Question

Consider an ideal gas in a box, n equilibrium at temperature T. The particles each have kinetic energy mv2/2 and are spinless point particles. They are at suficiently low density that their quantum statistics are unimportant. The box is made of a thin but impermeable material, and is surrounded by vacuum. (a) Find the normalized velocity distribution for the particles inside the sealed box, Now, suppose that a small hole of area a is made in the box, but where the hole diameter is much larger than the thickness of the material that the box is made of. Particles will start to escape from the box (b) What is the velocity distribution for the particles that escape from the box, just after the hole is made? (suppose that the direction coming out of the hole is the +z direction) (c) What is the average (vector) velocity and the average energy per particle that escapes? (d) What is the rate that particles escape from the box, just after the hole is made?

Answer 1

(a) Velocity distribution follows directly from Boltzmann distribution, mv 2TkBT Normalization is given by JdvP-1 (b) The particles come through the hole with the +z half of the distribution of (a), 3/2 The normalization is as in (a) (c) The average velocity of the particles coming out of the hole must be in the +z direction, by symmetry. This component averaged is 2TkBT 1/2 The average energy per particle that escapes from the box is just 3kBT/2, the average energy of particles in the box (d) The number of collisions with the hole over time interval Δι is computed by integrating over all possible vz's. For each value of vz, particles from a z-distance up to vAt will come out in Δt. The total number of particles coming out in time Δt is therefore mo dvz Uz At 2 So, the rate that particles come out at is