4.21.Show that the units in equation 4.19 are consistent on
either side of the equation.
4.22.Derive equations 4.35–4.37.
4.23.Derivatives of which of the following functions are ex-
act differentials?
(a)F(x, y) x y
(b)F(x, y) x^2 y^2
(c)F(x, y) xnyn, nany integer
(d)F(x, y) xmyn, mn, m, nany integer
(e)F(x, y) y sin(xy).
4.24.Show that ( S/ p)TV.
4.25.Starting with the natural variable equation for dH,
show that
H
pTV(1 T)4.26.When changes in the conditions of a system are infini-
tesimal, we use the or dsymbol to indicate a change in a
state variable. When they are finite, we use the symbol to
indicate the change. Rewrite the natural variable equations
4.14–4.17 in terms of finite changes.
4.27.Equation 4.19 says that
U
VSpIf we are considering the variation of U, the change in the
change of the internal energy, we can write that as (see the
previous problem for an analogous argument)
(
VU)
Sp
Show that this is entirely consistent with the first law of ther-
modynamics.
4.28.For an isentropic process, what is the approximate
change in Uif a system consisting of 1.0 mole of gas goes
from 7.33 atm and 3.04 L to 1.00 atm and 10.0 L? Hint:see
the previous problem.
4.29.Use the ideal gas law to demonstrate the cyclic rule of
partial derivatives.
4.30.Show that for an ideal gas,
Cp
U
TpH
pSp
TV^04.31.Show that
^
V
S
T 1where is the expansion coefficient and is the isothermal
compressibility.
4.32.Evaluate ( U/ V)Tfor an ideal gas. Use the expression
from Example 4.10. Does your answer make sense?
4.33.Determine an expression for ( p/ S)Tfor an ideal gas
and for a van der Waals gas.
4.7 Focus on G
4.34.Determine the value of the derivative {[ (G)]/ T}pfor
the solid-state reaction
2Al Fe 2 O 3 →Al 2 O 3 2Fe
(Hint:see exercise 3.37.)
4.35.Derive the equivalent of the Gibbs-Helmholtz equation,
but for the Helmholtz energy A.
4.36.A plot of 1/Tversus G/Thas what slope?
4.37.A 0.988-mole sample of argon expands from 25.0 L to
35.0 L at a constant temperature of 350 K. Calculate Gfor
this expansion.
4.38.Verify the manipulation of equation 4.41 into equation
4.42. Can you see how the chain rule of derivatives plays an im-
portant role in the derivation of the Gibbs-Helmholtz equation?
4.39.Use equation 4.45 as an example and find an expres-
sion for Aas the volume varies.4.8 & 4.9 The Chemical Potential and Fugacity
4.40.Why is there no nvariable in equation 4.54 like there is
in equation 4.45?
4.41.What is the change in the chemical potential of a system
if 1 mole of O 2 were added to a system already containing
1 mole of O 2? Probably the best answer is ‘’no change.’’ Why?
4.42.Is an extensive or intensive variable? What about the
partial molar volume? The partial molar entropy?
4.43.Write the fundamental equation of chemical thermody-
namics for a system that contains 1.0 mole of N 2 and 1.0 mole
of O 2.
4.44.Calculate the molar change in chemical potential of
an ideal gas that expands by 10 times its original volume at
(a)100 K, and (b)300 K.
4.45.Calculate the change in chemical potential of an ideal
gas that goes from 1.00 atm to 1.00 bar at 273.15 K. How
large an absolute amount of change do you think this is?
4.46.Can equation 4.61 be used to calculate for an ideal
gas? Why or why not?
4.47.Which of the following in each pair of systems do you
think has the greater chemical potential? (a)1.0 mole of H 2 O ()
at 100°C or 1.0 mole of H 2 O (g) at 100°C? (b)10.0 g of Fe
at 25°C or 10.0 g of Fe at 35°C? (c)25.0 L of air at 1 atm
pressure or the same amount of air but compressed isother-
mally to 100 atm pressure?
4.48.Use equation 4.46 to argue that the absolute chemical
potential for any substance has a positive value.
4.49.Of helium and oxygen gases, which one do you expect
to have a larger deviation from ideality at the same high pres-
sure? Is this the same gas that you would expect to have a
larger deviation from ideality at moderate pressure? How about
at very low pressure?116 Exercises for Chapter 4