Thermodynamics 11

A fundamental thermodynamic function provides a complete description of the thermodynamic state. In the case of a pure substance with two independent properties, the fundamental thermodynamic function can take one of the following four forms: u u1s, y2 h h1s, p2 (11.37) c c1T, y2 g g1T, p2 Of the four fundamental functions listed in Eqs. 11.37, the Helmholtz function and the Gibbs function g have the greatest importance for subsequent discussions (see Sec. 11.6.2). Accordingly, let us discuss the fundamental function concept with reference to and g. In principle, all properties of interest can be determined from a fundamental thermodynamic function by differentiation and combination.

An alternative expression of the additive volume rule in terms of compressibility factors can be obtained. Since component i is considered to be at the pressure and temperature of the mixture, the compressibility factor Zi for this component is Zi pVi/niRT, so the volume Vi is Vi ZiniRT p Similarly, for the mixture V ZnRT p Substituting these expressions into Eq. 11.100a and reducing gives Z a j i1 yiZi4p,T (11.100b) The compressibility factors Zi are determined assuming that component i exists at the temperature T and pressure p of the mixture. The next example illustrates alternative means for estimating the pressure of a gas mixture.

An essential ingredient for the calculation of properties such as the specific internal energy, enthalpy, and entropy of a substance is an accurate representation of the relationship among pressure, specific volume, and temperature. The p--T relationship can be expressed alternatively: There are tabular representations, as exemplified by the steam tables. The relationship also can be expressed graphically, as in the p--T surface and compressibility factor charts. Analytical formulations, called equations of state, constitute a third general way of expressing the p--T relationship. Computer software such as Interactive Thermodynamics: IT also can be used to retrieve p--T data. The virial equation and the ideal gas equation are examples of analytical equations of state introduced in previous sections of the book. Analytical formulations of the p--T relationship are particularly convenient for performing the mathematical operations required to calculate u, h, s, and other thermodynamic properties. The object of the present section is to expand on the discussion of p--T relations for simple compressible substances presented in Chap. 3 by introducing some commonly used equations of state. 11.1.1 Getting Started Recall from Sec. 3.11 that the virial equation of state can be derived from the principles of statistical mechanics to relate the p--T behavior of a gas to the forces between molecules. In one form, the compressibility factor Z is expanded in inverse powers of specific volume as Z 1 B1T2 y C1T2 y2 D1T2 y3 . . . (11.1) The coefficients B, C, D, etc. are called, respectively, the second, third, fourth, etc. virial coefficients. Each virial coefficient is a function of temperature alone. In principle, the virial coefficients are calculable if a suitable model for describing the forces of interaction between the molecules of the gas under consideration is known. Future advances in refining the theory of molecular interactions may allow the virial coefficients to be predicted with considerable accuracy from the fundamental properties of the molecules involved. However, at present, just the first few coefficients can be calculated and only for gases consisting of relatively simple molecules. Equation 11.1 also can be used in an empirical fashion in which the coefficients become parameters whose magnitudes are determined by fitting p--T data in particular realms of interest. Only a few coefficients can be found this way, and the result is a truncated equation valid only for certain states. In the limiting case where the gas molecules are assumed not to interact in any way, the second, third, and higher terms of Eq. 11.1 vanish and the equation reduces to Z 5 1. Since Z py/RT, this gives the ideal gas equation of state py R/T. The ideal gas equation of state provides an acceptable approximation at many states, including but not limited to states where the pressure is low relative to the critical pressure and/or the temperature is high relative to the critical temperature of the substance under consideration. At many other states, however, the ideal gas equation of state provides a poor approximation. Over 100 equations of state have been developed in an attempt to improve on the ideal gas equation of state and yet avoid the complexities inherent in a full virial series. In general, these equations exhibit little in the way of fundamental physical significance and are mainly empirical in character. Most are developed for gases, but some describe the p--T behavior of the liquid phase, at least qualitatively. Every

By comparison with Eq. 11.18, we conclude that T a 0u 0s b y (11.24) p a 0u 0y b s (11.25) The differential of the function h 5 h(s, p) is dh a 0h 0s b p ds a 0h 0p b s dp By comparison with Eq. 11.19, we conclude that T a 0h 0s b p (11.26) y a 0h 0p b s (11.27) Similarly, the coefficients 2p and 2s of Eq. 11.22 are partial derivatives of (, T) p a 0c 0y b T (11.28) s a 0c 0Tb y (11.29) and the coefficients and 2s of Eq. 11.23 are partial derivatives of g(T, p) y a 0g 0p b T (11.30) s a 0g 0Tb p (11.31) As each of the four differentials introduced above is exact, the second mixed partial derivatives are equal. Thus, in Eq. 11.18, T plays the role of M in Eq. 11.14b and 2p plays the role of N in Eq. 11.14b, so a 0T 0y b s a 0p 0s b y (11.32) In Eq. 11.19, T and play the roles of M and N in Eq. 11.14b, respectively. Thus, a 0T 0p b s a 0y 0s b p (11.33) Similarly, from Eqs. 11.22 and 11.23 follow

Consider a reference state where component i of a multicomponent system is pure at the temperature T of the system and a reference-state pressure pref. The difference in the chemical potential of i between a specified state of the multicomponent system and the reference state is obtained with Eq. 11.125 as mi mi RT ln fi f i (11.139) where the superscript 8 denotes property values at the reference state. The fugacity ratio appearing in the logarithmic term is known as the activity, ai, of component i in the mixture. That is, ai fi f i (11.140) For subsequent applications, it suffices to consider the case of gaseous mixtures. For gaseous mixtures, pref is specified as 1 atm, so mi and f i in Eq. 11.140 are, respectively, the chemical potential and fugacity of pure i at temperature T and 1 atm. Since the chemical potential of a pure component equals the Gibbs function per mole, Eq. 11.139 can be written as mi gi RT ln ai (11.141) where gi is the Gibbs function per mole of pure component i evaluated at temperature T and 1 atm: gi gi (T, 1 atm). For an ideal solution, the Lewis-Randall rule applies and the activity is ai yi fi f i (11.142) where fi is the fugacity of pure component i at temperature T and pressure p. Introducing Eq. 11.142 into Eq. 11.141 mi gi RT ln yi fi f i or mi gi RT ln c afi pbapref f i b yi p pref d 1ideal solution2 (11.143) In principle, the ratios of fugacity to pressure shown underlined in this equation can be evaluated from Eq. 11.124 or the generalized fugacity chart, Fig. A-6, developed from it. If component i behaves as an ideal gas at both T, p and T, pref, we have fi /p f i /pref 1; Eq. 11.143 then reduces to mi gi RT ln yip pref 1ideal gas2

Considering the equation of state p 5 p(T, ), the differential dp can be expressed as dp 0p 0Tb y dT 0p 0y b T dy Eliminating dp between the last two equations and collecting terms gives c 1cp cy2 Ta 0y 0Tb p a 0p 0Tb y d dT Tc a 0y 0Tb p a 0p 0y b T a 0p 0Tb y d dy Since temperature and specific volume can be varied independently, the coefficients of the differentials in this expression must vanish, so cp cy Ta 0y 0Tb p a 0p 0Tb y (11.66) a 0p 0Tb y a 0y 0Tb p a 0p 0y b T (11.67) Introducing Eq. 11.67 into Eq. 11.66 gives cp cy Ta 0y 0Tb 2 p a 0p 0y b T (11.68) This equation allows c to be calculated from observed values of cp knowing only p--T data, or cp to be calculated from observed values of c.

DEVELOPING THE ENTROPY DEPARTURE. The following Maxwell relation gives the variation of entropy with pressure at fixed temperature: a 0s 0p b T a 0y 0Tb p (11.35) Integrating from pressure p9 to pressure p at fixed temperature T gives s1T, p2 s1T, p¿2 p p¿ a 0y 0Tb p dp (11.87) For an ideal gas, 5 RT/p, so 10y/0T2p R/p. Using this in Eq. 11.87, the change in specific entropy assuming ideal gas behavior is s*1T, p2 s*1T, p¿2 p p¿ R p dp (11.88) Subtracting Eq. 11.88 from Eq. 11.87 gives 3s1T, p2 s*1T, p24 3s1T, p¿2 s*1T, p¿24 p p¿ c R p a 0y 0Tb p d dp (11.89) Since the properties of a substance tend to merge into those of its ideal gas model as pressure tends to zero at fixed temperature, we have lim p¿S0 3s1T, p¿2 s*1T, p¿24 0 Thus, in the limit as p9 tends to zero, Eq. 11.89 becomes s1T, p2 s*1T, p2 p 0 c R p a 0y 0Tb p d dp (11.90) Using p--T data only, Eq. 11.90 can be evaluated at states 1 and 2 and thus the correction term of Eq. 11.86 evaluated. Equation 11.90 can be expressed in terms of the compressibility factor Z and the reduced properties TR and pR. The result, on a per mole basis, is the entropy departure s*1T, p2 s1T, p2 R h*1T2 h1T, p2 RTRTc pR 0 1Z 12 dpR pR (11.91) The right side of Eq. 11.91 depends only on the reduced temperature TR and reduced pressure pR. Accordingly, the quantity 1s* s2/R, the entropy departure, is a function only of these two reduced properties. As for the enthalpy departure, the entropy departure can be evaluated with a computer using a generalized equation of state giving Z as a function of TR and pR. Alternatively, tabular data from the literature or the graphical representation provided in Fig. A-5 can be employed. EVALUATING ENTROPY CHANGE. The change in specific entropy between two states can be evaluated by expressing Eq. 11.86 in terms of the entropy departure as s2 s1 s2 * s1 * R c as* s R b 2 a s* s R b 1 d (11.92) The first underlined term in Eq. 11.92 represents the change in specific entropy between the two states assuming ideal gas behavior. The second underlined term is the correction that must be applied to the ideal gas value for entropy change to obtain the actual value for the entropy change. The quantity 1s* s21/R appearing in Eq. 11.92 can be evaluated from the generalized entropy departure chart, Fig. A-5, using the reduced temperature TR1 and reduced pressure pR1 corresponding to the temperature T1 and pressure p1 at the initial state, respectively. Similarly, 1s* s22/R can be evaluated from Fig. A-5 using TR2 and pR2. The use of Eq. 11.92 is illustrated in the next example.

