Physical Chemistry , 1st ed.

We recognize that pressure is defined as force per unit area. If we are moving
in the xdimension, then the wall we are colliding with has dimensions bc.
(Verify this by looking at Figure 19.2.) The pressure exerted in the xdimen-
sion,px, is therefore

pxf

a

o

r

r

e

c

a

eFa

b

vg



,tota

c

l,xN

a





m

b





v^2 av

c

g,x

Finally, we note two things. First, in the denominator of the above equation,
abcis the volume of the container, so we will substitute Vfor the vol-
ume in the denominator. Next, we extend the velocity to three dimensions.
Using a Pythagorean-theorem approach to velocity in three dimensions, we
can show that

v^2 avgv^2 avg,xv^2 avg,yv^2 avg,z

where v^2 avgis the square of the overall, three-dimensional average velocity.
Furthermore, since there is no reason to favor one dimension over the other,
the components of the average velocity must be equal to each other. Thus

v^2 avg 3 v^2 avg,x

Using these two ideas, the expression for pressure becomes

p

Nm

3 V

v^2 avg

 (19.8)

This is the basic expression in kinetic theory for the pressure of an ideal gas.
In equation 19.8, pressure has standard SI units of kg/(ms^2 ), or N/m^2 (which
is consistent with the original definition of pressure as a force per unit area).
We define the unit pascal(abbreviation Pa) as 1 N/m^2. The pascal is the basic
SI unit of pressure, although the bar(100,000 Pa) and atmosphere(1.01325 bar)
are commonly used.

Example 19.1

For 1 mole of He gas, a volume of 25.00 L of gas exerts a pressure of

0.8770 bar. What is the average velocity of the helium atoms in the system?

Solution

We can substitute the various values directly into the expression in equation

19.8, recognizing that the product of the variables Nmis the molarmass of

He, or 0.004003 kg (in standard units), and that 25.00 L is 25.00(0.001/1) 

0.02500 m^3 (again, in standard units):

87,700 

m

N

 2 

Since 1 N 1 kgm/s^2 , we have

87,700 

m

k



g

s^2



(0

3

.0

(

0

0

4

.0

0

2

0

5

k

00

g)

m

(v

3

2

a

)

vg)

The various units cancel to give us

1.643  106 

m

s^2

2

v^2 avg

(0.004003 kg)(v^2 avg)



3(0.02500 m^3 )

19.2 Postulates and Pressure 655