PlanetPhysics/Dalton's Law

The gases are mixable with each other in all proportions.\, Since the [[../IdealGasLaw/|ideal gas law]]

$\begin{array}{c}pV=nRT\end{array}$

is valid for any ideal gas, one may think that it's insignificant whether the mole number $n$ concerns one single gas or several gases.\, It is true, which can be shown experimentally.

Let's think that we mix the [[../Volume/|volumes]] $V_1$, $V_2$, ..., $V_k$ of different gases having an equal pressure $p$ and an equal [[../BoltzmannConstant/|temperature]] $T$.\, If one measures the volume $V$ of the mixture in the same pressure and temperature, one notices that $V=V_1+V_2+...+V_k.$ Each of the gases satisfies an equation\, $p{V}_i=n_{i}RT$,\, and thus

$\begin{array}{c}pV=p{V}_1+p{V}_2+...+p{V}_k=(n_1+n_2+...+n_k)RT.\end{array}$

This is similar as the general equation (1).\, If we think that the same volume $V$ would be filled by any of the gases alone, we had an equation $p_{i}V=n_{i}RT$ for each gas; here the pressure $p_i$, i.e. $n_i\frac{RT}{V}$, is called the partial pressure of the gas $i$.\, By (2), we have $p=(n_1+n_2+...+n_k)\frac{RT}{V}=n_1\frac{RT}{V}+n_2\frac{RT}{V}+...+n_k\frac{RT}{V}=p_1+p_2+...+p_k.$ Accordingly we have obtained the

Dalton's law. \, The pressure of a gas mixture is equal to the sum of the partial pressures of the component gases.

This law was invented by J. Dalton in 1801.