The macroscopic quantity we call pressure

Pressure is a scalar quantity: it measures the normal force exerted per unit area on a surface. In symbols,

One pascal is one newton per square metre.

Don’t skip the word perpendicular. When a gas molecule hits a wall, only the component of force normal to the wall contributes to the pressure. A shoe on a floor, water pushing on a dam, and gas in a cylinder all use the same definition; the microscopic origin of the force is what changes.

For a solid resting on a surface, the force is usually its weight spread over the contact area. Keep the force the same but make the contact area smaller, and the pressure increases — that’s why a sharp pin works so well. In a liquid, pressure at depth comes from the weight of liquid above and acts in all directions. In a gas, pressure comes from countless tiny collisions of molecules with the container walls. Each collision is brief, but there are so many of them that the wall feels a steady force.

A useful scale to remember is atmospheric pressure near sea level: about $1.0\times10^5\ \text{Pa}$. Everyday pressure differences can be much smaller or much larger than this; the equation above links those differences to measurable forces.

Counting particles without counting one by one

Amount of substance is a base physical quantity for the number of elementary entities present, expressed in moles. It is calculated using

The Avogadro constant has the exact value $N_A = 6.02214076\times10^{23}\ \text{mol}^{-1}$.

A mole contains exactly $6.02214076\times10^{23}$ specified elementary entities. These could be atoms, molecules, ions or electrons, so say clearly what you’re counting. One mole of helium atoms and one mole of nitrogen molecules have the same number of particles, but not the same mass.

Molar mass is the mass of one mole of a substance. In chemistry-style calculations the common unit is $g\ mol^{-1}$, while SI gas-law calculations usually work more safely with $kg\ mol^{-1}$. For example, for a molecular gas made of two atoms per molecule, the molecular molar mass refers to the whole molecule, not one atom.

This counting method connects the microscopic picture of a gas with its macroscopic behaviour. A pressure gauge doesn’t report on a single molecule; it shows the collective effect of roughly $10^{23}$ particles per mole.

Why we use an ideal gas model

Kinetic theory is a microscopic model that explains bulk gas behaviour by looking at many particles moving randomly and colliding with one another and with the container walls. Its strength is the link it makes between familiar mechanics — velocity, momentum, kinetic energy and force — and thermodynamic quantities such as pressure and temperature.

An ideal gas is a theoretical gas model in which the particles have negligible volume, move in continuous random motion, collide elastically, and exert no intermolecular forces except during collisions. A real gas is a physical gas whose particles have finite size and can exert attractive or repulsive intermolecular forces, especially when close together.

For any gas sample, we describe the macroscopic state using pressure $P$, volume $V$ ($\text{m}^3$), thermodynamic temperature $T$ (K), and amount of substance $n$. The ideal gas model predicts these bulk quantities from the average behaviour of a vast number of particles, rather than by trying to track every molecule separately.

The assumptions of the kinetic model are kept deliberately simple:

• the gas contains a very large number of identical particles;
• the particles move randomly in all directions;
• the total particle volume is negligible compared with the container volume;
• collisions between particles and with the walls are elastic;
• intermolecular forces are ignored except during collisions;
• collision times are negligible compared with the time between collisions;
• external forces, such as gravity, are ignored.

Evidence for random molecular motion is indirect, but it is strong. Diffusion is the spreading of particles from a region of higher concentration to lower concentration due to random molecular motion. Brownian motion is the irregular motion of small visible particles suspended in a fluid, caused by collisions with invisible molecules. These observations make the kinetic model feel less like a mathematical trick and more like a physical picture of what the gas is doing.

Models matter in physics because they simplify. That simplification has a cost: every model has a domain of validity. The ideal gas is not “true”; it is useful when its assumptions are close enough to reality.

The three restricted gas laws

The early gas laws were empirical laws: general relationships worked out from experiment before scientists had a detailed microscopic explanation. They apply to fixed amounts of gas, with one variable controlled at a time.

