Physical Chemistry Third Edition

3.5 The Third Law of Thermodynamics and Absolute Entropies 141

for low temperatures, an approximate theory of Debye^6 can be used in lieu of data.

According to this theory, the vibrations of a crystal of a monatomic substance produce

a contribution toCVat low temperatures that is proportional toT^3. An approximate

theory for the motion of mobile electrons in metals gives a contribution proportional

toT,^7 so that for sufficiently low temperature

CV, maT^3 +bT (valid at low temperature) (3.5-3)

whereaandbare parameters that can be determined from experimental data. For

electrical insulators the parameterbis equal to zero. Equation (3.5-3) is quite reliable

up to temperatures of about 15 K. Above this temperature data are usually available.

Since the difference betweenCPandCVis numerically small for solids, Eq. (3.5-3) is

usually used forCPas well as forCV.

Peter J. W. Debye, 1884–1966, was a
Dutch-American physicist and chemist
who received the Nobel Prize in
chemistry in 1936 for his work on the
dipole moments of molecules and who
made numerous other important
contributions.

Exercise 3.17

a.Show that if Eq. (3.5-3) is valid between zero temperature and some temperatureT 1 and if

b0, the value of the molar entropy atT 1 is given by

Sm(T 1 )

aT 13

3



CV, m(T 1 )

3

(3.5-4)

b.Find the expression forSm(T 1 )ifbis not equal to zero.

If a phase transition occurs between zero temperature and the temperature of interest,

Eq. (3.5-2) must be modified to include the entropy change of the phase transition. If

the substance is a liquid at temperatureT 1 , the analogue of Eq. (3.5-2) is

Sm(T 1 )−

∫Tf

0

CP, m(s)

T

dT+

∆fusHm

Tf

+

∫T 1

Tf

CP, m(l)

T

dT (liquid system)

(3.5-5)

whereTfis the reversible melting (fusion) temperature,∆fusHmis the molar enthalpy

change of fusion (melting),CP, m(s) is the molar heat capacity of the solid, andCP, m(l)

is the molar heat capacity of the liquid.

Exercise 3.18

Write the equation analogous to Eq. (3.5-5) that applies to a gaseous substance.

Calculation of Entropy Changes for Chemical Reactions

Since absolute (third-law) entropies can be calculated from experimental data, tables

of their values have been created. Some values for substances in their standard states

(^6) P. Debye,Ann. Physik, 17 (4), 817 (1911). See Section 22.3.
(^7) J. S. Blakemore,Solid State Physics, 2nd ed., Saunders, Philadelphia, 1974, p. 176ff.