Physical Chemistry , 1st ed.

correction and is related to the magnitude of the interactions between gas par-

ticles. The van der Waals constant bis the volume correction and is related to

the size of the gas particles. Table 1.6 lists van der Waals constants for various

gases, which can be determined experimentally. Unlike a virial equation, which

fits behavior of real gases to a mathematical equation, the van der Waals equa-

tion is a mathematical model that attempts to predict behavior of a gas in terms

of real physical phenomena (that is, interaction between gas molecules and the

physical sizes of atoms).

Example 1.5

Consider a 1.00-mole sample of sulfur dioxide, SO 2 , that has a pressure of

5.00 atm and a volume of 10.0 L. Predict the temperature of this sample of

gas using the ideal gas law and the van der Waals equation.

Solution

Using the ideal gas law, we can set up the following expression:

(5.00 atm)(10.0 L) (1.00 mol)0.08205 

m

L

o

a

l

t



m

K

(T)

and solve for Tto get T609 K. Using the van der Waals equation, we first

need the constants aand b. From Table 1.6, they are 6.714 atmL^2 /mol^2 and

0.05636 L/mol. Therefore, we set up

5.00 atm  (10.0 L 1.00 mol)0.05636 m

L

ol



(1.00 mol)0.08205 

m

L

o

a

l

t



m

K

(T)

Simplifying the left-hand side of the equation:

(5.00 atm 0.06714 atm)(10.0 L 0.05636 L)

(1.00 mol)0.08205 

m

L

o

a

l

t



m

K

(T)

(5.067 atm)(9.94 L) (1.00 mol)0.08205 

m

L

o

a

l

t



m

K

(T)

Solving for T, one finds T613 K for the temperature of the gas, 4° higher

than the ideal gas law.

The different equations of state are not always used independently of each

other. We can derive some useful relationships by comparing the van der Waals

equation with the virial equation. If we solve for pfrom the van der Waals

equation and substitute it into the definition of compressibility, we get

Z

p

R

V

T



V

V

b



RT

a

V

 (1.21)

which can be rewritten as

Z

1 

1

b/V



RT

a

V



6.714 

a

m

tm

o



l

L

2

2

(1 mol)^2



(10 L)^2

14 CHAPTER 1 Gases and the Zeroth Law of Thermodynamics

Table 1.6 Van der Waals parameters for

various gases

ab

Gas (atmL^2 /mol^2 ) (L/mol)

Acetylene, C 2 H 2 4.390 0.05136

Ammonia, NH 3 4.170 0.03707

Carbon dioxide, 3.592 0.04267

CO 2

Ethane, C 2 H 6 5.489 0.0638

Ethylene, C 2 H 4 4.471 0.05714

Helium, He 0.03508 0.0237

Hydrogen, H 2 0.244 0.0266

Hydrogen chloride, 3.667 0.04081

HCl

Krypton, Kr 2.318 0.03978

Mercury, Hg 8.093 0.01696

Methane, CH 4 2.253 0.0428

Neon, Ne 0.2107 0.01709

Nitric oxide, NO 1.340 0.02789

Nitrogen, N 2 1.390 0.03913

Nitrogen dioxide, 5.284 0.04424

NO 2

Oxygen, O 2 1.360 0.03183

Propane, C 3 H 8 8.664 0.08445

Sulfur dioxide, 6.714 0.05636

SO 2

Xenon, Xe 4.194 0.05105

Wa t e r, H 2 O 5.464 0.03049

Source:D. R. Lide, ed.,CRC Handbook of Chemistry and

Physics,82nd ed., CRC Press, Boca Raton, Fla., 2001.