Boyle’s law is the empirical law that, for a fixed amount of gas at constant temperature, pressure is inversely proportional to volume:

On a pressure–volume graph, an isothermal change gives a curve. Compress the gas, and the pressure rises. If $P$ is plotted against $1/V$, the same relationship becomes a straight line.

Charles’s law is the empirical law that, for a fixed amount of gas at constant pressure, volume is directly proportional to thermodynamic temperature:

The pressure law is the empirical law that, for a fixed amount of gas at constant volume, pressure is directly proportional to thermodynamic temperature:

Temperature has to be in kelvin. Celsius temperature differences are fine, but Celsius temperatures themselves don't work in ratios such as $V/T$ or $P/T$. It's a small detail, but it can spoil an otherwise good answer.

The combined gas law

For a fixed amount of gas, the three restricted laws combine to give

Use this to compare two states of the same gas sample when pressure, volume and temperature all change. It is still not the full ideal gas equation, since the amount of gas is held constant.

Graph signatures for the empirical gas laws with fixed amount of gas.

Pressure–volume diagrams

A pressure–volume diagram is a graph showing changes of state of a gas, with pressure on the vertical axis and volume on the horizontal axis. For an ideal gas:

• an isothermal change appears as a curved line, with higher-temperature isotherms lying above lower-temperature ones;
• a constant-pressure change is a horizontal line;
• a constant-volume change is a vertical line.

Work done by a gas is an energy transfer that occurs when gas pressure moves a boundary such as a piston. On a pressure–volume diagram, expansion to the right shows the gas doing work on its surroundings; compression to the left shows work being done on the gas. This links directly to heat engines: high kinetic energy gas particles can push a piston or turbine blade, transferring energy from a hot source towards a colder sink while useful work is obtained.

Verifying the laws experimentally

To verify Boyle’s law, trap a fixed mass of gas, change its volume slowly, measure the pressure, and give the gas time to return to room temperature after each compression. If temperature has really been controlled, a plot of $P$ against $1/V$ should be linear.

To verify Charles’s law, keep the gas pressure constant, vary the temperature using a water bath, and use the length of a gas column as a quantity proportional to volume. A graph of length against temperature should be linear and can be extrapolated to estimate absolute zero.

To verify the pressure law, trap a fixed volume of gas, vary its temperature, and measure the pressure once thermal equilibrium is reached. The connecting tube to the pressure gauge should have a small volume, because gas in that tube may not be at the water-bath temperature.

Good data need a sensible range of temperatures or volumes, enough readings to show the trend, controlled environmental conditions, and graphs interpreted as lines or curves rather than just collections of points. Interpolation is safer than extrapolation; extrapolating towards absolute zero is useful historically, but it assumes the gas remains gaseous and ideal over a huge range.

One equation of state

An equation of state relates the state variables of a system in thermal equilibrium. For an ideal gas, the molar form is

This one equation includes Boyle’s law, Charles’s law and the pressure law as special cases. If $n$ and $T$ are constant, then $PV=\text{constant}$; if $n$ and $P$ are constant, then $V/T=\text{constant}$; if $n$ and $V$ are constant, then $P/T=\text{constant}$.

There is also a molecular form:

The two constants are related by

So $R$ gives the gas constant per mole, while $k_B$ gives it per particle. It’s a neat piece of physics: the same gas law works for a laboratory sample measured in moles or for a collection described molecule by molecule.

The product $PV$ has units of energy, since $\text{Pa m}^3=(\text{N m}^{-2})\text{m}^3=\text{N m}=\text{J}$. That isn’t a coincidence. In the kinetic model, $PV$ is linked to molecular kinetic energy.

In calculations, use SI units unless a ratio means conversion is not needed. Pressure in pascals, volume in cubic metres, and temperature in kelvin keep $R$ and $k_B$ consistent.