During a phase change at fixed temperature, the pressure is independent of specific volume and is determined by temperature alone. Thus, the quantity 10p/0T2y is determined by the temperature and can be represented as a 0p 0Tb y a dp dTb sat where "sat" indicates that the derivative is the slope of the saturation pressure- temperature curve at the point determined by the temperature held constant during the phase change (Sec. 11.2). Combining the last two equations gives a 0s 0y b T a dp dTb sat Since the right side of this equation is fixed when the temperature is specified, the equation can be integrated to give sg sf a dp dTb sat 1yg yf2 Introducing Eq. 11.38 into this expression results in the Clapeyron equation a dp dTb sat hg hf T1yg yf2 (11.40) Equation 11.40 allows (hg 2 hf) to be evaluated using only p--T data pertaining to the phase change. In instances when the enthalpy change is also measured, the Clapeyron equation can be used to check the consistency of the data. Once the specific enthalpy change is determined, the corresponding changes in specific entropy and specific internal energy can be found from Eqs. 11.38 and 11.39, respectively. Equations 11.38, 11.39, and 11.40 also can be written for sublimation or melting occurring at constant temperature and pressure. In particular, the Clapeyron equation would take the form a dp dTb sat h- h¿ T1y- y¿2 (11.41) where 0 and 9 denote the respective phases, and (dp/dT)sat is the slope of the relevant saturation pressure-temperature curve. The Clapeyron equation shows that the slope of a saturation line on a phase diagram depends on the signs of the specific volume and enthalpy changes accompanying the phase change. In most cases, when a phase change takes place with an increase in specific enthalpy, the specific volume also increases, and (dp/dT)sat is positive. However, in the case of the melting of ice and a few other substances, the specific volume decreases on melting. The slope of the saturated solid-liquid curve for these few substances is negative, as pointed out in Sec. 3.2.2 in the discussion of phase diagrams. An approximate form of Eq. 11.40 can be derived when the following two idealizations are justified: (1) f is negligible in comparison to g, and (2) the pressure is low enough that g can be evaluated from the ideal gas equation of state as g 5 RT/p. With these, Eq. 11.40 becomes a dp dTb sat hg hf RT 2 /p which can be rearranged to read a d ln p dT b sat hg hf RT2 (11.42) Equation 11.42 is called the Clausius-Clapeyron equation. A similar expression applies for the case of sublimation.

EVALUATING ENTHALPY CHANGE. The change in specific enthalpy between two states can be evaluated by expressing Eq. 11.80 in terms of the enthalpy departure as h2 h1 h2 * h1 * RTc c ah* h RTc b 2 a h* h RTc b 1 d (11.85) The first underlined term in Eq. 11.85 represents the change in specific enthalpy between the two states assuming ideal gas behavior. The second underlined term is the correction that must be applied to the ideal gas value for the enthalpy change to obtain the actual value for the enthalpy change. Referring to the engineering literature, the quantity 1h* h2/RTc at states 1 and 2 can be calculated with an equation giving Z(TR, pR) or obtained from tables. This quantity also can be evaluated at state 1 from the generalized enthalpy departure chart, Fig. A-4, using the reduced temperature TR1 and reduced pressure pR1 corresponding to the temperature T1 and pressure p1 at the initial state, respectively. Similarly, 1h* h2/RTc at state 2 can be evaluated from Fig. A-4 using TR2 and pR2. The use of Eq. 11.85 is illustrated in the next example

EVALUATING FUGACITY. Let us consider next how the fugacity can be evaluated. With Z py/RT, Eq. 11.122 becomes RT a 0 ln f 0p b T RT Z p or a 0 ln f 0p b T Z p Subtracting 1/p from both sides and integrating from pressure p9 to pressure p at fixed temperature T 3ln f ln p4 p p¿ p p¿ 1Z 12d ln p or cln f p d p p¿ p p¿ 1Z 12d ln p Taking the limit as p9 tends to zero and applying Eq. 11.123 results in ln f p p 0 1Z 12d ln p When expressed in terms of the reduced pressure, pR 5 p/pc, the above equation is ln f p pR 0 1Z 12d ln pR (11.124) Since the compressibility factor Z depends on the reduced temperature TR and reduced pressure pR, it follows that the right side of Eq. 11.124 depends on these properties only. Accordingly, the quantity ln f/p is a function only of these two reduced properties. Using a generalized equation of state giving Z as a function of TR and pR, ln f/p can readily be evaluated with a computer. Tabular representations are also found in the literature. Alternatively, the graphical representation presented in Fig. A-6 can be employed. to illustrate the use of Fig. A-6, consider two states of water vapor at the same temperature, 4008C. At state 1 the pressure is 200 bar, and at state 2 the pressure is 240 bar. The change in the chemical potential between these states can be determined using Eq. 11.121 as m2 m1 RT ln f2 f1 RT ln a f2 p2 p2 p1 p1 f1 b Using the critical temperature and pressure of water from Table A-1, at state 1 pR1 5 0.91, TR1 5 1.04, and at state 2 pR2 5 1.09, TR2 5 1.04. By inspection of Fig. A-6, f1/p1 5 0.755 and f2/p2 5 0.7. Inserting values in the above equation m2 m1 18.31421673.152 ln c 10.72a240 200ba 1 0.755b d 597 kJ/kmol For a pure component, the chemical potential equals the Gibbs function per mole, g h T s. Since the temperature is the same at states 1 and 2, the change in the chemical potential can be expressed as m2 m1 h2 h1 T1s2 s12. Using steam table data, the value obtained with this expression is 597 kJ/kmol, which agrees with the value determined from the generalized fugacity coefficient chart. b b b b b Multicomponent Systems The fugacity of a component i in a mixture can be defined by a procedure that parallels the definition for a pure component. For a pure component, the development

EXTENSIVE PROPERTY CHANGES ON MIXING. Let us conclude the present discussion by evaluating the change in volume on mixing of pure components at the same temperature and pressure, a result for which an application is given in the discussion of Eq. 11.135. The total volume of the pure components before mixing is Vcomponents a j i1 niyi where yi is the molar specific volume of pure component i. The volume of the mixture is Vmixture a j i1 niVi where Vi is the partial molal volume of component i in the mixture. The volume change on mixing is ¢Vmixing Vmixture Vcomponents a j i1 niVi a j i1 niyi or ¢Vmixing a j i1 ni1Vi yi2 (11.105) Similar results can be obtained for other extensive properties, for example, ¢Umixing a j i1 ni1Ui ui2 ¢Hmixing a j i1 ni1Hi hi2 (11.106) ¢Smixing a j i1 ni1Si si2 In Eqs. 11.106, ui, hi , and si denote the molar internal energy, enthalpy, and entropy of pure component i, respectively. The symbols Ui, Hi , and Si denote the respective partial molal properties. 11.9.2 Chemical Potential Of the partial molal properties, the partial molal Gibbs function is particularly useful in describing the behavior of mixtures and solutions. This quantity plays a central role in the criteria for both chemical and phase equilibrium (Chap. 14). Because of its importance in the study of multicomponent systems, the partial molal Gibbs function of component i is given a special name and symbol. It is called the chemical potential of component i and symbolized by i: mi Gi 0G 0ni b T, p, nl (11.107) Like temperature and pressure, the chemical potential i is an intensive property. Applying Eq. 11.103 together with Eq. 11.107, the following expression can be written:

Eliminating dy between these two equations results in c1 a 0x 0y b z a 0y 0x b z d dx c a 0x 0y b z a 0y 0z b x a 0x 0z b y d dz (11.17) Since x and z can be varied independently, let us hold z constant and vary x. That is, let dz 5 0 and dx 0. It then follows from Eq. 11.17 that the coefficient of dx must vanish, so Eq. 11.15 must be satisfied. Similarly, when dx 5 0 and dz 0, the coefficient of dz in Eq. 11.17 must vanish. Introducing Eq. 11.15 into the resulting expression and rearranging gives Eq. 11.16. The details are left as an exercise. b b b b b APPLICATION. An equation of state p 5 p(T, ) provides a specific example of a function of two independent variables. The partial derivatives 10p/0T2y and 10p/0y2T of p(T, ) are important for subsequent discussions. The quantity 10p/0T2y is the partial derivative of p with respect to T (the variable being held constant). This partial derivative represents the slope at a point on a line of constant specific volume (isometric) projected onto the p-T plane. Similarly, the partial derivative 10p/0y2T is the partial derivative of p with respect to (the variable T being held constant). This partial derivative represents the slope at a point on a line of constant temperature (isotherm) projected on the p- plane. The partial derivatives 10p/0T2y and 10p/0y2T are themselves intensive properties because they have unique values at each state. The p--T surfaces given in Figs. 3.1 and 3.2 are graphical representations of functions of the form p 5 p(, T). Figure 11.1 shows the liquid, vapor, and two-phase regions of a p--T surface projected onto the p- and p-T planes. Referring first to Fig. 11.1a, note that several isotherms are sketched. In the single-phase regions, the partial derivative 10p/0y2T giving the slope is negative at each state along an isotherm except at the critical point, where the partial derivative vanishes. Since the isotherms are horizontal in the two-phase liquid-vapor region, the partial derivative 10p/0y2T vanishes there as well. For these states, pressure is independent of specific volume and is a function of temperature only: p 5 psat(T). Figure 11.1b shows the liquid and vapor regions with several isometrics (constant specific volume lines) superimposed. In the single-phase regions, the isometrics are

