Page 76 - Essentials of physical chemistry

38 Essentials of Physical Chemistry

Y

L

X

L
L
Z
FIGURE 3.1 A single particle in a cubical box undergoing elastic collision with a wall.

Even though collisions will occur on more than one face, we can calculate how long it takes before
another collision will occur at this face in terms of the velocity and the distance. This is based on the
familiar equation (distance) ¼ (rate)(time), so that the time is the distance=rate in the denominator,
#collisions v x
which flips up as the inverse, ¼ . Then, the total number of collisions per unit time
time 2L
on area A yields force=area ¼ pressure. So far, so good. The next step

2 2
dp=dt v x 2mv x mv x mv x force
A ¼ 2L L 2 ¼ L 3 ¼ V ¼ area

is to multiply by the number of atoms to obtain the pressure as
2
f x Nmv x
N ¼ P ¼ :
L 2 V
The following step is an approximation usually used in a freshman presentation in that we note that
the motion of the atom is random and the velocity is a vector with three components in general. The
2
motion is random so all three components are favored equally, but at the moment we only need v .
x
If we use the dot product, we can get the square of the velocity containing all three equally weighted
components as

^
^
^
^
^
^
2
2
2
2
v ¼ (v x i þ v y j þ v z k) (v x i þ v y j þ v z k) ¼ v þ v þ v :
x y z
2
2
2
2
2
Then, we say that if the motion is really random v ¼ v ¼ v and implies that v ¼ 3v and then,
x y z x
2
Nm(v =3) 2
or interestingly PV ¼ (1=3)Nmv .
we finally have the pressure as P ¼
V
This is a lot like the ideal gas law except for the right side of the equation. Next, we recall from
2
physics that the average kinetic energy of an atom=molecule can be written as ke ¼ mv =2, so we
2
2 mv 2
N ¼ N (ke). We can also let N ¼ nN Av , so the arbitrary number of
can write PV ¼
3 2 3
atoms=molecules, N, can be rewritten as a number of moles n times Avogadro’s number N Av . Then,
if we define a molar kinetic energy as N Av (ke) ¼ (KE), we have the molar expression as
2 2
2 2 N Av m v 2 M v
n n . Note, we have introduced an average
PV ¼ n(KE) ¼ ¼
3 3 2 3 2
2
square of the velocity as v and the molar mass as M. At this point, we make an assumption