Dalton's Law of Partial Pressures

Fill a rigid 10.0 L tank with 0.20 mol of nitrogen and 0.10 mol of oxygen at 298 K. Each gas behaves as if the other were not there. Each contributes its own share of pressure, and the total is simply their sum. This is Dalton's Law, one of the oldest quantitative statements in gas chemistry. The law matters because almost no gas sample chemists actually handle is pure. Air, exhaled breath, combustion products, and gas collected over water are all mixtures. Dalton's Law lets us treat each component independently using the Ideal Gas Law, then combine results by simple addition. This article builds the law from kinetic theory and from the Ideal Gas Law. It defines partial pressure precisely. It works two full numerical examples, including the classic "gas collected over water" correction. It closes by pointing toward mole fractions, the natural next tool for describing mixtures.

Key takeaways

Dalton's Law: in an ideal gas mixture, the total pressure equals the sum of the partial pressures of each component.
The partial pressure of gas $i$ is the pressure it would exert alone in the same container at the same temperature.
The law follows directly from the Ideal Gas Law because ideal molecules do not interact with each other.
A common application is correcting measured pressure for water vapor when gas is collected over water.
Dalton's Law assumes ideal behavior; real gases show small deviations from strict additivity at high pressure.
Key idea: Non-interacting gas particles each generate pressure independently, so total pressure is a simple sum.

The core insight

Because ideal gas particles occupy negligible volume and exert no forces on each other, each species in a mixture contributes pressure exactly as if it alone filled the container, and the observed total pressure is just the arithmetic sum of these independent contributions.

The physical picture

Picture a sealed flask holding a mixture of helium and argon gas at some temperature $T$ . Both gases share the same volume $V$ and the same temperature, because they are mixed together and in thermal equilibrium. Each type of molecule bombards the container walls independently of the other. A helium atom does not "know" that argon atoms are also present, since ideal gas molecules are treated as point particles that never collide with each other in a way that changes the overall momentum transfer statistics. Pressure arises from the rate and force of molecular collisions with the walls. If you could somehow track only the helium collisions, you would measure a certain pressure — call it $P_{\text{He}}$ . If you tracked only the argon collisions, you would measure $P_{\text{Ar}}$ . The total pressure the manometer reads is the sum of both, because both sets of collisions are happening simultaneously and independently, each adding its own momentum flux to the wall. This is the essence of Dalton's Law of Partial Pressures, first proposed by John Dalton in 1801, decades before the kinetic theory of gases existed. Dalton reasoned empirically from experiments on moist air and gas mixtures. Modern kinetic theory later explained why the law works: for point particles with no intermolecular forces, the statistical mechanics of a mixture factors cleanly into independent contributions from each species. The partial pressure of a gas is not something you measure with a separate instrument inserted only for that gas. It is a defined quantity: the pressure that gas would produce if it alone occupied the entire container at the same temperature. This definition is what makes the law both simple and directly connected to the Ideal Gas Law you already know.

The core relations

P_{\text{tot}} = P_1 + P_2 + P_3 + \cdots = \sum_i P_i

where: $P_i$ is the partial pressure of component $i$ , and $P_{\text{tot}}$ is the measured total pressure.

P_i = \dfrac{n_i R T}{V}

where: $n_i$ is moles of component $i$ , $R$ is the gas constant, $T$ is temperature, $V$ is the shared container volume.

P_i = x_i P_{\text{tot}}

where: $x_i = n_i / n_{\text{tot}}$ is the mole fraction of component $i$ in the mixture.

Derivation AFrom the Ideal Gas Law to Dalton's Law

Goal: show that for a mixture of ideal gases sharing volume $V$ and temperature $T$ , the total pressure is the sum of the partial pressures each gas would exert alone.

Step 1. Set up the mixture. Suppose a rigid container of volume $V$ at temperature $T$ holds $n_1$ mol of gas 1, $n_2$ mol of gas 2, up through $n_k$ mol of gas $k$ , all mixed uniformly. The total number of moles is

n_{\text{tot}} = n_1 + n_2 + \cdots + n_k

Step 2. Apply the Ideal Gas Law to the whole mixture. Because an ideal mixture behaves as a single ideal gas with $n_{\text{tot}}$ moles, the measured total pressure obeys

