Dalton's law

The gases are mixable with each other in all proportions. Since the ideal gas law

$\displaystyle pV = nRT$ (1)

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 volumes $ V_1$ , $ V_2$ , ..., $ V_k$ of different gases having an equal pressure $ p$ and an equal temperature $ T$ . If one measures the volume $ V$ of the mixture in the same pressure and temperature, one notices that

$\displaystyle V = V_1\!+\!V_2\!+\!...\!+\!V_k.$

Each of the gases satisfies an equation $ pV_i = n_iRT$ , and thus

$\displaystyle pV = pV_1\!+\!pV_2\!+\!...\!+\!pV_k = (n_1\!+\!n_2\!+\!...\!+\!n_k)RT.$ (2)

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

$\displaystyle p_iV = n_iRT$

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

$\displaystyle p = (n_1\!+\!n_2\!+\!...\!+\!n_k)\frac{RT}{V} =
n_1\frac{RT}{V}\!+\!n_2\frac{RT}{V}\!+\!...\!+\!n_k\frac{RT}{V} =
p_1\!+\!p_2\!+\!...\!+\!p_k.$

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.



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