Modern Engineering Thermodynamics

400 CHAPTER 11: More Thermodynamic Relations
Problems (* indicates problems in SI units)
1.* Calculate the specific Gibbs and Helmholtz functions of
saturated water vapor at 100.°C.
2. You are given an unknown material whose boiling point at
atmospheric pressure is 50.0°F. You are also given the following
property values at this pressure and temperature: S f = 0.310 Btu/
(lbm·R), s g = 1.76 Btu/(lbm· R), and h g = 940. Btu/lbm.
Calculate the following quantities for this material at this state
(a) h fg, (b) h f, (c) g f, and (d) g g .
3. Using Eq. (11.3), show that
c p =
∂h
∂T p
4. Beginning with Eq. (11.1), show that
c v
T =
∂s
∂T v
5. Beginning with Eq. (11.3), show that
c p
T =
∂s
∂T p
= T ∂s
∂T p
= − ∂2 f
∂T 2
v
= − ∂2 g
∂T 2
p
6. Using Eq. (11.1), show that another definition of
thermodynamic pressure is
p = T ∂s
∂v u
7. Using Eqs. (11.8) and (11.10) and the Gibbs phase equilibrium
conditions, show that
dp
= s fg
= h fg
dT
sat
v fg T sat v fg
8.* Calculate s fg for water at 100.°C using the Gibbs phase
equilibrium conditions and compare it with the value listed in
the steam tables.
9. Using the h fg data from the steam tables at p sat = 14.7, 100.,
200., and 300. psia, calculate the value of s fg at these
temperatures from the Gibb’s phase equilibrium condition and
compare your results with the s fg values listed in the steam
tables.
10. Determine whether or not any of the following are properties
a. dM = 7 3 u3 sdu+ 1 2 u2 s 3 ds
b. dN = ðh/TÞdT + lnð1/TÞdh
Z
c. X = ½ðp/2Þ ds − ðs/2Þ dpŠ
Z
d. Y = ðvdpÞ
11. Beginning with Eq. (11.1) and the condition that s = s(u, v),
show that, for an ideal gas,
∂T
= 0
∂v u
and thus that T = T(u) oru = u(T) only.
12. Beginning with Eq. (11.1) and using the appropriate Maxwell
equation, show that
∂u
= T ∂p
− p
∂v T ∂T
v
13. Using the results of Problem 12, show that it can be further
reduced to
a. ∂u
∂v
= ∂ðp/TÞ
T T2 ∂T
v
b. ∂u
∂v
= − ∂ðp/TÞ
T ∂ð1/TÞ
v
14. Beginning with Eq. (11.3) and using the appropriate Maxwell
equation, show that
∂h
= −T ∂v
+ v
∂p
T
∂T p
15. Using the results of Problem 14, show that it can be further
reduced to
a. ∂h
∂p
= −T 2 ∂ðv/TÞ
∂T
T
p
b. ∂h
∂p
= ∂ðv/TÞ
∂ð1/TÞ
T
p
16. Let the isentropic exponent k for an arbitrary substance be
defined by the process pv k = constant.
a. Show that k = −ðv/pÞð∂p/∂vÞ s :
b. Using Eqs. (11.24) and (11.26) and the classical definition
of an isentropic process (s = constant) along with the
appropriate Maxwell equations, show that part a reduces
to k = –(v/p)(∂p/∂v) T (c p /c v ).
c. Show that, for an ideal gas, part b reduces to k = c p /c v .
17. An empirical equation of state has been proposed of the form
pv = RT + pAðTÞ+ p 2 BT ð Þ
where A(T) andB(T) are empirically determined functions of
temperature. Beginning with Eq. (11.1) and using the appropriate
Maxwell thermodynamic property equation, show that, for this
material,
∂u
= − T dA
∂p
T
dT
− p B+ T
dB
dT
18. A simple magnetic substance has the following differential
equation of state:
du = Tds+ μ o vH.dM
where H is the strength of the applied magnetic field, M is the
magnetization vector, and μ o is the magnetic permeability of
free space (a constant). For this substance, show that the
thermodynamic temperature is defined by
T =
and that the Maxwell equation analogous to Eq. (11.13) is
∂T
= ∂μ
ovH
∂M s ∂s
M
19. A system involves both reversible expansion work (–∫pdv)and
reversible electrochemical work (∫ϕdq,whereϕ is the voltage and q
is the charge per unit mass). For such a system, its specific enthalpy
is now defined as h = u + pv – ϕq.
a. Find an expression for the differential change in specific
Gibbs free energy, dg, in terms of p, v, s, T, ϕ, and q.
b. Find the Maxwell equation (∂q/∂T) p,ϕ =?
∂u
∂s
M