The macroscopic quantity we call pressure

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

[ P=\frac{F}{A}, ]

where (P) is pressure (Pa), (F) is the force exerted perpendicular to the surface (N) and (A) is the area over which the force acts (m²). One pascal is one newton per square metre.

The word perpendicular matters. 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 source of the force is what changes.

For a solid resting on a surface, the force is usually its weight spread over the contact area. Make the contact area smaller, and the same force produces a larger pressure — 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 many tiny collisions of molecules with the container walls. The wall feels a steady force because there are so many collisions, even though each one lasts only a brief moment.

A useful scale to keep in mind is atmospheric pressure near sea level: about (1.0 imes10^5\ ext{Pa}). Everyday pressure differences may be much smaller or much larger than this; the equation above connects 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

[ n=\frac{N}{N_A}, ]

where (n) is amount of substance (mol), (N) is the number of molecules or particles (a pure count, dimensionless) and (N_A) is the Avogadro constant (mol⁻¹). The Avogadro constant has the exact value (N_A=6.02214076 imes10^{23}\ ext{mol}^{-1}).

A mole contains exactly (6.02214076 imes10^{23}) specified elementary entities. These could be atoms, molecules, ions or electrons, so name the thing being counted. 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, its common unit is g mol⁻¹. In SI gas-law calculations, kg mol⁻¹ is often safer. For example, a molecular gas made of two atoms per molecule has a molecular molar mass based on the whole molecule, not a single atom.

This counting idea connects the microscopic picture of a gas to the macroscopic one. A pressure gauge does not describe one 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 treating a gas as many particles moving randomly, colliding with each other and with the container walls. Its strength is that it links familiar mechanics — velocity, momentum, kinetic energy and force — with 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) (m³), 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 from tracking every molecule one by one.

The kinetic model uses deliberately simple assumptions:

• 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’s still convincing. 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 real physical picture.

Models matter in physics because they simplify. The cost is that 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 measured in experiments before scientists had a detailed microscopic explanation. They apply to a fixed amount of gas, with only one variable changed at a time.

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

[ P\propto \frac{1}{V}\quad ext{or}\quad PV= ext{constant}. ]

On a pressure–volume graph, an isothermal change gives a curve. Compress the gas and the pressure rises. If you plot (P) against (1/V) instead, the relationship becomes a straight line.

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

[ V\propto T\quad ext{or}\quad \frac{V}{T}= ext{constant}. ]

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

[ P\propto T\quad ext{or}\quad \frac{P}{T}= ext{constant}. ]

Temperature must be measured in kelvin. Celsius temperature differences can be used, but Celsius temperatures themselves do not work in ratios such as (V/T) or (P/T). It’s a small detail, but it ruins otherwise good answers.

The combined gas law

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

[ \frac{PV}{T}= ext{constant}. ]

Use this to compare two states of the same gas sample when pressure, volume and temperature all change. The amount of gas is still fixed, so this is not yet the full ideal gas equation.

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 happens when gas pressure moves a boundary, such as a piston. On a pressure–volume diagram, expansion to the right shows that the gas can do 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 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 wait for the gas to return to room temperature after each compression. If temperature has genuinely been controlled, a plot of (P) against (1/V) should be linear.

To verify Charles’s law, keep the gas pressure constant, vary temperature with 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 pressure after 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 reveal a 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, its molar form is

[ PV=nRT, ]

where (R) is the molar gas constant ((8.31\ ext{J mol}^{-1} ext{K}^{-1})). Boyle’s law, Charles’s law and the pressure law all sit inside this equation as special cases. If (n) and (T) stay constant, then (PV= ext{constant}); if (n) and (P) stay constant, then (V/T= ext{constant}); if (n) and (V) stay constant, then (P/T= ext{constant}).

There is also a molecular form:

[ PV=Nk_BT, ]

where (k_B) is the Boltzmann constant ((1.38 imes10^{-23}\ ext{J K}^{-1})). The two constants are connected by

[ k_B=\frac{R}{N_A}. ]

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 counted molecule by molecule.

The product (PV) has units of energy because ( ext{Pa m}^3=( ext{N m}^{-2}) ext{m}^3= ext{N m}= ext{J}). That’s not accidental. In the kinetic model, (PV) links to molecular kinetic energy.

In calculations, use SI units unless a ratio means you don’t need to convert. Pressure in pascals, volume in cubic metres, and temperature in kelvin keep (R) and (k_B) consistent.

Collisions, momentum and pressure

Gas pressure does not come from molecules continuously “pressing” on the wall. It comes from repeated collisions. When a molecule bounces off a wall, the component of its momentum perpendicular to the wall changes. The wall exerts a force on the molecule to change that 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 enormous numbers of random collisions, and the result is 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 key result is

[ P=\frac{1}{3}\rho v^2, ]

where (\rho) is the gas density (kg m⁻³) and (v) is the root mean square translational speed of the molecules (m s⁻¹).

Root mean square speed is found by squaring molecular speeds, averaging those squared values, and then taking the square root. It isn’t the same as mean velocity. In a gas that is at rest overall, the mean velocity is zero because the different directions cancel out. The rms speed is not zero, since 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 associated with the motion of a particle’s centre of mass through space. For an ideal gas molecule, its average translational kinetic energy is related to temperature by

[ E_k=\frac{1}{2}mv^2=\frac{3}{2}k_BT, ]

where (E_k) is the average translational kinetic energy per molecule (J) and (m) is the mass of one molecule (kg).

This idea matters a lot in this topic. Temperature is not just “how hot something feels”; for an ideal gas, it measures the average translational kinetic energy of the particles. A microscopic quantity — molecular kinetic energy — is then used in 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 includes no internal molecular rotation or vibration, and no intermolecular potential energy stores. So its internal energy is simply the total translational kinetic energy of all the atoms:

[ U=\frac{3}{2}Nk_BT ]

where (U) is the internal energy of the ideal monatomic gas (J). Equivalently,

[ U=\frac{3}{2}RnT. ]

For a temperature change,

[ \Delta U=\frac{3}{2}nR\Delta T, ]

where (\Delta U) is the change in internal energy (J) and (\Delta T) is the change in thermodynamic temperature (K). A temperature change in kelvin has the same numerical size as the temperature change in degrees Celsius.

Gas particles with high kinetic energy 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 transfer some energy to the surroundings. That’s the basic physics of a heat engine: energy transfers from a hot body to a cold body, with some of that transfer arranged to do useful mechanical work.