Physical Chemistry Third Edition

26.5 Miscellaneous Topics in Statistical Thermodynamics 1117

These equations let us interpret the chemical potential as a kind of benchmark energy.

Energies larger thanμare less probable than energies smaller thanμ. We will use

Eq. (26.5-4) in Chapter 28 when we discuss electrons in an electrical conductor.

The Interpretation of Heat and Work in Statistical

Mechanics

From Eq. (4.2-3) we have the relation for reversible changes in a closed simple system

dUTdS−PdV (26.5-6)

Since we identify the mechanical energy of the system,E, with its thermodynamic

energy,U, we can write a statistical mechanical expression fordU:

dUdEd

(

∑

i

Niεi

)



∑

i

εidNi+

∑

i

Nidεi (26.5-7)

We have assumed that the molecule energy eigenvalues depend only on the volume of

the system, so that

dU

∑

i

εidNi+

∑

i

Ni

dεi

dV

dV (26.5-8)

From the expression for the pressure in Eq. (26.1-10) we can write

dU

∑

i

εidNi−PdV

∑

i

εidNi+dwrev (26.5-9)

where we have used the fact thatdwrev−PdVfor a reversible process in a simple

system such as a dilute gas.

Comparison with the first law of thermodynamics shows that

dqrev

∑

i

εidNi (26.5-10)

Heat added to a system corresponds to a change in energy resulting from changes in

the numbers of molecules occupying the energy levels. Work done on a dilute gas must

be represented by the other term indE, corresponding to a change in energy resulting

from shifts of the molecular energy levels.

More on the Identification of the Statistical Entropy and

the Thermodynamic Entropy

We can now shed some additional light on the relation between statistical entropy and

thermodynamic entropy, and also on the identification of the parameterβwith 1/kBT.

Using the definition of the statistical entropy in Eq. (26.1-1) and replacing ln(Ω)byits

largest term, ln(Wmp), we obtain

dSstkBdln(Wmp)kB

∑

i

(

∂ln(Wmp)

∂Ni

)

dNi (26.5-11)