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Unformatted text preview: nnot. Therefore, it is necessary to develop some relations between these two groups so that the properties that cannot be measured directly can be evaluated. The derivations are based on the fact that properties are point functions, and the state of a simple, compressible system is completely specified by any two independent, intensive properties. The equations that relate the partial derivatives of properties P, v, T, and s of a simple compressible substance to each other are called the Maxwell relations. They are obtained from the four Gibbs equations, expressed as du dh da dg T ds T ds s dT s dT P dv v dP P dv v dP The Maxwell relations are a a a a 0T b 0v s 0T b 0P s 0s b 0v T 0s b 0P T a a a 0P b 0s v 0v b 0s P 0P b 0T v a 0v b 0T P The Clapeyron equation enables us to determine the enthalpy change associated with a phase change from a knowledge of P, v, and T data alone. It is expressed as a dP b dT sat hfg T vfg cen84959_ch12.qxd 4/5/05 3:58 PM Page 675 Chapter 12
For liquid–vapor and solid–vapor phase-change processes at low pressures, it can be approximated as ln a P2 b P1 sat h fg T2 T1 a b R T1T2 sat cp cv vTb2 a | 675 where b is the volume expansivity and a is the isothermal compressibility, defined as b 1 0v ab v 0T P and a 1 0v ab v 0P T The changes in internal energy, enthalpy, and entropy of a simple compressible substance can be expressed in terms of pressure, specific volume, temperature, and specific heats alone as 0P du cv dT cTa b P d dv 0T v 0v dh cp dT c v T a b d dP 0T P cv 0P dT a b dv ds T 0T v or ds cp T a a cp,T cp dT a 0v b dP 0T P 0 2P b 0T 2 v 0 2v b 0T 2 P
P The difference cp cv is equal to R for ideal gases and to zero for incompressible substances. The temperature behavior of a fluid during a throttling (h constant) process is described by the Joule-Thomson coefficient, defined as mJT a 0T b 0P h The Joule-Thomson coefficient is a measure of the change in temperature of a substance with pressure during a constantenthalpy process, and it can also be expressed as mJT 1 cv cp Ta 0v bd 0T P For specific heats, we have the following general relations: 0 cv b 0v T 0 cp 0P b Ta T Ta T The enthalpy, internal energy, and entropy changes of real gases can be determined accurately by utilizing generalized enthalpy or entropy departure charts to account for the deviation from the ideal-gas behavior by using the following relations: h2 u2 s2 h1 u1 s1 1 h2 1 h2 1 s2 h1 2 ideal h1 2 Ru 1 Z2T2 Z1T1 2 s 1 2 ideal Ru 1 Zs2 Zs1 2 RuTcr 1 Zh2 Zh1 2 cp0,T cv 0 0 2v a 2 b dP 0T P Ta 0v 2 0P ba b 0T P 0v T where the values of Zh and Zs are determined from the generalized charts. REFERENCES AND SUGGESTED READINGS
1. G. J. Van Wylen and R. E. Sonntag. Fundamentals of Classical Thermodynamics. 3rd ed. New York: John Wiley & Sons, 1985. 2. K. Wark and D. E. Richards. Thermodynamics. 6th ed. New York: McGraw-Hill, 1999. PROBLEMS*
Partial Derivatives and Associated Relations
12–1C Consider the function z(x, y). Plot a differential surface on x-y-z coordinates and indicate x, dx, y, dy, ( z)x, ( z)y, and dz. 12–2C What is the difference between partial differentials and ordinary diff...
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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