Ideal gas equation

An ideal gas is an idealised model: point-like molecules with negligible volume, no forces between them except during instantaneous elastic collisions. Real gases such as hydrogen or helium approach this behaviour closely at low pressure and moderate-to-high temperature. Three separate experimental laws describe how such a gas behaves when one variable is held fixed.

Boyle's law (constant temperature, constant amount): pressure and volume are inversely proportional,

Charles's law (constant pressure, constant amount): volume is directly proportional to absolute temperature,

Avogadro's law (same $p$, $V$, $T$): equal volumes of any gas contain equal numbers of molecules, so at fixed $p$ and $T$, volume is proportional to the amount of gas, $V \propto n$. Multiplying these three proportionalities together gives the ideal gas equation, also called the equation of state of a perfect gas:

where: $p$ is the pressure (Pa), $V$ is the volume ($\text{m}^3$), $n$ is the number of moles of gas, $T$ is the absolute temperature in kelvin, and $R = 8.314\ \text{J/mol}\,\cdot\,\text{K}$ is the universal gas constant, the same value for every ideal gas regardless of its identity.

Why absolute temperature. The Kelvin scale is built so that $0\ \text{K}$ is the point at which an ideal gas would, in principle, reach zero pressure and volume. Boyle's and Charles's laws only hold as clean proportionalities when temperature is measured on this absolute scale, so Celsius values must always be converted first:

Substituting Celsius directly into $pV = nRT$ silently breaks the proportionality and gives a wrong answer, even though the arithmetic looks fine.

For a fixed amount of gas ($n$ constant) taken from one state $(p_1, V_1, T_1)$ to another $(p_2, V_2, T_2)$, the quantity $nR$ stays constant, so a very useful combined form follows directly from $pV = nRT$:

This combined gas law is the workhorse for "before and after" problems where the amount of gas does not change.

The goal is to combine Boyle's law, Charles's law, and Avogadro's law into the single relation $pV = nRT$, using only the rule that if a quantity is separately proportional to several variables, it is jointly proportional to their product, and then fixing the proportionality constant by naming it $R$.

Step 1. State the three laws as proportionalities with everything else held fixed. Boyle: $V \propto 1/p$ (at fixed $T$, $n$). Charles: $V \propto T$ (at fixed $p$, $n$). Avogadro: $V \propto n$ (at fixed $p$, $T$). Each law isolates how $V$ responds to one variable while the other two are frozen.

Step 2. Combine the three proportionalities into one joint statement. Since $V$ is separately proportional to $1/p$, to $T$, and to $n$, it must be jointly proportional to their product:

This step is the entire physical content of the combination — no new experiment is needed once you accept that independent proportionalities multiply.

Step 3. Turn the proportionality into an equation by inserting a constant of proportionality, call it $R$:

Experiments (comparing gases at known $p$, $V$, $n$, $T$) show that $R$ has the same numerical value, $8.314\ \text{J/mol}\,\cdot\,\text{K}$, for every gas that behaves ideally — this universality is itself an experimental fact, not something the algebra above guarantees on its own.

Step 4. Note what this derivation does not do: it never appeals to molecular collisions, forces, or velocities. Kinetic theory (studied later, in Unit 9) separately derives an expression for pressure from molecular motion, $p = \tfrac{1}{3}\rho\overline{c^2}$, and that expression can be shown to be consistent with $pV = nRT$ — but $pV = nRT$ itself was established here purely by combining three experimental gas laws.

A cylinder contains $2.0\ \text{mol}$ of an ideal gas at a pressure of $3.0 \times 10^5\ \text{Pa}$ and a temperature of $27^\circ\text{C}$. Find the volume of the gas, then find the new pressure if the gas is compressed to half this volume while its temperature is raised to $77^\circ\text{C}$.

Step 1. Convert temperature to kelvin: $T_1 = 27 + 273 = 300\ \text{K}$.

Step 2. Apply the ideal gas equation to find the initial volume:

That is $16.6$ litres, using $1\ \text{m}^3 = 1000$ litres.

Step 3. Now compress the same amount of gas to $V_2 = V_1/2 = 8.3\times10^{-3}\ \text{m}^3$ and heat it to $T_2 = 77 + 273 = 350\ \text{K}$. Since $n$ is unchanged, use the combined gas law:

Step 4. Substitute, noting $V_1/V_2 = 2$:

Final answer: the initial volume is $\mathbf{1.66\times10^{-2}\ \text{m}^3 = 16.6\ \text{litres}}$, and after halving the volume and heating to $350\ \text{K}$ the pressure rises to $\mathbf{7.0\times10^5\ \text{Pa}}$ — more than double, because compression and heating both push pressure up together.

Competitive stretch

If instead the gas were allowed to expand until pressure returned to its original $3.0\times10^5\ \text{Pa}$ while temperature stayed at $350\ \text{K}$, the new volume follows from $pV = nRT$ directly: $V = nRT/p = (2.0)(8.314)(350)/(3.0\times10^5) = 1.94\times10^{-2}\ \text{m}^3$, about $17\%$ larger than the original $16.6$ litres — consistent with Charles's law, since only temperature rose at constant pressure.

• Forgetting to convert to kelvin. Plugging Celsius values directly into $pV = nRT$ or the combined gas law is the single most common numerical error — always add $273$ first.
• Mixing pressure and volume units. With $R = 8.314\ \text{J/mol}\,\cdot\,\text{K}$, pressure must be in pascals and volume in $\text{m}^3$. Litres are fine only in the combined form, where volumes appear as a ratio and cancel.
• Treating $pV = nRT$ as derived from kinetic theory. It is the reverse: the equation is experimental, built from Boyle's, Charles's, and Avogadro's laws; kinetic theory is later shown to be consistent with it.
• Confusing moles with mass. If a problem gives mass and molar mass, convert to moles, $n = m/M$, before substituting.

Ask the tutor: Want to practice picking between $pV = nRT$ and the combined gas law across a mix of problems, or see how this equation connects to the kinetic theory of gases covered next?