Modern Thermodynamics

- Chapter 1

33

F =

Momentum imparted

∆t

=

2mvxavg∆xA

∆t

n

2

=

mvxavg∆xAn

(∆x/vxavg )

= mvxavgnA.

2

(1.6.4)

Pressure, p, which is the force per unit area, is thus:

p =

F

A

= mvxavgn

2

(1.6.5)

Since the direction x is arbitrary, it is better to write this expression in terms of the average speed of the

molecule rather than its x-component. By using (1.6.2) and the definitions (1.6.3), we can write the

pressure in terms the macroscopic variables M, V and N:

p =

1

3

mnvavg

2

=

1

3

M

N

V

vavg

2

(1.6.6)

This expression relates the pressure to the square of the average speed. A rigorous description of the

random motion of molecules leads to the same expression for the pressure with the understanding that

v2avg is to be interpreted as the average of the square of the molecular velocity – a distinction that will

become clear when we discuss the Maxwell velocity distribution. When Daniel Bernoulli published the

above result in 1738, he did not know how to relate the molecular velocity to temperature; that connection

had to wait until Avogardo's stated his hypothesis in 1811 and the formulation of the ideal gas law based

on an empirical temperature that coincides with the absolute temperature that we use today (see Eqn,

(1.3.9)). On comparing expression (1.6.6) with ideal gas equation, pV=NRT, we see that:

RT =

1

3

Mvavg

2

(1.6.7)

Using the Boltzmann constant kB=R/NA=1.3807x10-23 J/K and noting m=M/NA, we can express (1.6.7) as

a relation between the kinetic energy and temperature:

1

2

mvavg

2

=

3

2

kBT (1.6.8)