The ideal gas law

Result

The ideal gas law

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Statement

Three separately discovered empirical proportionalities — Boyle's law (\(V\propto1/P\) at fixed \(n,T\)), Charles's law (\(V\propto T\) at fixed \(n,P\)), and Avogadro's law (\(V\propto n\) at fixed \(T,P\), gas-stoichiometry-avogadro) — combine into a single joint proportionality \(V\propto nT/P\), which becomes an equation once a proportionality constant \(R\) (the universal gas constant, empirically found to be the same for every gas in the ideal limit) is introduced: \(PV=nRT\).

Why it matters

The ideal gas law is the single equation of state used throughout gas-phase chemistry: given any three of pressure, volume, moles, and temperature, it predicts the fourth. It is the equation already used implicitly in gas-stoichiometry-avogadro's molar-volume calculation, and it is the equation whose molecular-level justification (why should macroscopic pressure, volume, and temperature relate this way at all?) is provided by kinetic-theory-pressure, the next result in this unit — and whose limits, where real gases depart from this idealisation, are quantified by van-der-waals-equation later in the same unit.

Hypotheses

Each empirical law's proportionality constant, measured while holding the other variables fixed, depends only on whichever variables were held fixed in that particular experiment — not on some additional hidden variable.This is what licenses combining three separately-measured, pairwise proportionalities into one joint proportionality \(V\propto nT/P\): if Boyle's constant (found at some fixed \(n,T\)) is itself observed to scale with \(n\) and \(T\) exactly as Avogadro's and Charles's laws separately predict, then the three laws are mutually consistent pieces of one larger relationship, rather than three unrelated coincidences that happen to hold only under their own specific fixed conditions. The combined proportionality constant \(R\) is universal — identical for every gas, not merely for the specific gas used in whichever experiment measured it.This is itself a substantial, independently testable empirical claim, not a logical necessity: it could have turned out that light gases and heavy gases, or chemically simple and complex gases, needed different constants. That a single \(R\) value fits essentially all gases well (in the low-pressure, moderate-temperature "ideal" regime) is a genuine experimental discovery, and its universality is exactly what kinetic-theory-pressure explains at the molecular level (gas pressure depends on molecular kinetic energy, which at fixed temperature is the same for any gas, regardless of molecular identity or mass).

Proof

V\propto\frac{1}{P}\ (n,T\text{ fixed}), \quad V\propto T\ (n,P\text{ fixed}), \quad V\propto n\ (T,P\text{ fixed})

Three independently, empirically discovered proportionalities: Boyle's law (1662), Charles's law (formalised by Gay-Lussac from Charles's unpublished observations, 1780s–1802), and Avogadro's law (1811, gas-stoichiometry-avogadro), each established by varying one pair of quantities while experimentally holding the others fixed. A

V \propto \frac{nT}{P}

Combining three pairwise proportionalities that are mutually consistent (Hypotheses) into a single joint proportionality: \(V\) increases with \(n\) and \(T\) and decreases with \(P\), exactly reproducing each individual law when the other two variables in \(nT/P\) are held fixed. A

V = R\left(\frac{nT}{P}\right) \quad \Longrightarrow \quad PV = nRT

Introducing the proportionality constant \(R\) turns Step 2's proportionality into an equation; \(R\) is determined empirically (by measuring \(P,V,n,T\) simultaneously for a real, close-to-ideal gas sample) and found, within experimental precision, to take the same value for essentially every gas under ordinary laboratory conditions (Hypotheses). A

R \approx 8.314\,\text{J/(mol}\cdot\text{K)} = 0.08206\,\text{L}\cdot\text{atm/(mol}\cdot\text{K)}

Two commonly used numerical values of \(R\), depending on which pressure/volume units are used in a given calculation; both describe the identical physical constant, related by the conversion \(1\,\text{L}\cdot\text{atm}=101.325\,\text{J}\). A

Result

PV = nRT, \qquad R \approx 8.314\,\text{J/(mol}\cdot\text{K)} = 0.08206\,\text{L}\cdot\text{atm/(mol}\cdot\text{K)}

Reading. A single equation of state, assembled purely by combining three independently discovered empirical proportionalities and introducing one universal constant, relates pressure, volume, moles, and (absolute) temperature for any ideal gas.