P_{\text{tot}} V = n_{\text{tot}} R T

Step 3. Apply the Ideal Gas Law to each component alone. Define the partial pressure $P_i$ as the pressure component $i$ would exert if it alone occupied volume $V$ at temperature $T$ . Since ideal molecules have zero volume and no intermolecular forces, each species genuinely behaves as though the others were absent. Hence for each $i$ ,

P_i V = n_i R T

Step 4. Sum the individual equations. Add the $k$ equations from Step 3 across all components:

\left(\sum_i P_i\right) V = \left(\sum_i n_i\right) R T = n_{\text{tot}} R T

Step 5. Compare with Step 2. The right-hand side is identical to $P_{\text{tot}} V$ from Step 2. Since $V$ is the same nonzero volume in both expressions, we can equate the left-hand sides directly:

\boxed{P_{\text{tot}} = \sum_i P_i}

This result holds exactly for ideal gases because the derivation never invoked any interaction between species — each component's pressure equation in Step 3 stood alone. Real gases deviate slightly because intermolecular forces between unlike molecules are generally different from those between like molecules.

Worked examples

Example 1 — Building total pressure from partial pressures (foundational). A rigid 5.00 L vessel at 300 K contains 0.120 mol of $\text{N}_2$ and 0.080 mol of $\text{O}_2$ . Find each partial pressure and the total pressure. Use $R = 0.08206 \text{ L atm mol}^{-1}\text{K}^{-1}$ . Partial pressure of $\text{N}_2$ :

P_{\text{N}_2} = \frac{n_{\text{N}_2} R T}{V} = \frac{(0.120 \text{ mol})(0.08206 \text{ L atm mol}^{-1}\text{K}^{-1})(300 \text{ K})}{5.00 \text{ L}}

P_{\text{N}_2} = \frac{2.9542 \text{ L atm}}{5.00 \text{ L}} = 0.5908 \text{ atm}

Partial pressure of $\text{O}_2$ :

P_{\text{O}_2} = \frac{(0.080 \text{ mol})(0.08206)(300)}{5.00} = \frac{1.96944}{5.00} = 0.3939 \text{ atm}

Total pressure:

P_{\text{tot}} = P_{\text{N}_2} + P_{\text{O}_2} = 0.5908 + 0.3939 = 0.9847 \text{ atm}

Sanity check: total moles are $0.200$ mol, so directly $P_{\text{tot}} = (0.200)(0.08206)(300)/5.00 = 0.9847$ atm — matches exactly, confirming additivity. Mole fractions: $x_{\text{N}_2} = 0.120/0.200 = 0.600$ and $x_{\text{O}_2} = 0.400$ . Check: $x_{\text{N}_2}P_{\text{tot}} = 0.600 \times 0.9847 = 0.5908$ atm, matching $P_{\text{N}_2}$ above.

Example 2 — Gas collected over water (applied, exam-level). A student generates hydrogen gas by reacting zinc with hydrochloric acid and collects it over water at 23 °C. The collected gas occupies 152 mL at a total (barometric) pressure of 748 mmHg. The vapor pressure of water at 23 °C is 21.1 mmHg. Find the moles of dry $\text{H}_2$ gas collected.

Step 1. Apply Dalton's Law to remove the water vapor contribution. The gas in the collection tube is a mixture of $\text{H}_2$ and water vapor, both saturating the space above the water surface. So

P_{\text{tot}} = P_{\text{H}_2} + P_{\text{H}_2\text{O}}

P_{\text{H}_2} = P_{\text{tot}} - P_{\text{H}_2\text{O}} = 748 \text{ mmHg} - 21.1 \text{ mmHg} = 726.9 \text{ mmHg}

Step 2. Convert units for use in the Ideal Gas Law.

P_{\text{H}_2} = \frac{726.9}{760} \text{ atm} = 0.9564 \text{ atm}

V = 0.152 \text{ L}, \qquad T = 23 + 273.15 = 296.15 \text{ K}

Step 3. Solve for moles using the Ideal Gas Law applied to $\text{H}_2$ alone.

n_{\text{H}_2} = \frac{P_{\text{H}_2} V}{RT} = \frac{(0.9564 \text{ atm})(0.152 \text{ L})}{(0.08206 \text{ L atm mol}^{-1}\text{K}^{-1})(296.15 \text{ K})}

n_{\text{H}_2} = \frac{0.14538}{24.30} = 5.984 \times 10^{-3} \text{ mol}

Sanity check: this is about 6.0 mmol, or roughly 0.13 mL(STP)-equivalent thinking aside, a plausible small-scale lab quantity — consistent with a 152 mL collection tube at close to room conditions. If we had mistakenly used the full 748 mmHg without subtracting water vapor, we would overestimate $n_{\text{H}_2}$ by about 2.9%, which is exactly the kind of systematic error Dalton's correction is designed to eliminate.

