JOURNAL OF RESEARCH of the National Bureau of Sta nda rds - A. Ph ysics and Chemistry Vol. 79A, No. 1, January-February 1975 Equation of State for Thermodynamic Properties of Fluids * Robert D. Goodwin** Institute for Basic Standards, National Bureau of Standards, Boulder, Colo. 80302 September 6 , 1974 This equation of state was de veloped from PVTc ompressibility data on methane and etha ne. The hi ghl y-constraine d form ori gi na tes on a given liquid-vapor cuexiste nce boundary (described by equations fur the vapor pressures a nd the urthobaric d e nsities). It the n requires onl y fiv e least-squa res coeffi ­ cient s . a nd e ns ures a qualitatively correct be havior uf the P (p, T) s urface a nd of its de riv atives. espe­ cial ly about the critical point. This nonanal ytic e quation yields a maximum in the specifi c heats C,,(p, T ) a t the criti cal point. Key words: Coex iste nce boundary: critical point; e tha ne; equation uf sta te; fluid s; methane; orthobaric d e ns it ies; s pecifi c heats; vapor pressures. Symbols and Units 1. Introduction Subscripts !2 and.!.. refe r to criti cal and liquid triple A proble m of im portance for the nat ural gas in ­ points dustry is th e prediction of thermodynamic properties Subscri pt q refe rs to Liquid-vapor coexiste nce of lique fi ed, m u Iti co m ponent mixtures. For t he wide a, b, [) nonlinear const.ants in the eq uation'of state range of co mpositi ons encountered , it may be neces­ 8(p), de nsity-de pe ndent coeffi cie nts in the equa- sary to utilize accurate properti es of the pure com­ C(p) tion of state ponents. We the refore be li eve that it will be helpfu l Cer (T), molal heat capacity for saturated Liquid to have a relatively simple and rational equati on of C~ (T), molal heat capacity for id eal gas states state of ide ntical form for each compone nt, s uch as C r (p , T) , molal heat capacity at constant de nsity the equation of state described below. C,,(p ,T) , molal heat capacity at constant pressure This equation ori ginates on a give n, liquid-vapor d, de nsitY , mol/l coexiste nce boundary, thus eliminating the long­ j, the Joule, l N-m standing problem of consiste ncy between equations l, the liter, 10- 3 m 3 of state a nd this inde pende ntly-derived e nvelope mol, 16.043 g of CH 4 ; 30.070 g of C 2H(;, (C 12 [261. It ensures a maximum in the s pecifi c heats scale) Ct. (p,T) approaching th e critical point, qualita­ P, pressure in bars, 1 bar= 10 5 N/m 2, (1 tively consiste nt with ex perime ntal behavior near, but atm = 1.01325 bar) not necessarily exactly at thi s pole [161, and it has Per(p) , the vapor pressure, bar only fiv e arbitrary, least-squares coeffi cients. Experi­ R, the gas constant, 8.31434 (J/mol}/K me ntal compressibility data for methane and ethane p, did"~ de nsity reduced at the liquid triple have been used at densities to the triple-point liquid point density, te mperatures to twice the c ritical, and er, d/dc, density reduced at the critical point pressures to 350 bar or greate r. T, te mpe ra utre, K, (IPTS-1968) Our objective in the present report is to give a co n­ Ter(p) , te mperature at liquid-vapor coexistence cise desc ription of this new type of equation of state () (p) , defin ed locus of te mperatures, figure 3 which, with full docume ntation, has been presente d U(p, T), T/T (T (p), temperature reduced at in previous publications [14 , 15J. coexistence For me thane we have shown in [141 that a n equation v, l/d, molal volume, l/mol similar to that described below yields calcul ated W(p ,T) , the speed of sound specifi c heats, C(J"(T), Cv(p,T), Cp(p,T), and speeds x (T), T/Te, temperature reduced at the critical of sound in acceptable agreement with experimental point data, without any weighting of the e quation of state x" (p), T rr( P )/Te, reduced te mperature at to those data. In a c urrent re port on ethane [15J coexist ence we compute provisional thermodynamic and related Z(P,p,T) Pv/RT, the "compressibility factor" properties by means of the simpler equation of state (1) described herein. *This work a l the National Bureau of Sta nd ard~ was supported by The American Cas Symbols and units used here are given in a List. Association. 