Question

Suppose that a 1.00 cm3 box in the shape of a cube is perfectly evacuated, except for a single particle of mass 1.23 10-3 g. The particle is initially moving perpendicular to one of the walls of the box at a speed of 415 m/s. Assume that the collisions of the particle with the walls are elastic.

(a) Find the mass density inside the box.

(b) Find the average pressure on the walls perpendicular to the particle's path.

(c) Find the average pressure on the other walls.

(d) Find the temperature inside the box in K

Discuss the assumption of elastic collisions, in light of this result.

Answer 1

a.)

Mass density = mass of the particle/ volume of the box = (1.23 * 10^-6) / (1.00 * 10^-6) = 1.23 kg/m³

b.)

Average pressure = force / area

force = mv² / L where m=mass = 1.23 * 10^-6 kg ,

L= length of side of cube = 1.00 * 10^-2 m ,

v = 415 m/s

pressure = force / area = mv²/ L³ = 1.23 * 10^-6 * (415)² / (1.00 * 10^-6) = 2.11 * 10⁵ Pa

c.)

Average pressure on the other walls is zero.

d.)

Temperature inside the box T

average translational kinetic energy = kT/2 (since number of degrees of freedom =1)

k is Boltzmann constant = 1.38 * 10^-23 J/K

mv²/2 = kT/2

T = 1.23 * 10^-6 * (415)^2 / (1.38*10^-23)

T = 1.535 * 10²² K

If the collisions are elastic, energy never gets transferred to the walls

so no thermodynamic equilibrium is not possible.