Limiting cases & common traps

Situation	What Dalton's Law says	Common trap
Single pure gas	$P_{\text{tot}} = P_1$ trivially	None — law is trivially satisfied
Gas collected over water	Subtract water's vapor pressure first	Forgetting to subtract $P_{\text{H}_2\text{O}}$ , overstating dry gas moles
High pressure, real gases	Deviations appear; law is only approximate	Assuming exact additivity at tens of atm
Reacting gas mixtures	Law applies only to non-reacting, inert-to-each-other mixtures	Applying it directly when species chemically interact (e.g. $\text{NO}_2$ / $\text{N}_2\text{O}_4$ equilibrium)
Partial pressure vs. concentration	$P_i \propto n_i$ at fixed $V,T$ , so partial pressure tracks mole fraction	Confusing partial pressure with mass fraction

The chemist behind it

John Dalton, the English chemist and schoolteacher better known for atomic theory, developed his law of partial pressures around 1801 while studying the composition of the atmosphere and the behavior of water vapor mixed with air. Dalton was deeply interested in meteorology — he kept detailed weather diaries for over 50 years — and this practical curiosity about humid air directly motivated his gas mixture experiments. His empirical law preceded the kinetic theory of gases by more than half a century, yet it turned out to be an exact consequence of treating gas particles as non-interacting points, a fact only rigorously justified once Ludwig Boltzmann and James Clerk Maxwell developed statistical mechanics later in the 19th century.

Origins · 1 minJohn Dalton and how this was discovered

John Dalton

1766–1844 · English chemist who unified atomic theory with quantitative gas laws

Dalton's law of partial pressures and his atomic theory together gave chemistry its first quantitative, particle-based description of gases and compounds.

Ask the tutor: How do mole fractions let us predict how partial pressures shift when we add or remove a gas from a mixture at constant volume?

Does Dalton's Law apply to liquids or only to gases?

Dalton's Law is specifically a gas-phase law, built on the assumption that molecules move independently and rarely interact except through brief elastic collisions. Liquids have strong, persistent intermolecular forces, so pressure and composition are related very differently there. However, a closely related idea, Raoult's Law, describes how the vapor pressure above a liquid mixture depends on mole fraction, and it is often used alongside Dalton's Law when analyzing vapor-liquid equilibria.

Why do we subtract water's vapor pressure when collecting gas over water?

When a gas bubbles up through water and collects in an inverted tube, the space above the water is never purely the collected gas. Water molecules constantly evaporate into that space until it reaches the equilibrium vapor pressure for the given temperature. The manometer reads the total pressure of this mixture. Dalton's Law tells us to subtract the known water vapor pressure at that temperature to isolate the partial pressure of the gas of actual interest.

Is Dalton's Law exact for real gas mixtures?

No, it is exact only for ideal gases. Real molecules exert intermolecular forces, and these forces between unlike molecules (say, $\text{N}_2$ and $\text{O}_2$ ) generally differ from forces between identical molecules. At low pressure and moderate temperature, deviations are typically well under 1%, so the law works excellently for everyday laboratory conditions. At high pressures, more advanced equations of state like van der Waals or virial expansions are needed for accurate mixture behavior.

How does Dalton's Law connect to mole fraction?

Since each partial pressure obeys $P_i = n_i RT/V$ and the total obeys $P_{\text{tot}} = n_{\text{tot}}RT/V$ , dividing these gives $P_i = x_i P_{\text{tot}}$ , where $x_i = n_i/n_{\text{tot}}$ is the mole fraction. This means partial pressure is just total pressure scaled by composition. This relationship is the natural bridge to a fuller treatment of mole fractions and gas mixtures, including mixing multiple containers and analyzing reacting systems.

Can Dalton's Law be used for gases that react with each other?

Not directly, and this is an important limitation. Dalton's Law assumes the amounts $n_i$ of each species are fixed and the species do not transform into one another. If gases react — like $\text{NO}$ combining with $\text{O}_2$ to form $\text{NO}_2$ — the mole numbers change over time, and total pressure must be tracked through reaction stoichiometry and possibly chemical equilibrium, not simple addition of fixed partial pressures.