You have now met the three gas laws, and each one is a bargain struck by holding one quantity still:

• Fix the temperature → pV = \text{constant} (Boyle).
• Fix the pressure → \dfrac{V}{T} = \text{constant} (Charles).
• Fix the volume → \dfrac{p}{T} = \text{constant} (the Pressure Law).

But real gases rarely do you the courtesy of holding anything still. Warm a gas and let it expand, or squeeze it and cool it, and no single law is enough. What we really want is one master rule that ties pressure p, volume V and temperature T together all at once — a single equation of state that lets any of the three roam free. Remarkably, all three laws fold into one clean line. That line is the ideal gas equation, and it is one of the most useful equations in all of physics.

Folding three laws into one

Watch the three laws snap together. Boyle says p \propto 1/V, Charles says V \propto T, and the Pressure Law says p \propto T. Combine them and, for a fixed amount of gas, the single combined statement is

This combined gas law already earns its keep — it handles any change of a sealed gas, with all three quantities free to move. Cover up any one of p, V or T and you recover the law you know: hold T constant and it collapses to p_1 V_1 = p_2 V_2; hold p constant and it becomes V_1/T_1 = V_2/T_2. The three separate rules were only ever this one rule, seen from three angles.

But "constant" hides a question: constant equal to what? Take twice as much gas and pV/T doubles too — so the constant must depend on how much gas there is. Pin that down and the combined law becomes an exact equation.

Counting the gas: the mole

"How much gas" is not really about mass or volume — it is about the number of particles doing the colliding. But a room's worth of air holds an absurd number of molecules, far too many to count one by one. So chemists and physicists bundle them into a convenient package called the mole.

One mole is simply a fixed number of particles — the way "a dozen" means 12, a mole means

where N_A is Avogadro's number. It is a preposterously large count because atoms are preposterously small: a single teaspoon of water contains about 1.7 \times 10^{23} molecules, roughly a third of a mole. If N is the actual number of molecules in a sample and n is the number of moles, the two are related by the same counting rule that turns dozens into eggs:

You can also reach n from a mass on a balance: divide the mass by the gas's molar mass M (the mass of one mole, in grams per mole), n = m/M. For example, carbon dioxide has M = 44\ \text{g/mol}, so 88\ \text{g} of it is exactly 2 moles.

The ideal gas equation

Now the constant in pV/T can be written properly. For n moles of gas it equals nR, where R is a single number that works for every gas — the molar gas constant,

Putting pV/T = nR and clearing the fraction gives the celebrated ideal gas equation:

The two forms say exactly the same thing. pV = nRT counts the gas in moles and pairs them with R; pV = NkT counts the gas in individual molecules and pairs them with k. Since one mole is N_A molecules and k = R/N_A, the bookkeeping matches: nR = (N/N_A)(kN_A) = Nk. Use whichever counting suits the problem — moles for a labelled cylinder, molecules when you care about a single particle's share.

What makes a gas "ideal"?

The equation is called the ideal gas equation because it describes a slightly idealised gas — a clean model that strips away the messy details. An ideal gas is a swarm of particles obeying three simplifying assumptions:

• The particles are point-like. Their own volume is negligible compared with the space they fly around in — the gas is almost entirely empty space.
• They exert no forces on one another except during a collision. No sticky attractions, no long-range pushes — between hits they travel in straight lines.
• All collisions are perfectly elastic and take no time. No kinetic energy is lost when particles strike the walls or each other, so the gas never "runs down".

Real molecules break all three rules a little: they have a real size, they do attract one another faintly, and so on. So pV = nRT is an approximation for a real gas — but a superb one, as long as the molecules are far apart and moving fast, so their size and mutual pull barely matter. That means a real gas behaves most nearly ideally at low pressure (particles spread thin) and high temperature (particles zipping past each other too fast to feel any attraction). Squeeze it hard or chill it towards liquefying, and the deviations grow — which is exactly why gases eventually condense into liquids, something an ideal gas could never do.

Feel the equation move

Here is exactly one mole of an ideal gas in a box (so n = 1 and R = 8.31 are fixed). You control the temperature and the volume; the pressure is not yours to set — it is forced on you by the equation, p = \dfrac{nRT}{V}, and the gauge shows the live result. Warm the gas and the pressure climbs; give the gas more room and the pressure drops as the particles thin out. Every reading on the gauge is pV = nRT balancing itself in real time.

Try to hold the pressure steady while changing both knobs: raise T and you must give the gas more V in the same proportion — that is Charles's Law living inside the master equation. The one equation contains all three gas laws at once.

Worked examples — kelvin first, every time

Example 1 — the pressure in a cylinder (find p). A rigid cylinder of volume V = 0.020\ \text{m}^3 holds n = 3.0\ \text{mol} of an ideal gas at 27\,^\circ\text{C}. What is the pressure?

Step 1 — convert to kelvin. T = 27 + 273 = 300\ \text{K}. Step 2 — rearrange pV = nRT for the unknown: p = \dfrac{nRT}{V}. Step 3 — substitute.

Notice the volume stays in cubic metres and the temperature in kelvin — feed the equation degrees Celsius or litres and the answer is simply wrong.

Example 2 — how many moles? (find n). A container of volume V = 0.050\ \text{m}^3 holds gas at p = 200\ \text{kPa} and T = 300\ \text{K}. How many moles are inside — and how many molecules?

Rearrange for n = \dfrac{pV}{RT}, keeping SI units (200\ \text{kPa} = 2.00\times10^{5}\ \text{Pa}):

That is N = nN_A = 4.0 \times 6.02\times10^{23} \approx 2.4\times10^{24} molecules — a reminder of just how many particles even a modest box of gas contains.

Example 3 — a change of state (use the two-state form). A fixed mass of gas occupies 2.0\ \text{m}^3 at 100\ \text{kPa} and 300\ \text{K}. It is heated to 400\ \text{K} and compressed to 1.0\ \text{m}^3. Find the new pressure.

The amount of gas n is fixed, so \dfrac{p_1V_1}{T_1} = \dfrac{p_2V_2}{T_2}. Rearranging,

Both effects push the same way here — halving the volume doubles the pressure (Boyle), and heating raises it further (the Pressure Law) — and the master equation books them both at once, without ever needing to know n or R.