Physical Chemistry Third Edition

134 3 The Second and Third Laws of Thermodynamics: Entropy

We now specify the positions of the particles in a new way. We mentally divide

the volume of the box into a numberMof rectangular cells of equal size stacked in a

three-dimensional array (lattice) as shown schematically in Figure 3.11. Since the cell

boundaries are imaginary the molecules pass freely through them. Instead of giving the

values of three coordinates to specify the location of one particle we specify which cell

it occupies. This“coarse-grained” descriptiongives less precise information about the

location of the particles than specifying exact coordinate values, but we can increase

the precision by making the cells smaller. We continue to specify the velocity of each

particle by specifying the values of three velocity components.

Figure 3.11 The Lattice Gas.

We assume that the probability that a randomly chosen molecule has a particular

velocity is independent of the probability that it has a particular position. It is a fact of

probability theory that the number of ways of accomplishing two independent events

is the product of the number of ways of accomplishing each event. The thermodynamic

probability is therefore the product of two factors, one for the coordinates and one for

the velocities:

ΩΩcoordΩvel (3.4-2)

We now seek a formula to representΩcoordfor our lattice gas. The number of possible

coordinate states for a single molecule is equal to the number of cells,M. Since the

molecules are mass points, the presence of one molecule in a cell does not keep other

molecules from occupying the same cell. Any state of a second molecule can occur

with any state of the first molecule, so the number of possible coordinate states for two

molecules isM^2. Any state of a third molecule can occur with any state of the first pair

of molecules, so the number of possible states for three particles isM^3. For a system

ofNmolecules,

ΩcoordMN (3.4-3)

EXAMPLE3.13

Calculate the value ofΩcoordand ln(Ωcoord) for 1.000 mol of an ideal gas in a volume of

24.4 L if cells of 0.500 nm on a side are taken.

Solution

Vcell(0. 500 × 10 −^9 m)^3  1. 25 × 10 −^28 m^3

M

V

Vcell



0 .0244 m^3

1. 25 × 10 −^28 m^3

 1. 95 × 1026

ΩcoordMN(1. 95 × 1026 )^6.^02 ×^10

23

 101.^58 ×^10

25

This is such a large number that we are not able to give any significant digits.

ln(Ωcoord) 6. 02 × 1023 ln(1. 95 × 1026 ) 3. 65 × 1025

The value ofΩcoordin Example 3.10 is calculated on the assumption that the particles

are distinguishable from each other. When quantum mechanics is studied, it is found

that identical particles must be treated as inherently indistinguishable from each other.

The number of ways of rearrangingNdistinguishable objects isN!(Nfactorial), which