Expressions for the internal energy, enthalpy, and Helmholtz function can be obtained from Eq. 11.108, using the definitions H 5 U 1 pV, G 5 H 2 TS, and C 5 U 2 TS. They are U TS pV a j i1 nimi H TS a j i1 nimi (11.109) ° pV a j i1 nimi Other useful relations can be obtained as well. Forming the differential of G(T, p, n1, n2, . . . , nj) dG 0G 0p b T, n dp 0G 0T b p, n dT a j i1 a 0G 0ni b T, p, nl dni (11.110) The subscripts n in the first two terms indicate that all n's are held fixed during differentiation. Since this implies fixed composition, it follows from Eqs. 11.30 and 11.31 (Sec. 11.3.2) that V a 0G 0p b T, n and S a 0G 0T b p, n (11.111) With Eqs. 11.107 and 11.111, Eq. 11.110 becomes dG V dp S dT a j i1 mi dni (11.112) which for a multicomponent system is the counterpart of Eq. 11.23. Another expression for dG is obtained by forming the differential of Eq. 11.108. That is, dG a j i1 ni dmi a j i1 mi dni Combining this equation with Eq. 11.112 gives the Gibbs-Duhem equation a j i1 ni dmi V dp S dT (11.113) 11.9.3 Fundamental Thermodynamic Functions for Multicomponent Systems A fundamental thermodynamic function provides a complete description of the thermodynamic state of a system. In principle, all properties of interest can be determined from such a function by differentiation and/or combination. Reviewing the developments of Sec. 11.9.2, we see that a function G(T, p, n1, n2, . . . , nj) is a fundamental thermodynamic function for a multicomponent system. Functions of the form U(S, V, n1, n2, . . . , nj ), H(S, p, n1, n2, . . . , nj ), and C(T, V, n1, n2, . . . , nj ) also can serve as fundamental thermodynamic functions for multicomponent systems. To demonstrate this, first form the differential of each of Eqs. 11.109 and use the Gibbs-Duhem equation, Eq. 11.113, to reduce the resultant expressions to obtain

For multicomponent systems, these are the counterparts of Eqs. 11.18, 11.19, and 11.22, respectively. The differential of U(S, V, n1, n2, . . . , nj) is dU 0U 0S b V, n dS 0U 0Vb S, n dV a j i1 a 0U 0ni b S,V, nl dni Comparing this expression term by term with Eq. 11.114a, we have T 0U 0S b V, n , p 0U 0Vb S, n , mi 0U 0ni b S,V, nl (11.115a) That is, the temperature, pressure, and chemical potentials can be obtained by differentiation of U(S, V, n1, n2, . . . , nj). The first two of Eqs. 11.115a are the counterparts of Eqs. 11.24 and 11.25. A similar procedure using a function of the form H(S, p, n1, n2, . . . , nj) together with Eq. 11.114b gives T 0H 0S b p, n , V 0H 0p b S, n , mi 0H 0ni b S, p, nl (11.115b) where the first two of these are the counterparts of Eqs. 11.26 and 11.27. Finally, with C(S, V, n1, n2, . . . , nj) and Eq. 11.114c p 0° 0V b T, n , S 0° 0T b V, n , mi 0° 0ni b T,V, nl (11.115c) The first two of these are the counterparts of Eqs. 11.28 and 11.29. With each choice of fundamental function, the remaining extensive properties can be found by combination using the definitions H 5 U 1 pV, G 5 H 2 TS, C 5 U 2 TS. The foregoing discussion of fundamental thermodynamic functions has led to several property relations for multicomponent systems that correspond to relations obtained previously. In addition, counterparts of the Maxwell relations can be obtained by equating mixed second partial derivatives. For example, the first two terms on the right of Eq. 11.112 give 0V 0Tb p, n 0S 0p b T, n (11.116) which corresponds to Eq. 11.35. Numerous relationships involving chemical potentials can be derived similarly by equating mixed second partial derivatives. An important example from Eq. 11.112 is 0mi 0p b T, n 0V 0ni b T, p, nl Recognizing the right side of this equation as the partial molal volume, we have 0mi 0p b T, n Vi (11.117) This relationship is applied in the development of Eqs. 11.126. The present discussion concludes by listing four different expressions derived above for the chemical potential in terms of other properties. In the order obtained, they are mi 0G 0ni b T, p, nl 0U 0ni b S, V, nl 0H 0ni b S, p, nl 0° 0ni b T, V, nl (11.118) Only the first of these partial derivatives is a partial molal property, however, for the term partial molal applies only to partial derivatives where the independent

Forming the ratio of these equations gives cp cy 10y/0s2T 10T/0y2s 10p/0s2T 10T/0p2s (11.70) Since 10s/0p2T 1/10p/0s2T and 10p/0T2s 1/10T/0p2s, Eq. 11.70 can be expressed as cp cy c a 0y 0s b T a 0s 0p b T d ca 0p 0Tb s a 0T 0y b s d (11.71) Finally, the chain rule from calculus allows us to write 10y/0p2T 10y/0s2T10s/0p2T and 10p/0y2s 10p/0T2s10T/0y2s, so Eq. 11.71 becomes k cp cy a 0y 0p b T a 0p 0y b s (11.72) This can be expressed alternatively in terms of the isothermal and isentropic compressibilities as k k a (11.73) Solving Eq. 11.72 for 10p/0y2s and substituting the resulting expression into Eq. 9.36b gives the following relationship involving the velocity of sound c and the specific heat ratio k c 2ky2 10p/0y2T (11.74) Equation 11.74 can be used to determine c knowing the specific heat ratio and p--T data, or to evaluate k knowing c and 10p/0y2T

Generalized Entropy Departure Chart A generalized chart that allows changes in specific entropy to be evaluated can be developed in a similar manner to the generalized enthalpy departure chart introduced earlier in this section. The difference in specific entropy between states 1 and 2 of a gas (or liquid) can be expressed as the identity s1T2, p22 s1T1, p12 s*1T2, p22 s*1T1, p12 53s1T2, p22 s*1T2, p224 3s1T1, p12 s*1T1, p1246 (11.86) where 3s1T, p2 s*1T, p24 denotes the specific entropy of the substance relative to that of its ideal gas model when both are at the same temperature and pressure. Equation 11.86 indicates that the change in specific entropy between the two states equals the entropy change determined using the ideal gas model plus a correction (shown underlined) that accounts for the departure from ideal gas behavior. The ideal gas term can be evaluated using methods introduced in Sec. 6.5. Let us consider next how the correction term is evaluated in terms of the entropy departure

Generalized charts giving the compressibility factor Z in terms of the reduced properties pR, TR, and 9 R are introduced in Sec. 3.11. With such charts, estimates of p--T data can be obtained rapidly knowing only the critical pressure and critical temperature for the substance of interest. The objective of the present section is to introduce generalized charts that allow changes in enthalpy and entropy to be estimated. Generalized Enthalpy Departure Chart The change in specific enthalpy of a gas (or liquid) between two states fixed by temperature and pressure can be evaluated using the identity h1T2, p22 h1T1, p12 3h*1T22 h*1T124 53h1T2, p22 h*1T224 3h1T1, p12 h*1T1246 (11.80) The term [h(T, p) 2 h*(T)] denotes the specific enthalpy of the substance relative to that of its ideal gas model when both are at the same temperature. The superscript * is used in this section to identify ideal gas property values. Thus, Eq. 11.80 indicates that the change in specific enthalpy between the two states equals the enthalpy change determined using the ideal gas model plus a correction that accounts for the departure from ideal gas behavior. The correction is shown underlined in Eq. 11.80. The ideal gas term can be evaluated using methods introduced in Chap. 3. Next, we show how the correction term is evaluated in terms of the enthalpy departure. DEVELOPING THE ENTHALPY DEPARTURE. The variation of enthalpy with pressure at fixed temperature is given by Eq. 11.56 as a 0h 0p b T y Ta 0y 0Tb p Integrating from pressure p9 to pressure p at fixed temperature T

In single-phase regions, pressure and temperature are independent, and we can think of the specific volume as being a function of these two, 5 (T, p). The differential of such a function is dy a 0y 0Tb p dT a 0y 0p b T dp Two thermodynamic properties related to the partial derivatives appearing in this differential are the volume expansivity , also called the coefficient of volume expansion b 1 y a 0y 0Tb p (11.62) and the isothermal compressibility k 1 y a 0y 0p b T (11.63) By inspection, the unit for is seen to be the reciprocal of that for temperature and the unit for is the reciprocal of that for pressure. The volume expansivity is an indication of the change in volume that occurs when temperature changes while pressure remains constant. The isothermal compressibility is an indication of the change in volume that takes place when pressure changes while temperature remains constant. The value of is positive for all substances in all phases. The volume expansivity and isothermal compressibility are thermodynamic properties, and like specific volume are functions of T and p. Values for and are provided in compilations of engineering data. Table 11.2 gives values of these properties for liquid water at a pressure of 1 atm versus temperature. For a pressure of 1 atm, water has a state of maximum density at about 48C. At this state, the value of is zero. The isentropic compressibility is an indication of the change in volume that occurs when pressure changes while entropy remains constant: a 1 y a 0y 0p b s (11.64) The unit for is the reciprocal of that for pressur

In the preceding section we considered means for evaluating the p--T relation of gas mixtures by extending methods developed for pure components. The current section is devoted to the development of some general aspects of the properties of systems with two or more components. Primary emphasis is on the case of gas mixtures, but the methods developed also apply to solutions. When liquids and solids are under consideration, the term solution is sometimes used in place of mixture. The present discussion is limited to nonreacting mixtures or solutions in a single phase. The effects of chemical reactions and equilibrium between different phases are taken up in Chaps. 13 and 14. To describe multicomponent systems, composition must be included in our thermodynamic relations. This leads to the definition and development of several new concepts, including the partial molal property, the chemical potential, and the fugacity.

