The Foundations of Chemistry

impulse exerted by each collision and (2) the rate of collisions (number of collisions in a

given time interval).

P(impulse per collision)(rate of collisions)

Let us represent the mass of an individual molecule by mand its speed by u. The heavier

the molecule is (greater m) and the faster it is moving (greater u), the harder it pushes on

the wall when it collides. The impulse due to each molecule is proportional to its momentum,

mu.

Impulse per collision mu

The rate of collisions, in turn, is proportional to two factors. First, the rate of collision

must be proportional to the molecular speed; the faster the molecules move, the more often

they reach the wall to collide. Second, this collision rate must be proportional to the number

of molecules per unit volume, N/V. The greater the number of molecules, N, in a given

volume, the more molecules collide in a given time interval.

Rate of collisions (molecular speed)(molecules per unit volume)

or

Rate of collisions (u)


We can introduce these proportionalities into the one describing pressure, to conclude that

P (mu)u or P  or PV Nmu^2

At any instant not all molecules are moving at the same speed, u. We should reason in terms

of the averagebehavior of the molecules, and express the quantity u^2 in average terms as

u^2 , the mean-square speed.

PV Nmu^2

Not all molecules collide with the walls at right angles, so we must average (using calculus)

over all the trajectories. This gives a proportionality constant of ^13 , and

PV^13 Nmu^2

This describes the quantity PV (pressurevolume) in terms of molecular quantities—

number of molecules, molecular masses, and molecular speeds. The number of molecules,

N, is given by the number of moles, n, times Avogadro’s number, NAv, or NnNAv. Making

this substitution, we obtain

PV^13 nNAvmu^2

The ideal gas equation describes (pressurevolume) in terms of measurable quantities—

number of moles and absolute temperature.

PVnRT

So we see that the ideas of the kinetic–molecular theory lead to an equation of the same

form as the macroscopic ideal gas equation. Thus, the molecular picture of the theory is

consistent with the ideal gas equation and gives support to the theory. Equating the right-

hand sides of these last two equations and canceling ngives

^13 NAvmu^2 RT

Nmu^2



V

N



V

N



V

Recall that momentum is mass
speed.

468 CHAPTER 12: Gases and the Kinetic–Molecular Theory

(Enrichment, continued)

u^2 is the average of the squares of the
molecular speeds. It is proportional to
the square of the average speed, but
the two quantities are not equal.