Physical Chemistry , 1st ed.

Because of the various algebraic relationships between the virial coefficients

in equations 1.17 and 1.18, typically only one set of coefficients is tabulated

and the other can be derived. Again,B(or B ) is the most important virial co-

efficient, since its term makes the largest correction to the compressibility,Z.

Virial coefficients vary with temperature, as Table 1.4 illustrates. As such,

there should be some temperature at which the virial coefficient Bgoes to zero.

This is called the Boyle temperature, TB, of the gas. At that temperature, the

compressibility is

Z

p

R

V

T



V

0

  

where the additional terms will be neglected. This means that

Z 

p

R

V

T



and the real gas is acting like an ideal gas. Table 1.5 lists Boyle temperatures of

some real gases. The existence of Boyle temperature allows us to use real gases

to study the properties of ideal gases—if the gas is at the right temperature,

and successive terms in the virial equation are negligible.

One model of ideal gases is that (a) they are composed of particles so tiny

compared to the volume of the gas that they can be considered zero-volume

points in space, and (b) there are no interactions, attractive or repulsive, be-

tween the individual gas particles. However, real gases ultimately have behav-

iors due to the facts that (a) gas atoms and molecules dohave a size, and (b)

there is some interaction between the gas particles, which can range from min-

imal to very large. In considering the state variables of a gas, the volume of the

gas particles should have an effect on the volume Vof the gas. The interactions

between gas particles would have an effect on the pressure pof the gas. Perhaps

a better equation of state for a gas should take these effects into account.

In 1873, the Dutch physicist Johannes van der Waals (Figure 1.9) suggested

a somewhat corrected version of the ideal gas law. It is one of the simpler equa-

tions of state for real gases, and is referred to as the van der Waals equation:

p

a

V

n

2

2

(Vnb) nRT (1.20)

where nis the number of moles of gas, and aand bare the van der Waals con-

stantsfor a particular gas. The van der Waals constant arepresents the pressure

1.6 Nonideal Gases 13

Table 1.4 The second virial coefficient

B(cm^3 /mol) at various

temperatures

Temperature (K) He Ne Ar

20 3.34 — —

50 7.4 35.4 —

100 11.7 6.0 183.5

150 12.2 3.2 86.2

200 12.3 7.6 47.4

300 12.0 11.3 15.5

400 11.5 12.8 1.0

600 10.7 13.8 12.0

Source:J. S. Winn,Physical Chemistry,HarperCollins, New

York, 1994

Table 1.5 Boyle temperatures for

various gases

Gas TB(K)

H 2 110

He 25

Ne 127

Ar 410

N 2 327

O 2 405

CO 2 713

CH 4 509

Source:J. S. Winn,Physical Chemistry,Harper

Collins, New York, 1994

Figure 1.9 Johannes van der Waals (1837–1923),

Dutch physicist who proposed a new equation

of state for gases. He won a 1910 Nobel Prize for

his work.

Photo by Gen. Stab. Lit. Anst, courtesy AIP Emilio Segre Visual Archives, W. F. MeggersGallery of Nobel Laureates and Weber Collection