In the present discussion we introduce the concept of a partial molal property and illustrate its use. This concept plays an important role in subsequent discussions of multicomponent systems. DEFINING PARTIAL MOLAL PROPERTIES. Any extensive thermodynamic property X of a single-phase, single-component system is a function of two independent intensive properties and the size of the system. Selecting temperature and pressure as the independent properties and the number of moles n as the measure of size, we have X 5 X(T, p, n). For a single-phase, multicomponent system, the extensive property X must then be a function of temperature, pressure, and the number of moles of each component present, X 5 X(T, p, n1, n2, . . . , nj). If each mole number is increased by a factor , the size of the system increases by the same factor, and so does the value of the extensive property X. That is, aX1T, p, n1, n2, . . . , nj2 X1T, p, an1, an2, . . . , anj2 Differentiating with respect to while holding temperature, pressure, and the mole numbers fixed and using the chain rule on the right side gives X 0X 01an12 n1 0X 01an22 n2 . . . 0X 01anj2 nj This equation holds for all values of . In particular, it holds for 5 1. Setting 5 1 X a j i1 ni 0X 0ni b T, p, nl (11.101) where the subscript nl denotes that all n's except ni are held fixed during differentiation. The partial molal property Xi is by definition Xi 0X 0ni b T, p, nl (11.102) The partial molal property Xi is a property of the mixture and not simply a property of component i, for Xi depends in general on temperature, pressure, and mixture composition: Xi1T, p, n1, n2, . . . , nj2. Partial molal properties are intensive properties of the mixture. Introducing Eq. 11.102, Eq. 11.101 becomes X a j i1 niXi (11.103) This equation shows that the extensive property X can be expressed as a weighted sum of the partial molal properties Xi . Selecting the extensive property X in Eq. 11.103 to be volume, internal energy, enthalpy, and entropy, respectively, gives V a j i1 niVi, U a j i1 niUi, H a j i1 niHi, S a j i1 niSi (11.104) where Vi, Ui, Hi, Si denote the partial molal volume, internal energy, enthalpy, and entropy. Similar expressions can be written for the Gibbs function G and the Helmholtz function C. Moreover, the relations between these extensive properties: H 5 U 1 pV, G 5 H 2 TS, C 5 U 2 TS can be differentiated with respect to ni while holding temperature, pressure, and the remaining n's constant to produce corresponding relations

In this section, general relations are obtained for the difference between specific heats (cp 2 c) and the ratio of specific heats cp/c. EVALUATING (cp 2 c). An expression for the difference between cp and c can be obtained by equating the two differentials for entropy given by Eqs. 11.48 and 11.57 and rearranging to obtain

In this section, several important property relations are developed, including the expressions known as the Maxwell relations. The concept of a fundamental thermodynamic function is also introduced. These results, which are important for subsequent discussions, are obtained for simple compressible systems of fixed chemical composition using the concept of an exact differential. 666 Chapter 11 Thermodynamic Relations 11.3.1 Principal Exact Differentials The principal results of this section are obtained using Eqs. 11.18, 11.19, 11.22, and 11.23. The first two of these equations are derived in Sec. 6.3, where they are referred to as the T ds equations. For present purposes, it is convenient to express them as du T ds p dy (11.18) dh T ds y dp (11.19) The other two equations used to obtain the results of this section involve, respectively, the specific Helmholtz function defined by c u Ts (11.20) and the specific Gibbs function g defined by g h Ts (11.21) The Helmholtz and Gibbs functions are properties because each is defined in terms of properties. From inspection of Eqs. 11.20 and 11.21, the units of and g are the same as those of u and h. These two new properties are introduced solely because they contribute to the present discussion, and no physical significance need be attached to them at this point. Forming the differential d dc du d1Ts2 du T ds s dT Substituting Eq. 11.18 into this gives dc p dy s dT (11.22) Similarly, forming the differential dg dg dh d1Ts2 dh T ds s dT Substituting Eq. 11.19 into this gives dg y dp s dT (11.23) 11.3.2 Property Relations from Exact Differentials The four differential equations introduced above, Eqs. 11.18, 11.19, 11.22, and 11.23, provide the basis for several important property relations. Since only properties are involved, each is an exact differential exhibiting the general form dz 5 M dx 1 N dy considered in Sec. 11.2. Underlying these exact differentials are, respectively, functions of the form u(s, ), h(s, p), (, T), and g(T, p). Let us consider these functions in the order given. The differential of the function u 5 u(s, ) is

In words, Eqs. 11.14 indicate that the mixed second partial derivatives of the function z are equal. The relationship in Eqs. 11.14 is both a necessary and sufficient condition for the exactness of a differential expression, and it may therefore be used as a test for exactness. When an expression such as M dx 1 N dy does not meet this test, no function z exists whose differential is equal to this expression. In thermodynamics, Eq. 11.14 is not generally used to test exactness but rather to develop additional property relations. This is illustrated in Sec. 11.3 to follow. Two other relations among partial derivatives are listed next for which applications are found in subsequent sections of this chapter. These are

Like other partial differential coefficients introduced in this section, the Joule- Thomson coefficient is defined in terms of thermodynamic properties only and thus is itself a property. The units of J are those of temperature divided by pressure. A relationship between the specific heat cp and the Joule-Thomson coefficient J can be established by using Eq. 11.16 to write a 0T 0p b h a 0p 0h b T a 0h 0Tb p 1 The first factor in this expression is the Joule-Thomson coefficient and the third is cp. Thus, cp 1 mJ10p/0h2T With 10h/0p2T 1/10p/0h2T from Eq. 11.15, this can be written as cp 1 mJ a 0h 0p b T (11.76) The partial derivative 10h/0p2T, called the constant-temperature coefficient, can be eliminated from Eq. 11.76 by use of Eq. 11.56. The following expression results: cp 1 mJ cT a 0y 0Tb p yd (11.77) Equation 11.77 allows the value of cp at a state to be determined using p--T data and the value of the Joule-Thomson coefficient at that state. Let us consider next how the Joule-Thomson coefficient can be found experimentally. EXPERIMENTAL EVALUATION. The Joule-Thomson coefficient can be evaluated experimentally using an apparatus like that pictured in Fig. 11.3. Consider first Fig. 11.3a, which shows a porous plug through which a gas (or liquid) may pass. During operation at steady state, the gas enters the apparatus at a specified temperature T1 and pressure p1 and expands through the plug to a lower pressure p2, which is controlled by an outlet valve. The temperature T2 at the exit is measured. The apparatus is designed

Many systems of interest involve mixtures of two or more components. The principles of thermodynamics introduced thus far are applicable to systems involving mixtures, but to apply such principles requires that mixture properties be evaluated. Since an unlimited variety of mixtures can be formed from a given set of pure components by varying the relative amounts present, the properties of mixtures are available in tabular, graphical, or equation forms only in particular cases such as air. Generally, special means are required for determining mixture properties. In this section, methods for evaluating the p--T relations for pure components introduced in previous sections of the book are adapted to obtain plausible estimates for gas mixtures. In Sec. 11.9 some general aspects of property evaluation for multicomponent systems are introduced. To evaluate the properties of a mixture requires knowledge of the composition. The composition can be described by giving the number of moles (kmol or lbmol) of each component present. The total number of moles, n, is the sum of the number of moles of each of the components n n1 n2 . . . nj a j i1 ni (11.93) The relative amounts of the components present can be described in terms of mole fractions. The mole fraction yi of component i is defined as yi ni n (11.94) Dividing each term of Eq. 11.93 by the total number of moles and using Eq. 11.94 1 a j i1 yi (11.95) That is, the sum of the mole fractions of all components present is equal to unity. Most techniques for estimating mixture properties are empirical in character and are not derived from fundamental principles. The realm of validity of any particular technique can be established only by comparing predicted property values with empirical data. The brief discussion to follow is intended only to show how certain of the procedures for evaluating the p--T relations of pure components introduced previously can be extended to gas mixtures. MIXTURE EQUATION OF STATE. One way the p--T relation for a gas mixture can be estimated is by applying to the overall mixture an equation of state such as introduced in Sec. 11.1. The constants appearing in the equation selected would be mixture values determined with empirical combining rules developed for the equation. For example, mixture values of the constants a and b for use in the van der Waals and Redlich-Kwong equations would be obtained using relations of the form a a a j i1 yiai 1/2 b 2 , b a a j i1 yibib (11.96) where ai and bi are the values of the constants for component i and yi is the mole fraction. Combination rules for obtaining mixture values for the constants in other equations of state also have been suggested. KAY'S RULE. The principle of corresponding states method for single components introduced in Sec. 3.11.3 can be extended to mixtures by regarding the mixture as if it were a single pure component having critical properties calculated by one of several mixture rules. Perhaps the simplest of these, requiring only the determination of a mole fraction averaged critical temperature Tc and critical pressure pc, is Kay's rule