Scope. Requires temperature in kelvin (absolute temperature) throughout, since the underlying proportionalities (Step 1) were established using an absolute temperature scale; using Celsius or Fahrenheit directly breaks the proportionality (Common errors). Valid to good approximation for real gases at ordinary pressures and temperatures away from their liquefaction point; deviations are quantified in van-der-waals-equation.

Corollaries & converses

For a fixed amount of gas (\(n\) constant) undergoing a change between two states, \(\dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2}\) (the combined gas law), obtained by applying \(PV=nRT\) at both states and eliminating the shared \(nR\); this is often more directly useful than \(PV=nRT\) itself when \(n\) is not separately needed.
Each of the four historical empirical laws is recovered from \(PV=nRT\) as a special case: fixing \(n,T\) recovers Boyle's law; fixing \(n,P\) recovers Charles's law; fixing \(n,V\) gives \(P\propto T\) (Gay-Lussac's pressure law, a consequence of the combination rather than an independently required fourth input); fixing \(T,P\) recovers Avogadro's law.
Converse: a gas sample that measurably fails to satisfy \(PV=nRT\) under given conditions is behaving non-ideally under those conditions — informative in its own right about how close to (or far from) ideal behaviour that particular gas is at that pressure and temperature (van-der-waals-equation).

Fails without

Use a non-absolute temperature scale (Celsius, Fahrenheit) directly in place of \(T\): Charles's law's proportionality \(V\propto T\) requires \(T=0\) to correspond to (extrapolated) zero volume, which is true only for the kelvin (absolute) scale; using Celsius directly (where \(0^\circ\text{C}\) is an arbitrary reference, not absolute zero) breaks the proportionality entirely — doubling a Celsius temperature does not double volume at constant \(n,P\), even though doubling the corresponding kelvin temperature does.
Assume \(R\) is not truly universal, and use a value fitted to one specific gas for a different gas: this would silently reintroduce a gas-specific correction that the whole point of assembling one combined law was to eliminate; the ideal gas law's practical usefulness rests specifically on \(R\)'s universality holding well enough, for ordinary conditions, that a single tabulated value suffices for any gas.

Common errors

Forgetting to convert a given Celsius temperature to kelvin before substituting into \(PV=nRT\) (directly the Fails without trap, and one of the single most common numerical errors in introductory gas-law problems).
Mixing unit systems for \(R\), \(P\), and \(V\) inconsistently — e.g. using \(R=0.08206\,\text{L}\cdot\text{atm/(mol}\cdot\text{K)}\) with a pressure given in pascals, or \(R=8.314\,\text{J/(mol}\cdot\text{K)}\) with a volume given in litres without converting to cubic metres.
Applying the combined gas law (Corollaries) across a change that also involves a change in the amount of gas (\(n\) not held fixed), where it does not apply; the full \(PV=nRT\) (tracking \(n\) explicitly at each state) is required instead.

Discussion

The combined equation \(PV=nRT\) in essentially its modern form is generally credited to Benoit Paul Émile Clapeyron, who assembled it in 1834 from the already well-established individual laws of Boyle (1662), Charles/Gay-Lussac (1780s–1809), and Avogadro (1811) — a genuine synthesis of roughly 170 years of separately accumulated empirical observation into one compact equation of state, rather than a single new discovery.

Notably, this synthesis long preceded any molecular-level explanation of why the combined law should hold: Clapeyron's derivation was purely a mathematical combination of empirically verified proportionalities, with no reference to molecules, collisions, or kinetic energy at all. Only with the maturation of kinetic theory in the mid-to-late 19th century (largely through the work of Maxwell, Boltzmann, and Clausius) did a genuine molecular mechanism — developed fully in kinetic-theory-pressure — explain why gas pressure should relate to volume, moles, and temperature in exactly this way. This is the same historical pattern already seen twice in this curriculum: VSEPR (empirically discovered, later explained by orbital hybridisation and quantum mechanics) and molecular orbital theory's O\(_2\) paramagnetism (an empirical puzzle later explained by degenerate-orbital filling) both followed the identical arc of empirical pattern first, molecular mechanism later.

Common misconception: that "ideal gas" describes some special, rare category of real gas, rather than a useful idealisation (zero molecular volume, no intermolecular forces) that ordinary real gases approximate well under typical laboratory conditions and approximate poorly under extreme conditions (very high pressure, very low temperature, near a gas's own liquefaction point) — the ideal gas law is best understood as an excellent approximation with a known, quantifiable domain of validity, not as a law obeyed by only a special class of substances.

