For a mixture of non-reactive gases,derive the equation for total pressure.

The pressure of an ideal gas is given by $P = \frac{1}{3} n m \langle v^2 \rangle$.
Consider a unit volume containing a mixture of non-reactive gases. Let the number densities of molecules be $n_1, n_2, \ldots$,their masses be $m_1, m_2, \ldots$,and their mean square speeds be $\langle v_1^2 \rangle, \langle v_2^2 \rangle, \ldots$.
The total pressure $P$ of the mixture is the sum of the partial pressures of the individual gases:
$P = \frac{1}{3} n_1 m_1 \langle v_1^2 \rangle + \frac{1}{3} n_2 m_2 \langle v_2^2 \rangle + \ldots$ ....$(1)$
In thermal equilibrium,the average kinetic energy of the molecules of each gas is equal:
$\langle \frac{1}{2} m_1 v_1^2 \rangle = \langle \frac{1}{2} m_2 v_2^2 \rangle = \ldots = \frac{3}{2} k_B T$ ....$(2)$
From equation $(2)$,we have:
$m_1 \langle v_1^2 \rangle = m_2 \langle v_2^2 \rangle = \ldots = 3 k_B T$
Substituting this into equation $(1)$:
$P = \frac{1}{3} [n_1 (3 k_B T) + n_2 (3 k_B T) + \ldots]$
$P = (n_1 + n_2 + \ldots) k_B T$
This represents Dalton's law of partial pressure,where $n = n_1 + n_2 + \ldots$ is the total number density of molecules.