The five constants a, b, c, A, and B appearing in these equations are determined by curve fitting to experimental data. Benedict, Webb, and Rubin extended the Beattie-Bridgeman equation of state to cover a broader range of states. The resulting equation, involving eight constants in addition to the gas constant, has been particularly successful in predicting the p--T behavior of light hydrocarbons. The Benedict-Webb-Rubin equation is p RT y aBRT A C T2 b 1 y2 1bRT a2 y3 aa y6 c y3 T2 a1 g y2 b exp a g y2 b (11.12) Values of the constants appearing in Eq. 11.12 for five common substances are given in Table A-24 for pressure in bar, specific volume in m3 /kmol, and temperature in K. Values of the constants for the same substances are given in Table A-24E for pressure in atm, specific volume in ft3 /lbmol, and temperature in 8R. Because Eq. 11.12 has been so successful, its realm of applicability has been extended by introducing additional constants. Equations 11.10 and 11.12 are merely representative of multiconstant equations of state. Many other multiconstant equations have been proposed. With high-speed computers, equations having 50 or more constants have been developed for representing the p--T behavior of different substances. Benedict-Webb-Rubin equation 11.2 Important Mathematical Relations Values of two independent intensive properties are sufficient to fix the intensive state of a simple compressible system of specified mass and composition—for instance, temperature and specific volume (see Sec. 3.1). All other intensive properties can be determined as functions of the two independent properties: p 5 p(T, ), u 5 u(T, ), h 5 h(T, ), and so on. These are all functions of two independent variables of the form z 5 z(x, y), with x and y being the independent variables. It might also be recalled that the differential of every property is exact (Sec. 2.2.1). The differentials of nonproperties such as work and heat are inexact. Let us review briefly some concepts from calculus about functions of two independent variables and their differentials. The exact differential of a function z, continuous in the variables x and y, is dz a 0z 0x b y dx a 0z 0y b x dy (11.13a) This can be expressed alternatively as dz M dx N dy (11.13b) where M 10z/0x2y and N 10z/0y2x. The coefficient M is the partial derivative of z with respect to x (the variable y being held constant). Similarly, N is the partial derivative of z with respect to y (the variable x being held constant). If the coefficients M and N have continuous first partial derivatives, the order in which a second partial derivative of the function z is taken is immaterial. That is, 0 0y c a 0z 0x b y d x 0 0x c a 0z 0y b x d y (11.14a) or a 0M 0y b x a 0N 0x b y (11.14b) which can be called the test for exactness, as discussed next.

The isentropic compressibility is related to the speed at which sound travels in a substance, and such speed measurements can be used to determine . In Sec. 9.12.2, the velocity of sound, or sonic velocity, is introduced as c By2 a 0p 0y b s (9.36b) The relationship of the isentropic compressibility and the velocity of sound can be obtained using the relation between partial derivatives expressed by Eq. 11.15. Identifying p with x, with y, and s with z, we have a 0p 0y b s 1 10y/0p2s With this, the previous two equations can be combined to give c 1y/a (11.65) The details are left as an exercise.

The objective of the present section is to derive expressions for evaluating Ds, Du, and Dh between states in single-phase regions. These expressions require both p--T data and appropriate specific heat data. Since single-phase regions are under present

The objective of this section is to utilize the thermodynamic relations introduced thus far to describe how tables of thermodynamic properties can be constructed. The characteristics of the tables under consideration are embodied in the tables for water and the refrigerants presented in the Appendix. The methods introduced in this section are extended in Chap. 13 for the analysis of reactive systems, such as gas turbine and vapor power systems involving combustion. The methods of this section also provide the basis for computer retrieval of thermodynamic property data. Two different approaches for constructing property tables are considered: c The presentation of Sec. 11.6.1 employs the methods introduced in Sec. 11.4 for assigning specific enthalpy, specific internal energy, and specific entropy to states of pure, simple compressible substances using p--T data, together with a limited amount of specific heat data. The principal mathematical operation of this approach is integration.

The presentation to this point has been directed mainly at developing thermodynamic relations that allow changes in u, h, and s to be evaluated from measured property data. The objective of the present section is to introduce several other thermodynamic relations that are useful for thermodynamic analysis. Each of the properties considered has a common attribute: It is defined in terms of a partial derivative of some other property. The specific heats c and cp are examples of this type of property.

The use of the Clapeyron equation in any of the foregoing forms requires an accurate representation for the relevant saturation pressure-temperature curve. This must not only depict the pressure-temperature variation accurately but also enable accurate values of the derivative (dp/dT)sat to be determined. Analytical representations in the form of equations are commonly used. Different equations for different portions of the pressure- temperature curves may be required. These equations can involve several constants. One form that is used for the vapor-pressure curves is the four-constant equation ln psat A B T C ln T DT in which the constants A, B, C, D are determined empirically. The use of the Clapeyron equation for evaluating changes in specific entropy, internal energy, and enthalpy accompanying a phase change at fixed T and p is illustrated in the next example

This equation is not altered fundamentally by adding and subtracting h*(T) on the left side. That is, 3h1T, p2 h*1T24 3h1T, p¿2 h*1T24 p p¿ c y T a 0y 0Tb p d dp (11.81) As pressure tends to zero at fixed temperature, the enthalpy of the substance approaches that of its ideal gas model. Accordingly, as p9 tends to zero lim p¿S0 3h1T, p¿2 h*1T24 0 In this limit, the following expression is obtained from Eq. 11.81 for the specific enthalpy of a substance relative to that of its ideal gas model when both are at the same temperature: h1T, p2 h*1T2 p 0 c y T a 0y 0Tb p d dp (11.82) This also can be thought of as the change in enthalpy as the pressure is increased from zero to the given pressure while temperature is held constant. Using p--T data only, Eq. 11.82 can be evaluated at states 1 and 2 and thus the correction term of Eq. 11.80 evaluated. Let us consider next how this procedure can be conducted in terms of compressibility factor data and the reduced properties TR and pR. The integral of Eq. 11.82 can be expressed in terms of the compressibility factor Z and the reduced properties TR and pR as follows. Solving Z 5 p/RT gives y ZRT p On differentiation a 0y 0Tb p RZ p RT p a 0Z 0Tb p With the previous two expressions, the integrand of Eq. 11.82 becomes y T a 0y 0Tb p ZRT p T c RZ p RT p a 0Z 0Tb p d RT 2 p a 0Z 0Tb p (11.83) Equation 11.83 can be written in terms of reduced properties as y T a 0y 0Tb p RTc pc T 2 R pR a 0Z 0TR b pR Introducing this into Eq. 11.82 gives on rearrangement h*1T2 h1T, p2 RTc T2 R pR 0 a 0Z 0TR b pR dpR pR Or, on a per mole basis, the enthalpy departure is h*1T2 h1T, p2 RTc T2 R pR 0 a 0Z 0TR b pR dpR pR (11.84) The right side of Eq. 11.84 depends only on the reduced temperature TR and reduced pressure pR. Accordingly, the quantity 1h* h2/RTc, the enthalpy departure, is a function only of these two reduced properties. Using a generalized equation of state giving Z as a function of TR and pR, the enthalpy departure can readily be evaluated with a computer. Tabular representations are also found in the literature. Alternatively, the graphical representation provided in Fig. A-4 can be employed

To fit the p--T data of gases over a wide range of states, Beattie and Bridgeman proposed in 1928 a pressure-explicit equation involving five constants in addition to the gas constant. The Beattie-Bridgeman equation can be expressed in a truncated virial form as

With the introduction of the Maxwell relations, we are in a position to develop thermodynamic relations that allow changes in entropy, internal energy, and enthalpy to be evaluated from measured property data. The presentation begins by considering relations applicable to phase changes and then turns to relations for use in single-phase regions. 11.4.1 Considering Phase Change The objective of this section is to develop relations for evaluating the changes in specific entropy, internal energy, and enthalpy accompanying a change of phase at fixed temperature and pressure. A principal role is played by the Clapeyron equation, which allows the change in enthalpy during vaporization, sublimation, or melting at a constant temperature to be evaluated from pressure-specific volume-temperature data pertaining to the phase change. Thus, the present discussion provides important examples of how p--T measurements can lead to the determination of other property changes, namely Ds, Du, and Dh for a change of phase. Consider a change in phase from saturated liquid to saturated vapor at fixed temperature. For an isothermal phase change, pressure also remains constant, so Eq. 11.19 reduces to dh T ds Integration of this expression gives sg sf hg hf T (11.38) Hence, the change in specific entropy accompanying a phase change from saturated liquid to saturated vapor at temperature T can be determined from the temperature and the change in specific enthalpy. The change in specific internal energy during the phase change can be determined using the definition h 5 u 1 p. ug uf hg hf p1yg yf2 (11.39) Thus, the change in specific internal energy accompanying a phase change at temperature T can be determined from the temperature and the changes in specific volume and enthalpy. CLAPEYRON EQUATION. The change in specific enthalpy required by Eqs. 11.38 and 11.39 can be obtained using the Clapeyron equation. To derive the Clapeyron equation, begin with the Maxwell relation

among partial molal properties: Hi Ui pVi, Gi Hi TSi, °i Ui TSi , where Gi and °i are the partial molal Gibbs function and Helmholtz function, respectively. Several additional relations involving partial molal properties are developed later in this section. EVALUATING PARTIAL MOLAL PROPERTIES. Partial molal properties can be evaluated by several methods, including the following: c If the property X can be measured, Xi can be found by extrapolating a plot giving 1¢X/¢ni2T, p, nl versus Dni. That is, Xi a 0X 0ni b T, p, nl lim ¢niS0 a ¢X ¢ni b T, p, nl c If an expression for X in terms of its independent variables is known, Xi can be evaluated by differentiation. The derivative can be determined analytically if the function is expressed analytically or found numerically if the function is in tabular form. c When suitable data are available, a simple graphical procedure known as the method of intercepts can be used to evaluate partial molal properties. In principle, the method can be applied for any extensive property. To introduce this method, let us consider the volume of a system consisting of two components, A and B. For this system, Eq. 11.103 takes the form V nAVA nBVB where VA and VB are the partial molal volumes of A and B, respectively. Dividing by the number of moles of mixture n V n yAVA yBVB where yA and yB denote the mole fractions of A and B, respectively. Since yA 1 yB 5 1, this becomes V n 11 yB2VA yBVB VA yB1VB VA2 This equation provides the basis for the method of intercepts. For example, refer to Fig. 11.5, in which V/n is plotted as a function of yB at constant T and p. At a specified value for yB, a tangent to the curve is shown on the figure. When extrapolated, the tangent line intersects the axis on the left at VA and the axis on the right at VB . These values for the partial molal volumes correspond to the particular specifications for T, p, and yB. At fixed temperature and pressure, VA and VB vary with yB and are not equal to the molar specific volumes of pure A and pure B, denoted on the figure as yA and yB , respectively. The values of yA and yB are fixed by temperature and pressure only.

