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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 87
Assuming the heat capacity to be constant over the temperature range of inter-
est, the change in entropy as a function of temperature is given by [3B.7–90],
S(Tf) = S(Ti) + Cp ln (Tf/Ti).�erefore
S−○m(200 K) = (29.79 JK−1mol−1) + (24.44 JK−1mol−1) × ln(200 K
100 K
)
= 46.73 JK−1mol−1 .
�e di�erence is slight as expected because Cp ,m does not vary signi�cantly in
the given temperature range.
P3C.10 �e temperature dependence of the entropy is given by [3C.1a–92], S(T2) =
S(T1) = ∫
T2
T1 (Cp ,m/T)dT . Given that the heat capacity at the lowest tempera-
tures is the sum of theDebye, aT3, and electronic, bT , contributions, themolar
heat capacity is Cp ,m(T) = aT3 + bT .�erefore, the molar entropy from zero
as a function of temperature is given by
Sm(T) − Sm(0) = ∫
T
0
Cp ,m
T ′
dT ′ = ∫
T
0
aT ′3 + bT ′
T ′
dT ′
= a
3
T3 + bT
Setting the contributions to the entropy corresponding to each term to be equal
gives
a
3
T3 = bT
hence T =
√
3b
a
�erefore this temperature is
T =
¿
ÁÁÀ3 × 1.38 × 10−3 J K−2 mol−1
0.507 × 10−3 JK−4mol−1
= 2.86 K .
Because the Debye term in the entropy expression has a cubic dependence on
T , as the temperature increases it will dominate over the linear electronic term.
�us, the Debye contribution .
3D Concentrating on the system
Answers to discussion questions
D3D.2 As is discussed in detail in Topic 3D the criteria for spontaneity at constant
volume and temperature is expressed in terms of theHelmholtz energy, dA ≤ 0,
and at constant pressure and temperature in terms of the Gibbs energy, dG ≤ 0.
Both the Helmholtz and Gibbs energies refer to properties of the system alone.
However, because of the way they are de�ned these quantities e�ectively allow