Worked examples

P=2.50\,\text{atm},\ V=5.00\,\text{L},\ T=300.\,\text{K}: \quad n=\frac{PV}{RT}=\frac{2.50\times5.00}{0.08206\times300}\approx0.508\,\text{mol}

A direct, one-state application of \(PV=nRT\) solved for moles; the same equation, given any three quantities, solves for whichever fourth is unknown. A

\text{Fixed } n,V:\ P_1=1.00\,\text{atm at }T_1=273\,\text{K}; \text{ heated to } T_2=546\,\text{K} \Rightarrow P_2=P_1\frac{T_2}{T_1}=1.00\times2=2.00\,\text{atm}

A direct application of the Corollaries' combined gas law (here reducing to Gay-Lussac's special case, \(V\) fixed): doubling absolute temperature at constant volume exactly doubles pressure, illustrating one of the four historical laws recovered as a special case of the general result. A

n\approx0.508\,\text{mol (single-state)}; \qquad P_2=2.00\,\text{atm (doubling }T\text{ at fixed }V\text{ doubles }P\text{)}

Reading. The same equation of state handles both a direct single-state calculation and a two-state comparison at fixed \(n\), the latter reducing transparently to one of the historical special-case laws.

Scope. Both calculations require absolute (kelvin) temperature throughout, per the Result's scope note.

Problems

A gas sample occupies \(3.00\,\text{L}\) at \(1.50\,\text{atm}\) and \(298\,\text{K}\). If the same amount of gas is compressed to \(1.00\,\text{atm}\) and heated to \(350\,\text{K}\), find its new volume.
Solution
Using the combined gas law (fixed \(n\)): \(V_2=\dfrac{P_1V_1T_2}{T_1P_2}=\dfrac{1.50\times3.00\times350}{298\times1.00}\approx5.29\,\text{L}\).
A student measures \(1.00\,\text{L}\) of gas at \(1.00\,\text{atm}\) and \(27^\circ\text{C}\), and computes moles using \(T=27\) directly (in the mistaken units of "kelvin"). Determine both the (incorrect) result from this error and the correct result using properly converted temperature, and state the size of the error.
Solution
Correct: \(T=27+273.15=300.15\,\text{K}\); \(n=\dfrac{1.00\times1.00}{0.08206\times300.15}\approx0.0406\,\text{mol}\). Incorrect (using \(T=27\,\text{K}\) directly): \(n_{\text{wrong}}=\dfrac{1.00\times1.00}{0.08206\times27}\approx0.451\,\text{mol}\) — more than ten times too large, since \(27\,\text{K}\) is far colder than \(300\,\text{K}\) and the ideal gas law predicts far fewer moles are needed to reach the same pressure and volume at the (incorrectly) much lower assumed temperature. This dramatic size of error is exactly why the Celsius-to-kelvin conversion (Common errors) matters in practice, not just as a formality.
Verify, using only the SI value \(R=8.314\,\text{J/(mol}\cdot\text{K)}\) and the exact conversion \(1\,\text{L}\cdot\text{atm}=101.325\,\text{J}\), that \(R\approx0.08206\,\text{L}\cdot\text{atm/(mol}\cdot\text{K)}\).
Solution
\(R=\dfrac{8.314\,\text{J/(mol}\cdot\text{K)}}{101.325\,\text{J/(L}\cdot\text{atm)}}\approx0.08206\,\text{L}\cdot\text{atm/(mol}\cdot\text{K)}\), confirming the two commonly quoted values of \(R\) (Step 4) are the same physical constant expressed in different, internally consistent unit systems, not two independently measured numbers that happen to be close.
A rigid, sealed \(2.00\,\text{L}\) container holds \(0.100\,\text{mol}\) of gas at \(298\,\text{K}\). If the container is heated until the internal pressure doubles, find the new temperature (the container's rigidity fixes both \(n\) and \(V\)).
Solution
With \(n,V\) fixed, \(P\propto T\) (Gay-Lussac's special case, Corollaries): doubling \(P\) requires doubling \(T\) as well, so \(T_2=2\times298=596\,\text{K}\). (The initial pressure itself, \(P_1=\tfrac{nRT_1}{V}=\tfrac{0.100\times0.08206\times298}{2.00}\approx1.22\,\text{atm}\), is not actually needed to answer the question, since the doubling relationship depends only on the ratio, not the absolute pressure value.)