Physical Chemistry Third Edition

9.3 The Velocity Probability Distribution 399

This must agree with the macroscopic energy of a dilute gas in Eq. (9.3-24), so that

N

3 m

2 b



3

2

NkBT (9.3-30)

so that

bm/kBT (9.3-31)

The normalized probability distribution for thexcomponent of the velocity is now

f(vx)

(

m

2 πkBT

) 1 / 2

e−mv

(^2) x/ 2 kBT
(9.3-32)
We have derived this distribution for particles without rotation or vibration, but we now
assert that rotation, vibration, and electronic motion occur independently of translation
(the only motion of structureless particles) so that we can use this distribution for
the translational motion of any molecules in a dilute gas. The normalized probability
distribution is represented in Figure 9.7 for a velocity component of oxygen molecules
at 298 K. The most probable value of the velocity component is zero, and most of the
oxygen molecules have values of the velocity component between−400ms−^1 and
400ms−^1.
2600240022000 200 400 600 800
vx/ms^21
f(
vx
)
Figure 9.7 The Probability Distri-
bution for a Velocity Component of
Oxygen Molecules at 298 K. The distribution of Eq. (9.3-32) is an example of aGaussian distribution, also called
anormal distribution.The Gaussian distribution is represented by the formula
f(u)

1

√

2 πσ

e−(u−μ)

(^2) / 2 σ 2
(9.3-33)
whereμis the mean value ofuand whereσis called thestandard deviation.Ifa
normal distribution has a standard deviation equal to 1 it is called thestandard normal
distribution. A graph of a Gaussian distribution is shown in Figure 9.8a. This graph is
sometimes called abell curve. The shaded area in Figure 9.8b represents the probability
thatulies betweenμ−σandμ+σ.
f
(a) (b)
f
m21.96smm1 1. 96 s
x
m2s m1s m21.96smm1 1. 96 s
x
m2s m1s
Figure 9.8 The Gaussian (Normal) Probability Distribution.(a) The graph showing the
famous “bell-shaped” curve. (b) The probability that the variable deviates no more than one
standard deviation from its mean.