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Unformatted text preview: v a a 0u b dv 0v T Using the definition of cv, we have
du cv dT 0u b dv 0v T
(12–25) cen84959_ch12.qxd 4/5/05 3:58 PM Page 662 662  Thermodynamics
Now we choose the entropy to be a function of T and v; that is, s and take its total differential,
ds a 0s b dT 0T v a 0s b dv 0v T s(T, v) (12–26) Substituting this into the T ds relation du
du Ta 0s b dT 0T v 0s b 0T v 0u b 0v T 0u b 0v T cTa T ds
0s b 0v T P dv yields
P d dv
(12–27) Equating the coefficients of dT and dv in Eqs. 12–25 and 12–27 gives
a a cv T Ta 0s b 0v T 0P b 0T v 0P b 0T v P
(12–28) Using the third Maxwell relation (Eq. 12–18), we get
a Ta P Substituting this into Eq. 12–25, we obtain the desired relation for du:
du cv dT cTa P d dv
(12–29) The change in internal energy of a simple compressible system associated with a change of state from (T1, v1) to (T2, v2) is determined by integration:
T2 v2 u2 u1
T1 cv dT
v1 cTa 0P b 0T v P d dv (12–30) Enthalpy Changes
The general relation for dh is determined in exactly the same manner. This time we choose the enthalpy to be a function of T and P, that is, h h(T, P), and take its total differential,
dh a 0h b dT 0T P a a 0h b dP 0P T Using the definition of cp, we have
dh cp dT 0h b dP 0P T
(12–31) Now we choose the entropy to be a function of T and P; that is, we take s s(T, P) and take its total differential,
ds a 0s b dT 0T P cv a 0s b dP 0P T
(12–32) Substituting this into the T ds relation dh
dh Ta 0s b dT 0T P T ds
Ta v dP gives
(12–33) 0s b d dP 0P T cen84959_ch12.qxd 4/5/05 3:58 PM Page 663 Chapter 12
Equating the coefficients of dT and dP in Eqs. 12–31 and 12–33, we obtain
a a 0s b 0T P 0h b 0P T 0h b 0P T cp T v Ta 0s b 0P T 0v b 0T P 0v b d dP 0T P
(12–34)  663 Using the fourth Maxwell relation (Eq. 12–19), we have
a v Ta Ta Substituting this into Eq. 12–31, we obtain the desired relation for dh:
dh cp d T cv
(12–35) The change in enthalpy of a simple compressible system associated with a change of state from (T1, P1) to (T2, P2) is determined by integration:
T2 P2 h2 h1
T1 cp dT
P1 cv Ta 0v b d dP 0T P (12–36) In reality, one needs only to determine either u2 u1 from Eq. 12–30 or h2 h1 from Eq. 12–36, depending on which is more suitable to the data at hand. The other can easily be determined by using the definition of enthalpy h u Pv:
h2 h1 u2 u1 1 P2v2 P1v1 2
(12–37) Entropy Changes
Below we develop two general relations for the entropy change of a simple compressible system. The first relation is obtained by replacing the first partial derivative in the total differential ds (Eq. 12–26) by Eq. 12–28 and the second partial derivative by the third Maxwell relation (Eq. 12–18), yielding
ds cv dT T
T2 a 0P b dv 0T v
v2 (12–38) and
s2 s1
T1 cv dT T v1 a 0P b dv 0T v (12–39) The second relation is obtained by replacing the first partial derivative in the total differential of ds (Eq. 12–32) by Eq. 12–34, and the second partial derivative by the fourth Maxwell relation (Eq. 12–19), yielding
ds cP dT T
T2 a 0v b dP 0T P
P2 (12–40) and
s2 s1 cp T dT
P1 T1 a 0v b dP 0T P (12–41) Either r...
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This note was uploaded on 03/09/2009 for the course ME 430 taught by Professor Y during the Spring '09 term at CUNY City.
 Spring '09
 Y

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