Question

(1) (a) Let’s derive an ideal gas law. Let’s start with a cubic box with side-length L. Now assume we have a particle traveling perfectly horizontally towards a single wall. When it collides with that wall, it will turn around and hit the wall on the other side. It will continue to bounce back and forth in this way forever. What is the period of this motion? In other words, how much time does it take for the particle to hit a wall, travel to the other side of the box, and then return to the original wall?

(b) Use impulse to find the force in terms of the particle’s mass, m, velocity, v, and length of the box, L. Assume that the collision causes the particle to have the same speed in the perfectly opposite direction.

(c) What is the pressure exerted on the wall by this molecule? If there were N molecules, what is the pressure then? Remember that the pressure is the average force per unit area.

(d) In the real world, though, particles do not just move horizontally. They can move in all three dimensions equally. According to the equipartition theorem, what is the relation between temperature and average translational kinetic energy? Keep in mind that this will also triple the pressure (since we are considering forces and changes in momentum in all three dimensions too). Use this result for to replace the mv2 in your expression for the pressure.

Answer 1

a) If the velocity of particle is v than the time period is total distance covered by speed.

Total distance covered=2*L therefore time period =2*L/v

b) Impulse=Force*time=change in momentum.

In every time period there is one hit.

Change in momentum upon hitting the wall is =mv-(-mv)=2*m*v.

Therefore 2*m*v=Force*2*L/v

c) Pressure=Force/area

Force=m*v*v/L

area of wall =L*L

for N particles

d) According to equipartition theorem average translational kinetic energy=

Considering the 3D case

using above relation

Therefore

Thus ideal gas law is derived.