and 11.99a, we also see that the additive pressure rule is exact for ideal gas mixtures. This special case is considered further in Sec. 12.2 under the heading Dalton model. EVALUATING FUGACITY IN A MIXTURE. Let us consider next how the fugacity of component i in a mixture can be expressed in terms of quantities that can be evaluated. For a pure component i, Eq. 11.122 gives RTa 0 ln fi 0p b T yi (11.131) where yi is the molar specific volume of pure i. Subtracting Eq. 11.131 from Eq. 11.126a RT c 0 ln 1 fi /fi2 0p d T, n Vi yi (11.132) Integrating from pressure p9 to pressure p at fixed temperature and mixture composition RTc lna fi fi b d p p¿ p p¿ 1Vi yi2dp In the limit as p9 tends to zero, this becomes RTc ln a fi fi b lim p¿S0 ln a fi fi b d p 0 1Vi yi2dp Since fi S p¿ and fi S yip¿ as pressure p9 tends to zero lim p¿S0 ln a fi fi b S ln a yip¿ p¿ b ln yi Therefore, we can write RTcln a fi fi b ln yi d p 0 1Vi yi2dp or RT ln a fi yifi b p 0 1Vi yi2dp (11.133) in which fi is the fugacity of component i at pressure p in a mixture of given composition at a given temperature, and fi is the fugacity of pure i at the same temperature and pressure. Equation 11.133 expresses the relation between fi and fi in terms of the difference between Vi and yi , a measurable quantity. 11.9.5 Ideal Solution The task of evaluating the fugacities of the components in a mixture is considerably simplified when the mixture can be modeled as an ideal solution. An ideal solution is a mixture for which fi yifi 1ideal solution2 (11.134) Equation 11.134, known as the Lewis-Randall rule, states that the fugacity of each component in an ideal solution is equal to the product of its mole fraction and the fugacity of the pure component at the same temperature, pressure, and state of aggregation (gas, liquid, or solid) as the mixture. Many gaseous mixtures at low to moderate pressures are adequately modeled by the Lewis-Randall rule. The ideal gas mixtures

approach is that the van der Waals constants can be expressed in terms of the critical pressure pc and critical temperature Tc, as demonstrated next. For the van der Waals equation at the critical point pc RTc yc b a y2 c Applying Eqs. 11.3 with the van der Waals equation gives a 0 2 p 0y2 b T 2RTc 1yc b23 6a y4 c 0 a 0p 0y b T RTc 1yc b22 2a y3 c 0 Solving the foregoing three equations for a, b, and yc in terms of the critical pressure and critical temperature a 27 64 R2 T2 c pc (11.4a) b RTc 8pc (11.4b) yc 3 8 RTc pc (11.4c) Values of the van der Waals constants a and b determined from Eqs. 11.4a and 11.4b for several common substances are given in Table A-24 for pressure in bar, specific volume in m3 /kmol, and temperature in K. Values of a and b for the same substances are given in Table A-24E for pressure in atm, specific volume in ft3 /lbmol, and temperature in 8R. GENERALIZED FORM. Introducing the compressibility factorZ py/RT, the reduced temperature TR 5 T/Tc, the pseudoreduced specific volumeyR¿ pcy /RTc, and the foregoing expressions for a and b, the van der Waals equation can be written in terms of Z, 9 R, and TR as Z yR¿ yR¿ 1/8 27/64 TRyR¿ (11.5) or alternatively in terms of Z, TR, and pR as Z3 a pR 8TR 1bZ2 a 27pR 64T2 R bZ 27p2 R 512T3 R 0 (11.6) The details of these developments are left as exercises. Equation 11.5 can be evaluated for specified values of 9 R and TR and the resultant Z values located on a generalized compressibility chart to show approximately where the equation performs satisfactorily. A similar approach can be taken with Eq. 11.6. The compressibility factor at the critical point yielded by the van der Waals equation is determined from Eq. 11.4c as Zc pcyc RTc 0.375 Actually, Zc varies from about 0.23 to 0.33 for most substances (see Tables A-1). Accordingly, with the set of constants given by Eqs. 11.4, the van der Waals equation is inaccurate in the vicinity of the critical point. Further study would show inaccuracy in other regions as well, so this equation is not suitable for many thermodynamic evaluations. The van der Waals equation is of interest to us primarily because it is

begins with Eq. 11.119, and the fugacity is introduced by Eq. 11.121. These are then used to write the pair of equations, Eqs. 11.122 and 11.123, from which the fugacity can be evaluated. For a mixture, the development begins with Eq. 11.117, the counterpart of Eq. 11.119, and the fugacity fi of component i is introduced by mi RT ln fi Ci 1T2 (11.125) which parallels Eq. 11.121. The pair of equations that allow the fugacity of a mixture component, fi , to be evaluated is RT a 0 ln fi 0p b T, n Vi (11.126a) lim pS 0 a fi yipb 1 (11.126b) The symbol fi denotes the fugacity of component i in the mixture and should be carefully distinguished in the presentation to follow from fi, which denotes the fugacity of pure i. DISCUSSION. Referring to Eq. 11.126b, note that in the ideal gas limit the fugacity fi is not required to equal the pressure p, as for the case of a pure component, but to equal the quantity yip. To see that this is the appropriate limiting quantity, consider a system consisting of a mixture of gases occupying a volume V at pressure p and temperature T. If the overall mixture behaves as an ideal gas, we can write p nRT V (11.127) where n is the total number of moles of mixture. Recalling from Sec. 3.12.3 that an ideal gas can be regarded as composed of molecules that exert negligible forces on one another and whose volume is negligible relative to the total volume, we can think of each component i as behaving as if it were an ideal gas alone at the temperature T and volume V. Thus, the pressure exerted by component i would not be the mixture pressure p but the pressure pi given by pi niRT V (11.128) where ni is the number of moles of component i. Dividing Eq. 11.128 by Eq. 11.127 pi p niRT/V nRT/V ni n yi On rearrangement pi yi p (11.129) Accordingly, the quantity yip appearing in Eq. 11.126b corresponds to the pressure pi. Summing both sides of Eq. 11.129, we obtain a j i1 pi a j i1 yip pa j i1 yi Or, since the sum of the mole fractions equals unity p a j i1 pi (11.130) In words, Eq. 11.130 states that the sum of the pressures pi equals the mixture pressure. This gives rise to the designation partial pressure for pi. With this background, we now see that Eq. 11.126b requires the fugacity of component i to approach the partial pressure of component i as pressure p tends to zero. Comparing Eqs. 11.130

c The approach of Sec. 11.6.2 utilizes the fundamental thermodynamic function concept introduced in Sec. 11.3.3. Once such a function has been constructed, the principal mathematical operation required to determine all other properties is differentiation. 11.6.1 Developing Tables by Integration Using p--T and Specific Heat Data In principle, all properties of present interest can be determined using cp cp01T2 (11.78) p p1y, T2, y y1p, T2 In Eqs. 11.78, cp0(T) is the specific heat cp for the substance under consideration extrapolated to zero pressure. This function might be determined from data obtained calorimetrically or from spectroscopic data, using equations supplied by statistical mechanics. Specific heat expressions for several gases are given in Tables A-21. The expressions p(, T) and (p, T) represent functions that describe the saturation pressure-temperature curves, as well as the p--T relations for the single-phase regions. These functions may be tabular, graphical, or analytical in character. Whatever their forms, however, the functions must not only represent the p--T data accurately but also yield accurate values for derivatives such as 10y/0T2p and 1dp/dT2sat. Figure 11.4 shows eight states of a substance. Let us consider how values can be assigned to specific enthalpy and specific entropy at these states. The same procedures can be used to assign property values at other states of interest. Note that when h has been assigned to a state, the specific internal energy at that state can be found from u 5 h 2 p. c Let the state denoted by 1 on Fig. 11.4 be selected as the reference state for enthalpy and entropy. Any value can be assigned to h and s at this state, but a value of zero is usual. It should be noted that the use of an arbitrary reference state and arbitrary reference values for specific enthalpy and specific entropy suffices only for evaluations involving differences in property values between states of the same composition, for then datums cancel. c Once a value is assigned to enthalpy at state 1, the enthalpy at the saturated vapor state, state 2, can be determined using the Clapeyron equation, Eq. 11.40 h2 h1 T11y2 y12a dp dTb sat where the derivative (dp/dT)sat and the specific volumes 1 and 2 are obtained from appropriate representations of the p--T data for the substance under consideration. The specific entropy at state 2 is found using Eq. 11.38 in the form s2 s1 h2 h1 T1 c Proceeding at constant temperature from state 2 to state 3, the entropy and enthalpy are found by means of Eqs. 11.59 and 11.60, respectively. Since temperature is fixed, these equations reduce to give s3 s2 p3 p2 a 0y 0Tb p dp and h3 h2 p3 p2 c y T a 0y 0Tb p d dp With the same procedure, s4 and h4 can be determined

