Physical Chemistry Third Edition

424 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

whereVm/n, the molar volume, where we define the constantb,

b

Vexc

n



N

n

2

3

πd^3 NAv

2

3

πd^3 (9.8-5)

and whereNAvis Avogadro’s constant.

If we write Eq. (9.8-4) in the form

P

RT

Vm−b

(9.8-6)

we can see that the pressure of our hard-sphere gas is greater than that of an ideal gas

with the same values ofVmandT. Since our hard-sphere system has only repulsive

forces, this agrees with our earlier assertion that repulsive forces make a positive

contribution to the pressure. Equation (9.8-4) resembles the van der Waals equation

of state of Eq. (9.3-1) except for the absence of the term containing the parameter

a. The parameterarepresents the effect of attractive forces, which make a negative

contribution to the pressure. The argument has been advanced that attractive forces

must slow a particle down just before it strikes a wall because other particles will be

only on the side of the particle away from the wall as it strikes the wall. Similarly,

repulsive forces accelerate a particle as it strikes the wall.

EXAMPLE9.16

Assume that the parameterbin Eq. (9.8-6) can be identified with the van der Waals parameter

b. Calculate the radius of an argon atom from the value of the van der Waals parameterbin

Table A.3.

Solution

From Table A.3,b 3. 219 × 10 −^5 m^3 mol−^1

bNAv

2

3

πd^3

d^3 

3(3. 219 × 10 −^5 m^3 mol−^1 )

2 π(6. 022 × 1023 mol−^1 )

 2. 55 × 10 −^29 m^3

d 2. 94 × 10 −^10 m

From Table A.15,d 3. 61 × 10 −^10 m. This value is calculated from viscosity data taken

at 293 K. The values differ because atoms and molecules are not actually hard spheres, and

different kinds of measurements give different effective hard-sphere sizes.

The virial equation of state of Eq. (1.3-3) is a widely used equation of state for

nonideal gases. Classical statistical mechanics provides formulas for calculating the

virial coefficients of a nonideal monatomic gas. We present the following formula for

the second virial coefficient without derivation:

B 2 −

NAv

2

∫∞

0

e−u(r)/kBT− 14 πr^2 dr (9.8-7)

whereris the intermolecular distance andu(r) is the pair potential energy function.