Physical Chemistry Third Edition

400 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

EXAMPLE 9.3

Show that the Gaussian distribution of Eq. (9.3-33) is normalized.

Solution

∫∞

−∞

f(u)du

1

√

2 πσ

∫∞

−∞

e−(u−μ)

(^2) / 2 σ 2
du
1
√
2 πσ
∫∞
−∞
e−w
(^2) / 2 σ 2
dw

1
√
2 πσ
√
2 σ
∫∞
−∞
e−t
2
dt
1
√
2 πσ
√
2 σ
√
π 1
where we have looked up the integral in Appendix C.
The Gaussian curve is named for
Johann Carl Friedrich Gauss,
1777–1855, a great German
mathematician who made many
fundamental contributions to
mathematics.
The standard deviation of an arbitrary probability distribution is defined by
σu

(〈

u^2

〉

−〈u〉^2

) 1 / 2

(definition ofσu) (9.3-34)

The standard deviation is a convenient measure of the width of a probability distribution.

For a Gaussian distribution the probability thatulies betweenμ−σandμ+σis equal

to 0.683. For most other probability distributions the probability that the independent

variable lies within one standard deviation of the mean is approximately equal to

two-thirds.

Exercise 9.7

a.Show that〈u〉μfor the Gaussian distribution of Eq. (9.3-33).

b.Show that the definition in Eq. (9.3-34) when applied to the Gaussian probability distribution

leads toσuσ.

By comparison of Eq. (9.3-32) and Eq. (9.3-33), the standard deviationσvxfor the

probability distribution ofvxis given by

σvx

√

kBT

m



√

RT

M

(9.3-35)

In the second equalityMis the molar mass. The value ofσvxfor O 2 molecules at 298 K

is equal to 278 m s−^1 , which is equal to 623 miles per hour.

The standard deviation of a functionh(u) is defined by

σh

[〈

h(u)^2

〉

−〈h(u)〉^2

] 1 / 2

(definition) (9.3-36)

EXAMPLE 9.4

Obtain a formula forσκx, the standard deviation of thevxcontribution to the kinetic energy,

κxmv^2 x/2.