Physical Chemistry Third Edition

412 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

Consider a particle that strikes the wall at the right end of the box (perpendicular to

theyaxis), as shown in Figure 9.13. The values ofvxandvzdo not change because

no force is exerted in these directions. Because the kinetic energy does not change,v^2 y

does not change, although the sign ofvydoes change. Letv(i) be the initial velocity

andv(f) be the final velocity:

vx(f)vx(i), vy(f)−vy(i), vz(f)vz(i) (9.5-1)

As the particle strikes the wall, the force exerted on the wall rises suddenly and drops

Angle of

reflection

Angle of

incidence v(i)

v(f)

Figure 9.13 The Trajectory of a
Particle Colliding Specularly with a
Wall.

to zero just as suddenly. This makes the force hard to analyze mathematically, so we

deal with its time average. This seems physically reasonable, because any measuring

instrument requires a period of time, called theresponse time, to adjust to a sudden

change in the pressure. Consider the force exerted on the particle by the wall, which

by Newton’s third law is the negative of the force exerted on the wall by the particle.

The time average of theycomponent of the force on the particle over a period of time

from 0 toτis given by

〈

Fy

〉



1

τ

∫τ

0

Fy(t)dt (definition of the time average ofFy) (9.5-2)

We use the notation〈···〉to denote either a time average or a mean value. You must

determine from the context which type of average is meant in a particular case.

We choose the period of timeτso that it includes the time at which the particle

strikes the wall. From Newton’s second law, Eq. (9.2-4), the force equals the mass

times the acceleration:

〈

Fy

〉



1

τ

∫τ

0

Fydt

1

τ

∫τ

0

m

d^2 y

dt^2

dt

1

τ

∫τ

0

m

dvy

dt

dt

m

τ

∫v(τ)

v(0)

dvy



m

τ

[

vy(τ)−vy(0)

]



m

τ

[

vy(f)−vy(i)

]

(9.5-3)

Becausevy(f)=−vy(i),

〈

Fy

〉

−

2 mvy(i)

τ

(9.5-4)

By Newton’s third law the force on the wall,Fwall, is equal to−F, so that the average

value of the force on the wall due to one particle is

〈

Fwall,y

〉

−

〈

Fy

〉



2 mvy

τ

(9.5-5)

We use only initial velocities from now on and omit the (i) label.

The total time-average force on the areaAof the wall is the sum of the contributions

from all particles that strike this area in timeτ. Consider first the particles whose

velocities lie in the rangedvxdvydvz, abbreviated byd^3 v. The fraction of all particles

whose velocities lie in this range is

(fraction)g(v)dvxdvydvzg(v)d^3 v (9.5-6)

wherevis a velocity in the ranged^3 v. All of the particles with velocities in this range

are moving in the same direction with the same speed (within infinitesimal ranges).

Those particles in this set that will strike an areaAon the wall are contained in a prism

with base areaAand sides that are parallel to the direction of motion of the particles,

as shown in Figure 9.14.