Modern Thermodynamics

- Chapter 1

40

becomes broader as T increases. The speed v at which f(v) reaches its maximum is the most probable

speed.

With the above probability distributions we can calculate several average values. We shall use

the notation in which the average value of a quantity X is denoted by X. The average speed is given by

the integral:

v = vf(v)dv

0

∞

∫

(1.6.17)

For the probability distribution (1.6.15), such integrals can be calculated using integral tables or

Mathematica or Maple. While doing such calculations it is convenient to write the probability f(v) as:

f(v)dv =

4π

z

e−βv

2

v2dv

β =

M

2RT

,

1

z

=

M

2πRT

3 / 2

(1.6.18)

Using the appropriate integral in Appendix 1.2 at the end of this chapter, the average speed can be

obtained in terms of T and the molar mass M (exc 1.23).

v =

4π

z

v3e−βv

2

dv

0

∞

∫

=

4π

z

1

2β

2

=

8RT

πM

(1.6.19)

Similarly, one can calculate the average energy of a single molecule using m and kB instead of M and R

(exc 1.23):

1

2

mv2

=

m4π

2z

v4e−βv

2

dv

0

∞

∫

=

m2π

z

3 π

8β 5 / 2

=

3

2

k

B

T

(1.6.20)

A rigorous calculation of the pressure using Maxwell-Boltzmann velocity distribution leads to the

expression (1.6.6) in which v2avg = v2. Also, the value of v at which f(v) has a maximum is the most

probable speed. This can easily be determined by setting df/dv = 0, a calculation left as an exercise.

What do the above calculations tell us? First, we see that the average speed of a molecule is

directly proportional to the square root of the absolute temperature and inversely proportional to its molar

mass. This is one of the most important results of kinetic theory of gases. Another point to note is the