Physical Chemistry Third Edition

136 3 The Second and Third Laws of Thermodynamics: Entropy

The Statistical Entropy and the Thermodynamic Entropy

We now show that the change in the statistical entropy of our lattice gas for an isothermal

volume change is equal to the change in the thermodynamic entropy for an ideal gas

for the same volume change. Equation (3.3-3) gives the change in the thermodynamic

entropy for an isothermal volume change in an ideal gas:

∆SnRln(V 2 /V 1 ) (3.4-11)

whereV 2 is the final volume andV 1 is the initial volume. The change in the statistical

entropy for any process is

∆SstkBln(Ω 2 )−kBln(Ω 1 )kBln(Ω 2 /Ω 1 )kBln

(

Ωcoord(2)Ωvel(2)

Ωcoord(1)Ωvel(1)

)

(3.4-12)

The velocity of a molecule is independent of its position, so we assert that the

velocity factorΩvelwill not change in an isothermal change of volume. The velocity

factors in Eq. (3.4-11) cancel, and

∆SstkBln

(

Ωcoord(2)

Ωcoord(1)

)

(3.4-13)

Using Eq. (3.4-6) and the fact thatNis fixed,

∆SstkBln

(

MN 2

MN 1

)

kBln

((

M 2

M 1

)N)

NkBln

(

M 2

M 1

)

(3.4-14)

The correction for indistinguishability does not affect∆Sstfor our process since the

two divisors ofNcancel.

In order to maintain a given precision of position specification, the size of the cells

must be kept constant, so that the number of cells is proportional to the volume of the

system:

M 2 /M 1 V 2 /V 1 (3.4-15)

Therefore,

∆SstNkBln(V 2 /V 1 ) (3.4-16)

The change in the statistical entropy is identical to the change in thermodynamic entropy

given in Eq. (3.4-11) if we let

NkBnR (3.4-17)

which assigns the value to Boltzmann’s constant that we have already used:

kB

nR

N



R

NAv



8 .3145 J K−^1 mol−^1

6. 0221 × 1023 mol−^1

 1. 3807 × 10 −^23 JK−^1 (3.4-18)

We assert without further discussion that the equivalence between the statistical and

thermodynamic entropies for isothermal expansions in a lattice gas is typical of all

systems and all processes and write the general relation