Statistical Physics, Second Revised and Enlarged Edition

The connection to thermodynamics 69

asfollows:

S/kkkB=ln definition

=lnt∗ usual approximation

=ln

(∏

gnii/ni!

)

usingtttMB,( 5. 6 )

=

∑

(nilngi−nilnni+ni) Stirling’s approximation

=

∑

ni(lngi/ni+ 1 ) rearranging

=

∑

ni(lnZ−lnN+εi/kkkBT+ 1 ) MB result forni/gi,i.e.(6.1)

withA=N/Z

=N(lnZ−lnN+ 1 )+U/kkkBT identifyingU

Hence the result for any MB gas (in this proof, no assumptions are made about any
specific monatomicgas etc.)is

S=NkkkB(lnZ−lnN+ 1 )+U/T (6.14)

For the monatomic, spinless gas under particular consideration in this chapter, we
haveevaluatedZ,( 6. 6 ), andU,( 6 .12). Hence in this case we have

S=NkkkB(lnV−lnN+^32 lnT)+S 0 (6.15)

with the ‘entropyconstant’S 0 given by

S 0 =NkkkB[ 23 ln( 2 πMkkkB/h^2 )+^52 ] (6.1 5 a)

Equation (6.15) (the Sackur–Tetrode equation) is an interesting result. It is classical
inthe sense thatit cannotbe correctdown to theabsolute zero;lnT→−∞whereas
the physicalShas a lower limit of zero atT= 0. Nevertheless it contains Planck’s
constantinthe entropy constantS 0. Furthermore this constant canbecheckedby
experiment asfollows. We measure thespecificandthelatentheats ofaspecific
substance from (essentially)T=0 up to a temperature at which the substance is an
idealgas. This enables us to calculateScalorimetrically, using thefact thatS= 0 at
T= 0 , from an expression of the type

Scal=

∫

0

∫∫

(C/T)dT+L 1 /T 1 +L 2 /TTT 2

thesubscript1referring to thesolid–liquidtransition and2totheliquid–gas transition.
In this waythe value (6.15a) of the constantS 0 has been accuratelyverified.