c The isobar (constant-pressure line) passing through state 4 is assumed to be at a low enough pressure for the ideal gas model to be appropriate. Accordingly, to evaluate s and h at states such as 5 on this isobar, the only required information would be cp0(T) and the temperatures at these states. Thus, since pressure is fixed, Eqs. 11.59 and 11.60 give, respectively s5 s4 T5 T4 cp0 dT T and h5 h4 T5 T4 cp0 dT c Specific entropy and enthalpy values at states 6 and 7 are found from those at state 5 by the same procedure used in assigning values at states 3 and 4 from those at state 2. Finally, s8 and h8 are obtained from the values at state 7 using the Clapeyron equation. 11.6.2 Developing Tables by Differentiating a Fundamental Thermodynamic Function Property tables also can be developed using a fundamental thermodynamic function. It is convenient for this purpose to select the independent variables of the fundamental function from among pressure, specific volume (density), and temperature. This indicates the use of the Helmholtz function (T, ) or the Gibbs function g(T, p). The properties of water tabulated in Tables A-2 through A-6 have been calculated using the Helmholtz function. Fundamental thermodynamic functions also have been employed successfully to evaluate the properties of other substances in the Appendix tables. The development of a fundamental thermodynamic function requires considerable mathematical manipulation and numerical evaluation. Prior to the advent of highspeed computers, the evaluation of properties by this means was not feasible, and the approach described in Sec. 11.6.1 was used exclusively. The fundamental function approach involves three steps: 1. The first step is the selection of a functional form in terms of the appropriate pair of independent properties and a set of adjustable coefficients, which may number 50 or more. The functional form is specified on the basis of both theoretical and practical considerations. 2. Next, the coefficients in the fundamental function are determined by requiring that a set of carefully selected property values and/or observed conditions be satisfied in a least-squares sense. This generally involves the use of property data requiring the assumed functional form to be differentiated one or more times, such as p--T and specific heat data. 3. When all coefficients have been evaluated, the function is carefully tested for accuracy by using it to evaluate properties for which accepted values are known. These may include properties requiring differentiation of the fundamental function two or more times. For example, velocity of sound and Joule-Thomson data might be used. This procedure for developing a fundamental thermodynamic function is not routine and can be accomplished only with a computer. However, once a suitable fundamental function is established, extreme accuracy in and consistency among the thermodynamic properties is possible. The form of the Helmholtz function used in constructing the steam tables from which Tables A-2 through A-6 have been extracted is c1r, T2 c01T2 RT 3ln r rQ1r,t24 (11.79) where 0 and Q are given as the sums listed in Table 11.3. The independent variables are density and temperature. The variable denotes 1000/T. Values for pressure,

consideration, any two of the properties pressure, specific volume, and temperature can be regarded as the independent properties that fix the state. Two convenient choices are T, and T, p. T AND AS INDEPENDENT PROPERTIES. With temperature and specific volume as the independent properties that fix the state, the specific entropy can be regarded as a function of the form s 5 s(T, ). The differential of this function is ds a 0s 0Tb y dT a 0s 0y b T dy The partial derivative 10s/0y2T appearing in this expression can be replaced using the Maxwell relation, Eq. 11.34, giving ds a 0s 0Tb y dT a 0p 0Tb y dy (11.43) The specific internal energy also can be regarded as a function of T and : u 5 u(T, ). The differential of this function is du a 0u 0Tb y dT a 0u 0y b T dy With cy 10u/0T2y du cy dT a 0u 0y b T dy (11.44) Substituting Eqs. 11.43 and 11.44 into du 5 T ds 2 p d and collecting terms results in c a 0u 0y b T p Ta 0p 0Tb y d dy cT a 0s 0Tb y cy d dT (11.45) Since specific volume and temperature can be varied independently, let us hold specific volume constant and vary temperature. That is, let d 5 0 and dT fi 0. It then follows from Eq. 11.45 that a 0s 0Tb y cy T (11.46) Similarly, suppose that dT 5 0 and dy 0. It then follows that a 0u 0y b T Ta 0p 0Tb y p (11.47) Equations 11.46 and 11.47 are additional examples of useful thermodynamic property relations.

considered in Chap. 12 are an important special class of such mixtures. Some liquid solutions also can be modeled with the Lewis-Randall rule. As consequences of the definition of an ideal solution, the following characteristics are exhibited: c Introducing Eq. 11.134 into Eq. 11.132, the left side vanishes, giving Vi yi 0, or Vi yi (11.135) Thus, the partial molal volume of each component in an ideal solution is equal to the molar specific volume of the corresponding pure component at the same temperature and pressure. When Eq. 11.135 is introduced in Eq. 11.105, it can be concluded that there is no volume change on mixing pure components to form an ideal solution. With Eq. 11.135, the volume of an ideal solution is V a j i1 ni Vi a j i1 niyi a j i1 Vi 1ideal solution2 (11.136) where Vi is the volume that pure component i would occupy when at the temperature and pressure of the mixture. Comparing Eqs. 11.136 and 11.100a, the additive volume rule is seen to be exact for ideal solutions. c It also can be shown that the partial molal internal energy of each component in an ideal solution is equal to the molar internal energy of the corresponding pure component at the same temperature and pressure. A similar result applies for enthalpy. In symbols Ui ui, Hi hi (11.137) With these expressions, it can be concluded from Eqs. 11.106 that there is no change in internal energy or enthalpy on mixing pure components to form an ideal solution. With Eqs. 11.137, the internal energy and enthalpy of an ideal solution are U a j i1 niui and H a j i1 nihi 1ideal solution2 (11.138) where ui and hi denote, respectively, the molar internal energy and enthalpy of pure component i at the temperature and pressure of the mixture. Although there is no change in V, U, or H on mixing pure components to form an ideal solution, we expect an entropy increase to result from the adiabatic mixing of different pure components because such a process is irreversible: The separation of the mixture into the pure components would never occur spontaneously. The entropy change on adiabatic mixing is considered further for the special case of ideal gas mixtures in Sec. 12.4.2. The Lewis-Randall rule requires that the fugacity of mixture component i be evaluated in terms of the fugacity of pure component i at the same temperature and pressure as the mixture and in the same state of aggregation. For example, if the mixture were a gas at T, p, then fi would be determined for pure i at T, p and as a gas. However, at certain temperatures and pressures of interest a component of a gaseous mixture may, as a pure substance, be a liquid or solid. An example is an air-water vapor mixture at 208C (688F) and 1 atm. At this temperature and pressure, water exists not as a vapor but as a liquid. Although not considered here, means have been developed that allow the ideal solution model to be useful in such cases. 11.9.6 Chemical Potential for Ideal Solutions The discussion of multicomponent systems concludes with the introduction of expressions for evaluating the chemical potential for ideal solutions used in Sec. 14.3.3.

equation of state is restricted to particular states. This realm of applicability is often indicated by giving an interval of pressure, or density, where the equation can be expected to represent the p--T behavior faithfully. When it is not stated, the realm of applicability of a given equation can be approximated by expressing the equation in terms of the compressibility factor Z and the reduced properties pR, TR, 9 R and superimposing the result on a generalized compressibility chart or comparing with tabulated compressibility data obtained from the literature. 11.1.2 Two-Constant Equations of State Equations of state can be classified by the number of adjustable constants they include. Let us consider some of the more commonly used equations of state in order of increasing complexity, beginning with two-constant equations of state. van der Waals Equation An improvement over the ideal gas equation of state based on elementary molecular arguments was suggested in 1873 by van der Waals, who noted that gas molecules actually occupy more than the negligibly small volume presumed by the ideal gas model and also exert long-range attractive forces on one another. Thus, not all of the volume of a container would be available to the gas molecules, and the force they exert on the container wall would be reduced because of the attractive forces that exist between molecules. Based on these elementary molecular arguments, the van der Waals equation of state is p RT y b a y2 (11.2) The constant b is intended to account for the finite volume occupied by the molecules, the term a/y2 accounts for the forces of attraction between molecules, and R is the universal gas constant. Note than when a and b are set to zero, the ideal gas equation of state results. The van der Waals equation gives pressure as a function of temperature and specific volume and thus is explicit in pressure. Since the equation can be solved for temperature as a function of pressure and specific volume, it is also explicit in temperature. However, the equation is cubic in specific volume, so it cannot generally be solved for specific volume in terms of temperature and pressure. The van der Waals equation is not explicit in specific volume. EVALUATING a AND b. The van der Waals equation is a two-constant equation of state. For a specified substance, values for the constants a and b can be found by fitting the equation to p--T data. With this approach several sets of constants might be required to cover all states of interest. Alternatively, a single set of constants for the van der Waals equation can be determined by noting that the critical isotherm passes through a point of inflection at the critical point, and the slope is zero there. Expressed mathematically, these conditions are, respectively a 0 2 p 0y2 b T 0, a 0p 0y b T 0 1critical point2 (11.3) Although less overall accuracy normally results when the constants a and b are determined using critical point behavior than when they are determined by fitting p--T data in a particular region of interest, the advantage of this

here Tc,i, pc,i, and yi are the critical temperature, critical pressure, and mole fraction of component i, respectively. Using Tc and pc, the mixture compressibility factor Z is obtained as for a single pure component. The unknown quantity from among the pressure p, volume V, temperature T, and total number of moles n of the gas mixture can then be obtained by solving Z pV nRT (11.98) Mixture values for Tc and pc also can be used to enter the generalized enthalpy departure and entropy departure charts introduced in Sec. 11.7. ADDITIVE PRESSURE RULE. Additional means for estimating p--T relations for mixtures are provided by empirical mixture rules, of which several are found in the engineering literature. Among these are the additive pressure and additive volume rules. According to the additive pressure rule, the pressure of a gas mixture occupying volume V at temperature T is expressible as a sum of pressures exerted by the individual components: p p1 p2 p3 . . .4 T,V (11.99a) where the pressures p1, p2, and so on are evaluated by considering the respective components to be at the volume and temperature of the mixture. These pressures would be determined using tabular or graphical p--T data or a suitable equation of state. An alternative expression of the additive pressure rule in terms of compressibility factors can be obtained. Since component i is considered to be at the volume and temperature of the mixture, the compressibility factor Zi for this component is Zi piV/niRT, so the pressure pi is pi ZiniRT V Similarly, for the mixture p ZnRT V Substituting these expressions into Eq. 11.99a and reducing gives the following relationship between the compressibility factors for the mixture Z and the mixture components Zi Z a j i1 yiZi4T,V (11.99b) The compressibility factors Zi are determined assuming that component i occupies the entire volume of the mixture at the temperature T. ADDITIVE VOLUME RULE. The underlying assumption of the additive volume rule is that the volume V of a gas mixture at temperature T and pressure p is expressible as the sum of volumes occupied by the individual components: V V1 V2 V3 . . .4 p,T (11.100a) where the volumes V1, V2, and so on are evaluated by considering the respective components to be at the pressure and temperature of the mixture. These volumes would be determined from tabular or graphical p--T data or a suitable equation of state.

