Question

1 Ideal Gas Law The ideal gas law is familiar to anyone who has taken a college chemistry course: P V = νRT. This problem will show you why the ideal gas law has this form. We can arrive at this expression just by using classical mechanics! Consider a box of volume V containing N particles, each having mass m, that are moving horizontally with average speed v. The particles bounce back and forth between the end walls of the container, which have area A. They are spread out uniformly in the box, so that there are n = N/V particles per cubic meter.

Single Collision — Consider a single particle bouncing off a wall of the container. In an elastic collision, how much momentum is transferred to the wall of the container? (The wall has a much greater mass than a particle.)

Multiple Collisions — If there are N collisions like the one you analyzed in the preceding question, how much momentum will be transferred to the wall? Time Interval — How many collisions will occur in time ∆t? Express your result N in terms of the average speed v, the particle density n, the area of the wall A, and the time interval ∆t. (Remember: At any given time, only half the particles will be moving toward the wall; the other half will be moving away.)

Average Force — What is the average force on the wall due to the N collisions of the previous question? (Recall Newton’s Second Law: F = ∆p/∆t.)

Average Pressure — What is the average pressure on the wall?

Kinetic Energy — Rewrite your result in terms of the average kinetic energy of the particles: E = 1/ 2mv^2 .

Ideal Gas Law? — Rearrange your result to resemble the ideal gas law. That is, derive an expression that relates P, V , N, and E.

Average Energy — What value of the average energy E would give the physicists’ version of the ideal gas law, P V = N kBT?

Answer 1