Concise Physical Chemistry

c08 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come

THE THERMODYNAMIC FUNCTIONS 113

8.4 ENTROPY AND PROBABILITY

If you drop marbles randomly into a large box with 75 compartments painted green

and 25 compartments painted red, the ratio of occupation numbers of marbles in

compartments tends toward 75/25=3/1 over very many trials. If you put all the

marbles into the green compartments and shake the box, the distribution of marbles

in compartments will approach 3/1, as before, even though the potential energy of

the boxes in the gravitational field is the same (for a level box). The ratio of the

number of red compartments to the number of green compartments is the ratio of

microstates for the system. It is independent of the size of the system; 7500/2500

is the same as 75/25. Call this ratioW. Boltzmann made the connection between

statistics (probability ratio) and thermodynamics by defining the proportionality

S∝lnW

but this is just the way we found the A, B equilibrium constant except that it doesn’t

contain the temperatureT.Yo usupply the “thermal energy” by shaking the box. If

you don’t shake the box, it does not approach equilibrium (no reaction at 0 K).

The universal constant of proportionality betweenSand lnWis now designated

kBand called Boltzmann’s constant in honor of this great and tragic figure^1 :

S≡kBlnW

kB= 1. 38066 × 10 −^23 JK−^1

8.5 THE THERMODYNAMIC FUNCTIONS

If the number of particles at each energy level

ni=

ne−εi/kBT

Q

is multiplied by the energy at that levelεi, then the total energy is the sum of the

individual contributions

E=

∑

i

εini=

n

∑

i

e−εi/kBT

Q

Having the energy and entropy, we have all that is necessary to derive the rest of

the thermodynamic functions in terms of the partition functionQ. Irikura (1998) has

given the necessary equations in compact form, which we present as Table 8.1.

(^1) Suicide, 1906.