How can one rationalize Avogadro's hypothesis using the equation of kinetic molecular theory?
$$pV = \frac{1}{3} \cdot N \cdot m\overline{u^2} = \frac{2}{3} \cdot N \cdot \frac{1}{2}m\overline{u^2}$$
$N$ is the number of molecules and $u$ is the average speed.
Pressure is the number of collisions with the container per unit area. Imagine a particle in a box of length $L$. Assuming for the moment that it only moves in the $x$-direction, whenever it collides with a wall of the container the wall will gain momentum from the particle: \begin{align} \Delta p &= p_{\mathrm{final}, x} - p_{\mathrm{initial}, x} \\ &= p_{\mathrm{final}, x} - (- p_{\mathrm{final}, x})\\ &= 2p_{\mathrm{final}, x}\\ &= 2mu_x, \end{align} where $u_x$ is the $x$-component of velocity.
The particle hits a specific wall every $\Delta t = \frac{2L}{u_x}$, since it travels the length of the box twice. Force can be defined as momentum per unit time, therefore $$ F = \frac{\Delta p}{\Delta t} = \frac{mu_x^2}{L}$$ For $N$ number of particles, the expression becomes $$ F = \frac{Nm\overline {u_x^2}}{L},$$ where $\overline{u_x^2}$ is the mean square velocity of the $N$ particles.
Now, we can extend the situation so that our particle can move in three directions. Since the box is cubic, the $x$-, $y$-, and $z$-components are equivalent: $$ \overline{u_x^2} = \frac{\overline{u^2} }{ 3}$$ Therefore, $$ F = \frac{Nm\overline {u^2}}{3L}.$$ The area of any wall is $L^2$, so \begin{align} p &= \frac{F}{L^2}\\ &= \frac{Nm\overline {u^2}}{3L^3}\\ &= \frac{Nm\overline {u^2}}{3V} \end{align} This leads to the expression $$pV = \frac{1}{3}Nm\overline{u^2}$$
See also Wikipedia.
