Page 49 - Modern physical chemistry

38 Gases and Collective Properties

collisions tend to create complete chaos in the system. A gas at rest is thus made isotropic,
so that it exerts the same pressure P on each confining wall.
Consider an ideal gas containing N molecules, each of mass m, confined in a cubical
box of edge Z, at equilibrium under pressure P. Place rectangular axes along three edges
of the cube as figure 3.1 shows. Then the coordinates of the ith molecule are represented
as (Xi' Yi' Zi) at the given time t.
Let the velocity at which the ith molecule moves be u i • Also represent the time deriv-
atives of the coordinates as (Xi,'f/i,Zi)' Then with the Pythagorean theorem, we have
[3.1 J

In an ideal gas the only role of collisions is to randomize the motions. So if we suppose
that complete disorder reigns, we may neglect collisions in our derivation.
The intrinsic properties of a gas are independent of the nature of the walls confining
it. Here for simplicity, we will consider that each wall is smooth and elastic. Then on strik-
ing the wall that coincides with the yz plane, molecule i is reflected, withxi merely reversed
in sign and with iii and Zi unchanged. The only other wall where Xi is altered is the one
opposite, where Xi is also reversed in sign. Neglecting collisions with other molecules, we
find that the X component of the distance traveled by the molecule between strikes on
the yz plane equals 2l. If the molecule strikes the wall n i times a second, it travels dis-
tance 2Zni parallel to the X axis. Since speed is distance divided by time, we have

IXi 1 = 2Z~i . [3.2J

Each time molecule i strikes the yz plane, its momentum Pi changes by 2mlxil. In unit
time the change in momentum at this plane is

IlPi 2 I' 1 mi:f [3.3J
Ilt =ni mXi =-Z-,

where n i has been eliminated using equation (3.2) and IlPi is the momentum change in
time Ilt. Sununing over all N molecules yields

[3.4J

z

y

FIGURE 3.1 Cubical box confining the ideal gas.