4.2 Spontaneity Conditions
4.1.Explain why conditions for using S0 as a strict spon-
taneity condition imply that Uand Hboth equal zero.
4.2.Explain how the equation
dU^ TpdVdS 0
is consistent with the idea that spontaneous changes occur
with a decrease in energy and an increase in entropy.
4.3.Explain why the spontaneity conditions given in equa-
tions 4.3 and 4.4 are in terms of the general derivatives dU
and dHand not some partial derivative of Uand Hwith re-
spect to some other state variable.
4.4.Prove that the adiabatic free expansion of an ideal gas is
spontaneous.
4.3 Gibbs and Helmholtz Energies
4.5.Derive equation 4.6 from equation 4.5.
4.6.Derive equation 4.8 from equation 4.7.
4.7.The third part of equation 4.9 mentions a condition
called equilibrium, in which there is no net change in the
state of a system. What are the equilibrium conditions for dU,
dH, and dA?
4.8.Calculate Afor a process in which 0.160 mole of an
ideal gas expands from 1.0 L to 3.5 L against a constant pres-
sure of 880 mmHg at a temperature of 37°C.
4.9.What is the maximum amount of non-pVwork that can
be done by the reaction
2H 2 O 2 →2H 2 Oif fG(H 2 O) 237.13 kJ/mol, and fG(H 2 ) fG(O 2 ) 0?
4.10.Consider a piston whose compression ratio is 10:1; that
is, Vf 10 Vi. If 0.02 mole of gas at 1400 K expands re-
versibly, what is Afor one expansion of the piston?
4.11.When one dives, water pressure increases by 1 atm
every 10.55 m of depth. The deepest sea depth is 10,430 m.
Assume that 1 mole of gas exists in a small balloon at that
depth at 273 K. Assuming an isothermal and reversible process,
calculate w, q, U, H, A, and Sfor the gas after it rises to
the surface, assuming the balloon doesn’t burst!
4.12.Calculate G° (25°C) for the following chemical reac-
tion, which is the hydrogenation of benzene to make cyclo-
hexane:
C 6 H 6 () 3H 2 (g) →C 6 H 12 ()Would you predict that this reaction is spontaneous at con-
stant Tand p? Use data in Appendix 2.
4.13.Thermodynamic properties can also be determined for
ions. Determine H, S, and Gfor the following two reac-
tions, which are simply reactions of dissolution:
NaHCO 3 (s) →Na^ (aq) HCO 3 (aq)
Na 2 CO 3 (s) →2Na^ (aq) CO 32 (aq)
Assume standard conditions (standard concentration is 1 M
for ions in aqueous solution), and consult the table of ther-
modynamic properties in Appendix 2. What similarities and
differences are there?
4.14.Calculate Gin two different ways for the following
dimerization of NO 2 :
2NO 2 (g) →N 2 O 4 (g)
Are the two values equal?
4.15.Determine Gfor the following reaction at 0°C and
standard pressure:
H 2 O () →H 2 O (s)
Is the reaction spontaneous? Why are the thermodynamic val-
ues from Appendix 2 not strictly applicable to this reaction un-
der these conditions?
4.16.Batteries are chemical systems that can be used to gen-
erate electrical work, which is one form of non-pVwork. One
general reaction that might be used in a battery is
M (s) ^12 X 2 (s//g) →MX (crystal)
where M is an alkali metal and X 2 is a halogen. Using Appendix
2, construct a table that gives the maximum amount of work
that a battery can provide if it uses different alkali metals and
halogens. Do you know if any of these types of batteries are
actually produced?
4.17.Example 4.2 calculated Afor one step of a Carnot cy-
cle. What is Afor the entire Carnot cycle?4.4–4.6 Natural Variables, Partial Derivatives,
and Maxwell Relationships
4.18.Can CVand Cpbe easily defined using the natural vari-
able expressions for dUand dH? Why or why not?
4.19.Analogous to equation 4.26, what is the expression for
U, assuming one knows the behavior of Aas it varies with re-
spect to temperature and volume?
4.20.Show thatdSdV (( TS// pP))T
VdT
where is the thermal expansion coefficient and is the
isothermal compressibility. Hint:Write a natural variable ex-
pression for dSin terms of Vand Tand substitute for some of
the expressions. You will have to use Maxwell’s relationships
and the chain rule of partial derivatives.Exercises for Chapter 4 115EXERCISES FOR CHAPTER 4