Dalton's Law of Partial Pressures

The ideal gas equation, as introduced so far, applies to a single, pure gas. Dalton's law of partial pressures

extends the same idea to a mixture of two or more gases that do not chemically react with one another. It states

that the total pressure exerted by a mixture of non-reacting gases confined to a fixed volume at a fixed

temperature is equal to the sum of the partial pressures that each individual gas would exert if it alone

occupied that same volume at that same temperature:

This follows directly from the ideal gas equation and the kinetic theory's assumption that gas particles do not

interact with one another: each gas in the mixture behaves exactly as if the others were not present, contributing

its own share of collisions with the container walls, and the pressures simply add.

A convenient way to calculate an individual partial pressure uses the mole fraction, $x_i = \dfrac{n_i}{n_{\text{total}}}$,

the fraction of the total moles contributed by gas $i$:

Worked example. A closed vessel contains $2\ \text{mol}$ of $\text{N}_2$ and $3\ \text{mol}$ of $\text{O}_2$,

with the mixture at a total pressure of $5\ \text{atm}$. Total moles $= 2 + 3 = 5\ \text{mol}$. Mole fraction of

$\text{N}_2$: $x_{\text{N}_2} = 2/5 = 0.4$; mole fraction of $\text{O}_2$: $x_{\text{O}_2} = 3/5 = 0.6$. Partial

pressures: $P_{\text{N}_2} = 0.4 \times 5 = 2\ \text{atm}$, and $P_{\text{O}_2} = 0.6 \times 5 = 3\ \text{atm}$;

note that $2 + 3 = 5\ \text{atm}$, correctly recovering the total pressure.

Dalton's law has a particularly important practical application in the laboratory: gases are very often collected

by downward displacement of water, bubbling the gas of interest into an inverted, water-filled container. The

gas collected this way is never pure — it is always "wet," saturated with water vapour, because some liquid water

inevitably evaporates into the gas space. The measured total pressure of this wet gas is therefore the sum of the

partial pressure of the gas of interest and the partial pressure (vapour pressure) of water vapour at that

temperature: $P_{\text{total}} = P_{\text{dry gas}} + P_{\text{water vapour}}$. To find the pressure of the actual

gas of interest alone, the water's vapour pressure (a known, tabulated value at each temperature) must be …