1549380323-Statistical Mechanics Theory and Molecular Simulation

Ideal gas 91

P

kT

=

(

∂ln Ω

∂V

)

N,E

. (3.5.25)

Since Ω∼VN, ln Ω∼NlnV so that the derivative yields

P

kT

=

N

V

, (3.5.26)

or

P=

NkT

V

=ρkT, (3.5.27)

where we have introduced thenumber density,ρ=N/V, i.e., the number of particles

per unit volume. Eqn. (3.5.27) is the familiar ideal gas equation of state or ideal gas

law (cf. eqn. (2.2.1)), which can be expressed in terms of the number of moles by

multiplying and dividing by Avogadro’s number,N 0 :

PV=

N

N 0

N 0 kT=nRT. (3.5.28)

The productN 0 kyields thegas constantR, whose value is 8.314472 J·mol−^1 ·K−^1.

0 1 2 3 4 5

V

0

2

4

6

8

10

P

0 1 2 3 4 5

T

0

2

4

6

8

10

T 1 P

T 2

T 3

T 1 > T 2 > T 3

r 1

r 2

r 3

r 1 > r 2 > r 3

Fig. 3.2(Left) Pressure vs. volume for different temperatures (isothermsof the equation of

state (2.2.1)). (Right) Pressure vs. temperature for different densitiesρ=N/V.

Fig. 3.2 (left) is a plot ofP vs.V for different values ofT. The curves are known

as theisothermsof the ideal gas. From the figure, the inverse relationship between

pressure and volume can be clearly seen. Similarly, Fig. 3.2 (right) shows a plot ofP

vs.Tfor different densities. The lines are theisochoresof the ideal gas. Because of the

absence of interactions, the ideal gas can only exist as a gas underallthermodynamic

conditions.