1515 Wi lson Buule vard. Arlington, Va. 22209. **Cryogcnics Di vis ion. Natiunal Bureau of Standards. Boulde r. Colo. 80302. Fixed-point values from [14, 15] are given in table 1. 71 For equation of state (1) the gas constant IS This monotonic behavior (nonnegative slopes) is R ;= (0.0831434)· (dt ) bar/K, consistent with use of difficult to achieve from equations of state, yet is very the dimensionless density, p ;= d/dt• important for such computations as- TABLE 1. Fixed·point values from [14,15] W(p, T) = [Cpo (ap/ap)/Cv] 1/2. (c) Triple point Methane Ethane Densit y ................ mol/I.. Vapor ....................... 1.567 865.10- 2 1.35114'10- 6 Liquid ...................... 28.147 21.68 Te mperature ............. K .. 90.680 89.899 Pressure ............... bar" .. 0.1174 35675 1.009 906.10 - 5 Critical point I I I Density ............. mol/l.. 10.0 6.74 I I Temperature ........ .K .. 190.555 305.37 I I Pressure ............. bar.. 45.988 48.755 I I I I a Precision required for vapor-pressure equation. I I I I I ---------1I------------- Various methods for utilizing the equations of state I of pure components to derive properties of mixtures are described in a number of recent publications, e.g., Pc p- [1 , 4a, 17, 18, 19, 20,21, 21a, 22, 23,24,271. FIGURE 2. Bp.havior of the critical isother",. The liquid-vapor envelope, T ()" (p), figure 3, is an important boundary of the P(p, T) surface for the equa­ tion of state. We constrain the equation to this bound- Tc T- FIG URE 1. The locus of isochore inflection points. c D 2. The Equation of State The P(p, T) surface and equations of state are ~l described in several reports, e.g., [6,7,10,12,14,15, Ii 20, 24, 28J. Figure 1 shows the qualitative behavior of / I isochores as indicated by Rowlinson [20J, needed to / I / : give a calculated maximum in C v (p, T) at the critical ./ I ,./ I point via the isothermal co m putation- ,./ I - I -- 'A (a) TEMPERATURE Figure 2 shows the well-known zero slope and FIGURE 3. Behavior of the locus O(p). curvature of the critical isotherm at the critical point. Point C is the critical point. a nd Tu( p ) is the liquid-va pur coexiste nce e nvelope. 72 ary as follows, by use of the vapor pressures and the describe the sigmoid shape of isochores in the de nsity coexiste nce te mpe ratures T" (p) formulated in the range pc < p < 2·pc, figure 1. appendix. The first of these functions is- For any density (isochore), obtain the coexiste nce te mperature from the fun ction T,,(p) . Use this to <t> (p, T) == x ' /2 'ln [u(p , T)], (2) obtain the vapor pressure P,,[T,,(p)] as a function of density, thus defining the equation of state at coexist­ where u(p, T) == T/ T,,(p). It is lin ear (a 2cp /ap = O) ence. By subtraction one then obtains an equation of e verywhere on the coexiste nce boundary at u = 1, the type of eq (1) . The melting lin e is not a part of this figure 4. It therefore giv es a criti cal isochore whic h is equation of state linear at the criti cal point because C(p) = 0 by defini­ tion along this isochore, eq (6). The second of these functions, 'I'(p , T) , is defined such that eq (1) wiU yield a maximum in the specific where, heats Cv(p,T) at the criti cal point via eq (a). We first define the arguments- F(p , T) == B(p)'CP(p, T)+C(p)''I'(p , T). (1) W(p,T) == o· [T/fJ(p) -1]. The fun ctions B (p) and C)p) are polynomial coeffi­ cie nts to be found by least squares. The te mpe rature· wa(p) == o· [T,,(p) /(}(p) -1] , dependent functions cP (p , T) and 'I' (p, T) must be defin ed to be zero on the coexistence boundary at where 0 is an arbitrary constant, and fJ (p) is ou r T = Ta(P) , as s hown by fi gures 4 and 5. locus of tempe ratures in side the coexiste nce e n· velope, figure 3, fJ(p) == T,,(p) . exp [-ex ·f(u) ]. (3) The function feu) here is normalized to unity at the liquid triple-point density- ..... >8<0 wh ere u, == d,/dc is a co nstant. Function 'I' (p, T) now is defined as the difference, 'I'(p,T) == tf;(p,T) -tf;,,(p) , (4) such that 'I' = °at coexistence, T = T" (p). Com pone nt o T- functions, tf; (p , T) , are designed to give infinite c urva­ ture (a 2 tf;/ap) at the origin, w = O, FIGURE 4.