nearly straight or are slightly curved and the partial derivative 10p/0T2y is positive at each state along the curves. For the two-phase liquid-vapor states corresponding to a specified value of temperature, the pressure is independent of specific volume and is determined by the temperature only. Hence, the slopes of isometrics passing through the two-phase states corresponding to a specified temperature are all equal, being given by the slope of the saturation curve at that temperature, denoted simply as (dp/dT)sat. For these two-phase states, 10p/0T2y 1dp/dT2sat. In this section, important aspects of functions of two variables have been introduced. The following example illustrates some of these ideas using the van der Waals equation of state.

so that the gas undergoes a throttling process (Sec. 4.10) as it expands from 1 to 2. Accordingly, the exit state fixed by p2 and T2 has the same value for the specific enthalpy as at the inlet, h2 5 h1. By progressively lowering the outlet pressure, a finite sequence of such exit states can be visited, as indicated on Fig. 11.3b. A curve may be drawn through the set of data points. Such a curve is called an isenthalpic (constant enthalpy) curve. An isenthalpic curve is the locus of all points representing equilibrium states of the same specific enthalpy. The slope of an isenthalpic curve at any state is the Joule-Thomson coefficient at that state. The slope may be positive, negative, or zero in value. States where the coefficient has a zero value are called inversion states. Notice that not all lines of constant h have an inversion state. The uppermost curve of Fig. 11.3b, for example, always has a negative slope. Throttling a gas from an initial state on this curve would result in an increase in temperature. However, for isenthalpic curves having an inversion state, the temperature at the exit of the apparatus may be greater than, equal to, or less than the initial temperature, depending on the exit pressure specified. For states to the right of an inversion state, the value of the Joule-Thomson coefficient is negative. For these states, the temperature increases as the pressure at the exit of the apparatus decreases. At states to the left of an inversion state, the value of the Joule-Thomson coefficient is positive. For these states, the temperature decreases as the pressure at the exit of the device decreases. This can be used to advantage in systems designed to liquefy gases.

specific internal energy, and specific entropy can be determined by differentiation of Eq. 11.79. Values for the specific enthalpy and Gibbs function are found from h 5 u 1 p and g 5 1 p, respectively. The specific heat c is evaluated by further differentiation, cy 10u/0T2y. With similar operations, other properties can be evaluated. Property values for water calculated from Eq. 11.79 are in excellent agreement with experimental data over a wide range of conditions.

the simplest model that accounts for the departure of actual gas behavior from the ideal gas equation of state. Redlich-Kwong Equation Three other two-constant equations of state that have been widely used are the Berthelot, Dieterici, and Redlich-Kwong equations. The Redlich-Kwong equation, considered by many to be the best of the two-constant equations of state, is p RT y b a y1y b2T12 (11.7) This equation, proposed in 1949, is mainly empirical in nature, with no rigorous justification in terms of molecular arguments. The Redlich-Kwong equation is explicit in pressure but not in specific volume or temperature. Like the van der Waals equation, the Redlich-Kwong equation is cubic in specific volume. Although the Redlich-Kwong equation is somewhat more difficult to manipulate mathematically than the van der Waals equation, it is more accurate, particularly at higher pressures. The two-constant Redlich-Kwong equation performs better than some equations of state having several adjustable constants; still, two-constant equations of state tend to be limited in accuracy as pressure (or density) increases. Increased accuracy at such states normally requires equations with a greater number of adjustable constants. Modified forms of the Redlich-Kwong equation have been proposed to achieve improved accuracy. EVALUATING a AND b. As for the van der Waals equation, the constants a and b in Eq. 11.7 can be determined for a specified substance by fitting the equation to p--T data, with several sets of constants required to represent accurately all states of interest. Alternatively, a single set of constants in terms of the critical pressure and critical temperature can be evaluated using Eqs. 11.3, as for the van der Waals equation. The result is a a¿ R2 Tc 52 pc and b b¿ RTc pc (11.8) where a9 5 0.42748 and b9 5 0.08664. Evaluation of these constants is left as an exercise. Values of the Redlich-Kwong constants a and b determined from Eqs. 11.8 for several common substances are given in Table A-24 for pressure in bar, specific volume in m3 /kmol, and temperature in K. Values of a and b for the same substances are given in Table A-24E for pressure in atm, specific volume in ft3 /lbmol, and temperature in 8R. GENERALIZED FORM. Introducing the compressibility factor Z, the reduced temperature TR, the pseudoreduced specific volume 9 R, and the foregoing expressions for a and b, the Redlich-Kwong equation can be written as Z yR¿ y¿ R b¿ a¿ 1y¿ R b¿2T3/2 R (11.9) Equation 11.9 can be evaluated at specified values of 9 R and TR and the resultant Z values located on a generalized compressibility chart to show the regions where the equation performs satisfactorily. With the constants given by Eqs. 11.8, the compressibility factor at the critical point yielded by the Redlich-Kwong equation is Zc 5 0.333, which is at the high end of the range of values for most substances, indicating that inaccuracy in the vicinity of the critical point should be expected. In Example 11.1, the pressure of a gas is determined using three equations of state and the generalized compressibility chart. The results are compared.

Equations 11.32 through 11.35 are known as the Maxwell relations. Since each of the properties T, p, , s appears on the left side of two of the eight equations, Eqs. 11.24 through 11.31, four additional property relations can be obtained by equating such expressions. They are a 0u 0s b y a 0h 0s b p , a 0u 0y b s a 0c 0y b T a 0h 0p b s a 0g 0p b T , a 0c 0Tb y a 0g 0Tb p (11.36) Equations 11.24 through 11.36, which are listed in Table 11.1 for ease of reference, are 16 property relations obtained from Eqs. 11.18, 11.19, 11.22, and 11.23, using the concept of an exact differential. Since Eqs. 11.19, 11.22, and 11.23 can themselves be derived from Eq. 11.18, the important role of the first T dS equation in developing property relations is apparent. The utility of these 16 property relations is demonstrated in subsequent sections of this chapter. However, to give a specific illustration at this point, suppose the partial derivative 10s/0y2T involving entropy is required for a certain purpose. The Maxwell relation Eq. 11.34 would allow the derivative to be determined by evaluating the partial derivative 10p/0T2y, which can be obtained using p--T data only. Further elaboration is provided in Example 11.3

vEquations 11.32 through 11.35 are known as the Maxwell relations. Since each of the properties T, p, , s appears on the left side of two of the eight equations, Eqs. 11.24 through 11.31, four additional property relations can be obtained by equating such expressions. They are a 0u 0s b y a 0h 0s b p , a 0u 0y b s a 0c 0y b T a 0h 0p b s a 0g 0p b T , a 0c 0Tb y a 0g 0Tb p (11.36) Equations 11.24 through 11.36, which are listed in Table 11.1 for ease of reference, are 16 property relations obtained from Eqs. 11.18, 11.19, 11.22, and 11.23, using the concept of an exact differential. Since Eqs. 11.19, 11.22, and 11.23 can themselves be derived from Eq. 11.18, the important role of the first T dS equation in developing property relations is apparent. The utility of these 16 property relations is demonstrated in subsequent sections of this chapter. However, to give a specific illustration at this point, suppose the partial derivative 10s/0y2T involving entropy is required for a certain purpose. The Maxwell relation Eq. 11.34 would allow the derivative to be determined by evaluating the partial derivative 10p/0T2y, which can be obtained using p--T data only. Further elaboration is provided in Example 11.3

variables are temperature, pressure, and the number of moles of each component present. 11.9.4 Fugacity The chemical potential plays an important role in describing multicomponent systems. In some instances, however, it is more convenient to work in terms of a related property, the fugacity. The fugacity is introduced in the present discussion. Single-Component Systems Let us begin by taking up the case of a system consisting of a single component. For this case, Eq. 11.108 reduces to give G nm or m G n g That is, for a pure component the chemical potential equals the Gibbs function per mole. With this equation, Eq. 11.30 written on a per mole basis becomes 0m 0p b T y (11.119) For the special case of an ideal gas, y RT/p, and Eq. 11.119 assumes the form 0m* 0p b T RT p where the asterisk denotes ideal gas. Integrating at constant temperature m* RT ln p C1T2 (11.120) where C(T) is a function of integration. Since the pressure p can take on values from zero to plus infinity, the ln p term of this expression, and thus the chemical potential, has an inconvenient range of values from minus infinity to plus infinity. Equation 11.120 also shows that the chemical potential can be determined only to within an arbitrary constant. INTRODUCING FUGACITY. Because of the above considerations, it is advantageous for many types of thermodynamic analyses to use fugacity in place of the chemical potential, for it is a well-behaved function that can be more conveniently evaluated. We introduce the fugacity f by the expression m RT ln f C1T2 (11.121) Comparing Eq. 11.121 with Eq. 11.120, the fugacity is seen to play the same role in the general case as pressure plays in the ideal gas case. Fugacity has the same units as pressure. Substituting Eq. 11.121 into Eq. 11.119 gives RT a 0 ln f 0p b T y (11.122) Integration of Eq. 11.122 while holding temperature constant can determine the fugacity only to within a constant term. However, since ideal gas behavior is approached as pressure tends to zero, the constant term can be fixed by requiring that the fugacity of a pure component equals the pressure in the limit of zero pressure. That is, limpS0 f p 1 (11.123) Equations 11.122 and 11.123 then completely determine the fugacity function.

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