Physical Chemistry Third Edition

196 4 The Thermodynamics of Real Systems

whereμiis the chemical potential of componenti, and additional useful formulas were

obtained from this.

For any extensive quantity, represented byY, Euler’s theorem is

Y

∑c

i 1

niYi

whereYiis the partial molar quantity. The generalized Gibbs–Duhem relation is

∑c

i 1

nidYi

(

∂Y

∂T

)

P,n

dT+

(

∂Y

∂P

)

T,n

dP

and the original Gibbs–Duhem relation is

∑c

i 1

nidμi0 (constantTandP)

ADDITIONAL PROBLEMS

4.49 Label the following statements as true or false. If a
statement is true only under certain conditions, label it
as false.
a.The minimum of the Gibbs energy of the system
always corresponds to the maximum of the entropy of
the system plus surroundings.
b.The minimum of the Helmholtz energy of a system
always corresponds to the maximum of the entropy of
the system plus surroundings.
c.The minimum of the Gibbs energy always corresponds
to the minimum of the energy of the system plus
surroundings.
d.If a system is at constant temperature and pressure, the
minimum in the Gibbs energy of the system
corresponds to the maximum of the entropy of the
system plus surroundings.
e.If a system is at constant temperature and volume, the
minimum in the Helmholtz energy corresponds to the
maximum of the entropy of the system plus
surroundings.
f. In a reversible adiabatic process, the state of the
surroundings is unaffected.
g.In a reversible adiabatic process, the entropy of the
surroundings is unaffected.

h.In an irreversible adiabatic process, the entropy of the

surroundings is unaffected.

4.50 For each of the following processes, tell whether each of

the quantities∆S,∆G,q, andwis positive, zero, or

negative. Present your answers in the form of a table with

entries +, 0, or−:

a.Liquid water is vaporized at 100◦C and a partial

pressure of water vapor equal to 1.000 atm.

b.Liquid water is vaporized at 100◦C and a partial

pressure of water vapor equal to 0.900 atm.

c.Liquid water is frozen at− 10 ◦C and a pressure of

1.000 atm.

d.Liquid water is frozen at 0◦C and a pressure of

1.000 atm.

4.51 a.Find the value of the heat capacity at constant volume

for 1.000 mol of liquid water at 20.00◦C.

b.Find the value of the adiabatic compressibility of liquid

water at 25.00◦C.

c.Find the speed of sound in liquid water at 20.00◦C,

using the formula from Problem 4.12:

v^2 s

VmCP

MκTCV



Vmγ

MκT