Modern Engineering Thermodynamics

11.4 Maxwell Equations 369
then,
∂s
= −R/p
∂p
T
so that ds T = –R(dp/p) T , where the subscript T is used to indicate that the temperature is to be held constant. This can be
integrated for the constant temperature condition to give
ðs 2 − s 1 Þ T = −R
Z 2
1
ðdp/pÞ T + funcðTÞ = −R ln ðp 2 /p 1 Þ + funcðTÞ
where func(T ) is an arbitrary function of integration. The function of integration here depends on the temperature T, and for
an isothermal process, it is treated as a constant. Since we happen to know that the entropy relation for an ideal gas is in
fact s 2 – s 1 = c p ln(T 2 /T 1 ) – R ln(p 2 /p 1 ), it is easy to see that the function of integration here is simply c p ln(T 2 /T 1 ).
The following exercises reinforce the concepts presented in Example 11.5.
Exercises
13. Use the other Maxwell equation (Eq. 11.15) available for the solution of Example 11.5 to find a different ideal gas
entropy relation. Answer: (s 2 – s 1 ) T = R ln(v 2 /v 1 ) + func(T ).
14. Show that (∂s/∂) p = (∂p/∂T ) s . Answer: Invert Eq. (11.14).
15. Show that ðs 2 − s 1 Þ y
= −
Z 2
1
ð∂T/∂vÞ s
dp + funcðvÞ. Answer: Use Eq. (11.13).
Before we continue with our search for the illusive u, h, s equations in terms of p, v, T variables, the following
example shows that the form taken by the Maxwell equations depends on the type of reversible work mode present
in the system.
EXAMPLE 11.6
The equation of state for a nonlinear rubber band is given by
F = KTðL/L o − 1Þ 2
where F is the stretching force, L is the stretched length, L o is the initial length, K is the elastic constant, and T is the absolute
temperature of the material. Then,
a. Determine the Maxwell equations for this material.
b. Show that the internal energy of this material is a function of temperature only.
c. Determine the heat transfer required when the rubber band is stretched isothermally and reversibly from L o = 0.0700 m
to L = 0.200 m at T = 20.0°C when K = 0.150 N/K.
Solution
The unknowns here are the Maxwell equations for this material, showing that the internal energy of this material is a function
of temperature only and the heat transfer required when the rubber band is stretched isothermally and reversibly
between two states.
a. Since the reversible work mode involved in the stretching process is
ðdWÞ rev
= −FdL
the Maxwell equations for this material can be easily obtained from those derived in the text by replacing p with –F and
v with L/m = l, the specific length of the material. Then Eqs. (11.13) to (11.16) become
∂T
= ∂F
∂l s ∂s l
∂T
= − ∂l
∂F s ∂s F
and
∂F
∂T l
∂l
∂T F
= − ∂s
∂l T
= ∂s
∂F T
(Continued )