Page 129 - Essentials of physical chemistry

The Second and Third Laws of Thermodynamics 91

Once again, the verbal law is full of hidden meaning. Let us go back to Boltzmann’s basic equation
(carved into his tombstone) to see the physical implications:

S ¼ k ln W or S ¼ k ln V:

(Perhaps, we can now see how the letter ‘‘W ’’ evolved from the Greek letter ‘‘V’’?)
We know that the natural logarithm of the number 1 is zero. A perfectly crystalline pure
substance has a lattice structure that extends in all three dimensions as a perfectly repetitive pattern
of atoms or molecules such that one cannot distinguish an imperfection that would aid in defining
any list of alternate structures. A perfect crystal only has ‘‘one structure,’’ not a list of possibilities
where there might be some imperfection here or there. As far as the constant ‘‘k’’ is concerned,
Boltzmann applied his statistics at the atom=molecular level and was interested in the gas constant
per particle rather than the gas constant per mole so that basically k B is the gas constant for
1 atom=molecule. It is a precise number based on the gas constant and the best known value of
the Avogadro number:

R 8:314472 J= K mol 23 16

¼ 1:3806505 10 J= K ffi 1:38 10 erg= K:

k B ¼ ¼ 23 1
N Av 6:0221415 10 mol
So, now maybe we can combine your understanding of the third law. It says that entropy, S,is
basically statistical where ‘‘W’’ in Boltzmann’s equation refers to ‘‘the number of ways the system
can exist.’’ In the case of a perfectly crystalline system, W might evolve into a number more than 1 if
the atoms swing and sway during vibration, but at 08K, the vibrations will be minimized, although
maybe not completely. There are more pitfalls as we consider this further, but first let us look at the
simple interpretation of absolute entropy:

T mp T bp ð T

ð ð
C P (sol) dT DH fus C P (liq) dT DH vap C P (gas) dT
:
S tot ¼ þ þ þ þ
T T mp T T bp T
0 T mp T bp
The simple interpretation is that a perfectly crystalline solid increases in randomness with heat at
low temperatures until the lattice structure collapses at the melting point temperature where there is
a large change in disorder since the liquid is more random than the crystal lattice. Then the liquid
warms and becomes more random until the boiling point is reached. Then there is a much larger
increase in randomness as the vaporization occurs. After the gas phase is reached, it can still increase
its disorder at higher temperatures, so the last term represents the increase in gas entropy with further
heating. This is the overall picture, but there has been a lot of research at what happens at very low
temperatures near or below 18K. For practical room temperature thermodynamics, the above
equations suffice very well to describe solids, liquids, and gases.
As expected, there are always problems in the details. Let us think about trying to reach absolute
08K temperature. If we consider highly pure materials, a problem crops up in that many elements
have several nuclear isotopes. A really bad case would be to try to crystallize HCl. We would have
37
35
1
0
2
to somehow isolate either the Cl or the Cl isotope as well as sort out the isotopes of H, D, and T.
17 17 1 1 1
Then we would worry over whether the molecules were vibrating in a net symmetric or asymmetric
way (what physicists call quantized ‘‘phonon’’ lattice vibrations). We would also have to worry
about whether the nuclear spins are aligned. Experiments have actually been done where a cold
sample is put into a strong electromagnet to align the nuclear spins (should we worry about electron
spins?) and when the magnetic field is turned off, the spins randomize but absorb heat as they do so;
this so-called ‘‘adiabatic demagnetization’’ can be used to obtain very low temperatures of the order
of 0.0018K, but the remaining randomness prevents attainment of true 08K. Thus, for several