Derive Equation 12.5 by following these steps. (a) Consider a single molecule of mass $m$ traveling in the $+x$ -direction with velocity $v_{x}$. Show that when this molecule collides and rebounds elastically from the container wall in a time $\Delta t,$ it exerts a force $F=2 m v_{x} / \Delta t$ on the wall. (b) Assume that the container is a cube of side $L$, so the average time between collisions on a particular wall is $\Delta t=2 L / v_{x} .$ Show therefore that the average force in part (a) can be written $F=m v_{x}^{2} / L .$ (c) Let the area of each wall of the container be $A$. Use the fact that pressure $P=F / A,$ along with the fact that the container's volume is $V=A L,$ to show that the average pressure is $P=N m \overline{v_{x}^{2}} / V .$ (d) Use the fact that $v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}$ to argue that the average pressure is $P=N m \bar{v}^{2} / 3 V .$ Hint: You may assume from symmetry that, on average, $v_{x}^{2}=v_{y}^{2}=v_{z}^{2}$.