Deriving the pressure of a gas of particles using the root mean square velocity

Question

The pressure of a gas of particles is $P = \frac{N}{V}mv_x^2$, where $v_x$ is the root mean square velocity of the particles in the $x$-direction, $m$ is the mass of each particle, $N$ is the total number of particles in a volume $V$. How does one derive this from basic principles? I understand that $\frac{N}{V}m$ is the density of the gas. But why does multiplying it with the square of the RMS velocity give the pressure?

Answer 1

Well, We know that $$E=\frac{3}{2}k_BT=\frac{1}{2}m\langle v^2\rangle $$ $$PV=Nk_BT\Rightarrow PV=\frac{N}{3}m\langle v^2\rangle $$ $$\rightarrow P=\frac{N}{3V}m\langle v^2\rangle$$ Considering space is isotropic, $$\langle v_x^2\rangle =\langle v_y^2\rangle =\langle v_z^2\rangle =\frac{1}{3}\langle v^2\rangle $$ $$\rightarrow \boxed{P=\frac{N}{V}m\langle v_x^2\rangle }$$