when particles are far apart and interactions are relatively unimportant.

Common trap

The model does not say every particle has the same speed. It uses a distribution of speeds and averages over many particles.

One relation for a fixed amount of gas

The constant-pressure, constant-volume and constant-temperature observations combine to give

$$\frac{PV}{T}=\text{constant}$$

provided the amount of gas is unchanged.

Compare two states

$$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$

Temperatures must be absolute. If pressure and temperature stay fixed, volume is fixed; if volume and temperature stay fixed, pressure is fixed.

Use the constraint first

Name what is constant before choosing a simplified law: Boyle-type behaviour uses constant T, Charles-type behaviour uses constant P, and pressure–temperature behaviour uses constant V.

Common trap

Do not apply $$PV/T=\text{constant}$$ after gas has been added or removed unless the problem explicitly tracks the changed amount of gas.

Molar form

For an ideal gas,

$$PV=nRT$$

Use n in mol, T in kelvin and $$R=8.31\,\mathrm{J\,K^{-1}\,mol^{-1}}$$.

Particle form

Using the number of particles N, the same law is

$$PV=Nk_BT$$

where $$k_B$$ is the Boltzmann constant. The forms are equivalent because $$R=N_Ak_B$$ and $$N=nN_A$$.

Choose the form from the data

Use the molar form when amount of substance is given. Use the particle form when a question asks for the number of molecules or gives microscopic quantities. Rearrange before substituting, for example $$N=PV/(k_BT)$$.

Common trap

Do not use Celsius in either equation, and do not mix n with N. Pressure must be in pascals and volume in cubic metres for SI results.

Pressure from collisions

Gas particles collide with a wall and change momentum. The wall exerts a force on the particles; by Newton’s third law, the particles exert an equal and opposite force on the wall. Pressure is this normal force per unit area.

Kinetic-theory relation

For an ideal gas,

$$P=\frac13\rho\overline{v^2}$$

where ρ is gas density and $$\overline{v^2}$$ is the mean square molecular speed. The speed in this equation is not simply the square of the average speed.

Read the trends

Greater molecular speed increases momentum change per collision and collision rate, increasing pressure. At fixed speed, greater density means more mass per unit volume and therefore greater pressure.

Common trap

A single particle’s collision force is not the total gas force. Pressure is a statistical average over many collisions on the surface.

Internal energy model

For an ideal monatomic gas, internal energy is the total random translational kinetic energy of its particles:

$$U=\frac32Nk_BT=\frac32nRT$$

What is included

The model includes translational kinetic energy only. It neglects intermolecular potential energy and does not include rotational or vibrational molecular energy.

Read the dependence

At fixed amount of gas, U is proportional to T. At fixed temperature, U is proportional to N or n. Particle mass does not appear directly in $$U=\frac32Nk_BT$$.

Common trap

Equal mass samples of different monatomic gases do not necessarily have equal internal energy: compare their number of particles or moles at the same temperature.

Good approximation

A real gas is closest to the ideal-gas model at relatively low pressure and low density, where particles are far apart and intermolecular forces and particle volume are small compared with the container volume.

Temperature matters

Higher temperature gives particles more kinetic energy, making attractive interactions less important. Low temperature increases the importance of intermolecular attractions and can bring the gas closer to condensation.

Where it fails

At high pressure or high density, particles are crowded: their finite size and interactions matter. Near phase changes, the ideal model is especially unreliable.

Use the full condition

Do not state only “high temperature”. The reliable region is generally high temperature together with low pressure or low density.