Partial Pressure

Suppose that an ideal gas consists of $N$ distinct types of molecule. Let a molecule of type $i$ have a number density $n_i$, a mass $m_i$, and a velocity ${\bf v}_i$. If we repeat the analysis of Section 5.3.3, taking into account the different types of molecule, then it is easily shown that the total pressure of the gas is

$$p = \frac{1}{3}\sum_{i=1,N} n_i\,m_i\left\langle v_i^{\,2}\right\rangle.$$ (5.183)

However, the law of equipartition of energy (see the previous section) implies that

$$\frac{1}{2}\,m_i\left\langle v_i^{\,2}\right\rangle=\frac{3}{2}\,k_B\,T.$$ (5.184)

Moreover,

$$n_i = \frac{\nu_i\,N_A}{V},$$ (5.185)

where $\nu_i$ is the number of moles of molecules of type $i$ in the gas, and $V$ is the volume of the gas. [See Equation (5.177).] The previous three equations yield

$$p= \sum_{i=1,N} p_i,$$ (5.186)

where

$$p_i\,V = \nu_i\,R\,T.$$ (5.187)

We conclude that the total pressure of the gas is the sum of the pressures that a gas of each constituent type of molecule would exert independently. This result is known as Dalton's law, after John Dalton, who verified it experimentally in 1802. The quantity $p_i$ is known as the partial pressure of type-$i$ molecules. Thus, Dalton's law is equivalent to the statement that the total pressure of an ideal gas is the sum of the partial pressures of the individual gases from which it is composed.