Collisions, momentum and pressure

Gas pressure doesn’t come from molecules continuously “pressing” on the walls. It comes from collisions, over and over again. When a molecule bounces off a wall, the component of its momentum perpendicular to that wall changes. The wall has to exert a force on the molecule to change its momentum; by Newton’s third law, the molecule exerts an equal and opposite force on the wall.

Here, mechanics connects directly with thermodynamics. Force is the rate of change of momentum, and pressure is force per unit area. Add together the effects of huge numbers of random collisions, and they show up as a smooth macroscopic pressure.

A full derivation starts with a molecule in a cubic box. It resolves the molecule’s velocity into components, averages over many molecules, and uses the fact that random motion is equally likely in all directions. The result you need is

Root mean square speed

Root mean square speed is found by squaring molecular speeds, averaging the squared values, then taking the square root. It is not the same as mean velocity. For a gas at rest as a whole, the mean velocity is zero because the directions cancel out. The rms speed, though, is not zero, because squaring removes the signs.

This equation helps explain several macroscopic observations qualitatively:

• increasing density at the same rms speed increases pressure because more mass of gas collides with each unit area;
• increasing temperature increases molecular kinetic energy and hence rms speed, so pressure rises if volume is fixed;
• decreasing volume for a fixed gas sample increases collision frequency with the walls, so pressure rises if temperature is fixed.

The factor $1/3$ comes from the three spatial directions. On average, only one third of the molecular motion contributes to collisions with any one pair of opposite walls.

Temperature as molecular kinetic energy

Translational kinetic energy is the kinetic energy linked to the motion of a particle’s centre of mass through space. For an ideal gas molecule, the average translational kinetic energy is related to temperature by

This idea matters throughout the topic. Temperature is not just “how hot something feels”; for an ideal gas, it measures the average translational kinetic energy of the particles. In this way, a microscopic quantity — molecular kinetic energy — becomes part of the development of the gas laws.

A monatomic gas is a gas whose particles are single atoms rather than multi-atom molecules. In an ideal monatomic gas, this model does not include internal molecular rotation or vibration, and there are no intermolecular potential energy stores. So its internal energy is simply the total translational kinetic energy of all the atoms:

Equivalently,

For a temperature change,

A temperature change in kelvin has the same numerical size as the temperature change in degrees Celsius.

High kinetic energy gas particles can do work when their collisions move a boundary. In a cylinder, hot gas at high pressure pushes a piston outwards. As the boundary moves, the particles lose some energy to the surroundings. This is the basic physics behind a heat engine: energy is transferred from a hot body to a cold body, with some of that energy transfer arranged to do useful mechanical work.

The limits of the model

A real gas comes closest to an ideal gas when its particles are far apart and moving fast, so intermolecular attractions don’t make much difference. In practice, this means high temperature, low pressure and low density. Under these conditions, the particles take up very little volume compared with the gas as a whole, and they spend less time interacting strongly with one another.

The approximation gets worse at low temperature, high pressure and high density. The particles sit closer together, intermolecular forces start to matter, and the finite size of the molecules can’t be ignored anymore. Real gases can also liquefy. An ideal gas cannot, because the ideal model includes no attractive forces that could hold particles together as a liquid.

A simple way to put it: ideal behaviour is most likely when the gas is far from condensation. Close to condensation, or when the gas is strongly compressed, the assumptions of negligible molecular volume and negligible intermolecular forces break down.

The gap between ideal and real gases is not a failure of physics. It just shows that models are built for particular purposes. The ideal gas model is very good at showing the link between pressure, temperature, volume and molecular motion. More detailed models of real gases add corrections for molecular size and intermolecular forces, often using constants that depend on the particular gas.

The same pattern appears across physics. We use simplified models — point masses, frictionless pulleys, uniform fields, ideal waves, ideal circuits and quantized energy levels — because they make complicated phenomena easier to understand. The skill is not pretending that the model is reality; it is knowing what the model explains, what it leaves out, and when those missing details start to matter.