forced a so as to reproduce the vapor pressure slope near the critical temperature, whose values were estimated as a function of u:. He gave first a Iinear expression for a(T*), which he Later (hJiIson, L966) modified by introducing a multiplier LlTo*''2; vapor pressures calculated by such expressions were very accurate at reduced temperatures approaching unity, but they tended to diverge at the lower temperatures. Zudkevitch and Joffe (Zudkevitch, Joffe, L97O) assumed both O. and Q- to be temperature dependent, and determined the functionsD na(TR), nb(T*) for single pure compounds, by imposing that saturated-1Íquid and saturated-vapor fugacity coefficients calculated at the experimental vapor pressure r^rere equal to those estimated by a generalized correlation (Lyckman et aI.,1965) of the kind: log@s= ( togps )to'*. ( logpt )tn' parameter. The common choice hras to refer to its critical value, by multiplying it ot(T*,<r) replacing I/"î:'" in eq . (2) , while Tp oro ( TR,. ) =-n-R rR-Qb v*(v*+o') rnfs= z.-L-ln( Z"-nb) - (nalob) ln( I *frb/2") =-0 . 4o7o 349 the attractive parameter by a correction factor b was kept constant: (3) (4) In a following paper (Joffe et aI.,1970), na(TR) and Ob(TR) btere determined by imposing the saturated-Iiquid density value and the isofugacity constraintr Of=Of, regardless of the values of the single-phase fugacity coefficients and the saturated-vapor density. Both methods failed at the critical temperature, since there the values of Q.,QO are determined by the application of the critical constraints, and hence both the values of r" and Cs calculated through the equation of state are defined: (s) (6) SSV=VLV The impossibility of forcing a two-parameter equation of state to reproduce critical properties has been recognized definitely. AII trials to improve the results at the criticat temperature have been based on the use of a higher number of parameters. It has been 350 also seen that the inability of an equation of state to reproduce saturated-phase volumetric properties does not affect its phase-equilibrium performance; so most works have focused on the correration and prediction of phase equilibria, negJ-ecting the volumetric properties. The works by Joffe and Zudkevitch did not provide any generarized expression for the estimation of na(TR), nb(TR), which had to be determined for each substance by correlation of experimental vapor-pressure and liquid-density data. The first generalized, wide-range predictive equations were given in 1972 by thJ-s author (Soave,L972). tdhile nU hras assumed constant, Qa(TR ) was determined from experimental vapor pressures of non-polar substances and was expressed through a correction factor o(T*) of the criticat value: o. ( T* ) =a ( Tn )nu. ( o."= 0 . 427 48) By an examination of the curves of c.,; values (see Fig.4), it was seen that a good approximation a Iinear function (fis.5) (7) ot(T ) obtained for various . R' the square root of or was to of the square root of T* 05 07 09 reduoed temperature Sf,udre rccl of alpha 00 07 0B 09 square root or Tr Fig.4. aJ-pha vs/ reduced temperature fÍ9.5. The square root of alpha vsl the square root of T 351 6=1+mll-u5 ) ' R', The slope m r^ras assumed to be a factor o and an analytical expression (8) function of the acentric bras derived for it: m=0 . 480+ 1 . 574t^r-0 .1.7 6r.o2 The 1972 version of the R-K equation hras the first equation-of-state based method to be programmed on computers and applied widely for the design of hydrocarbon-treatment plants. It enabled the design of new process schemes and nehr working conditions where the old, activity-coefficient based methods failed. Before it, equations of state had found a limited applicatÍon and were considered a theoretical peculiarity. Its success proved the great potential of this kind of method and marked the beginning of a flourishing of refinements and new models, which lasts nowadays. It remains anlrway the first modern equation of state, and this is the reason for the titte of this paper. FURTHER DEVELOPMENTS OF THE REDLICH-KT^IONG EQUATION It is impossible to describe in one paper the refinements, modifications and additions which in the last years have been made to the original model. It is possÍble only to focus on a few significant improvements, which have found wider application. a) Pure-component vapor pressures Eq.(8), although widely applied for cubic EOS's, is only an approximation; fr is not completely rectitinear in fr*, and furthermore the calculated vapor pressures are very sensitive to any error in a at the lower temperatures. As a resuJ.t, calculated vapor pressures tend to diverge from the experimental ones as the temperature decreases. Besides, eq. (8) is not applicable to polar compounds, particularly associating compounds Iike alkanoIs. In order to extend the application to the low temperatures and to polar substances, many Authors (Soave,1980), (Harmens,Knapp, 1980), (Mathias,Copeman,l-983), (Mathias, l-983), (Stryjek,Vera, 1986), (e) 352 (MeJ-hem, 1989) have repraced eq. (8) by some two-parameter equation; the two parameters must be determined by correlation of experimental vapor pressures, which decreases the predictive character of the equation of state and requires experimental vapor pressure data. The last problem can be partly overcome by using for a an equation which can be reduced to eq. (8) when necessary; it is the case of (Soave,1991): a=1+m( 1-T* ) +n( 1 -fi*f an equation which becomes identical to eq.(8) when n=m(m+L) that case m can be determined from <o according to eq. (9) ). generally, fr,D are determined separately by correlation experimental vapor pressures, when available. ( 10 ) (in More of (11) of this modification is that the values of need not be changed: the effect is to coefficients in the equilibrium phases by a their ratio (the K values) are not affected. original equations for the calculation of b) Pure-component volumetric properties It is well known that the R-K equation underestimates the density of liquid phases by l-0-L5U (according to the value of the acentric factor and the reduced temperature). That drawback has been only reduced by using a different two-parameter equation of state; e.9., the Peng-Robinson EOS yields higher Iiquid densities, which are nearer to the average hydrocarbon densities, but are stilr fÍxed. The sorution can onry tie in the use of a third parameter. An elegant proposal, which has been applied to other two-parameter equations also, Ís the one by Peneloux and Rauzy (PeneIoux,Rauzy,7982); they modified the R-K equation by introducing a volume shift, not subjected to the critical constraints and adjustabre for each compound to reproduce experimental tiquid density values. rf c is the volume shift, the R-K equation is rewritten as: RT9-- v+c-b a(T)(v+c) (v+c+b) A great advantage parameters a(T) and b multiply the fugacity common factor, so that This allows use of the 353 parameters and K values, introducing the correction for density calculations only. The value of c depends on the substance considered and is usually assumed constant. Irlhen determined f rom saturated liquid densities, it varies only slightly with the temperature up to about T*=0.8, so it is usual practice to determine it from the liquid density at the boiling point or at T*=0.7 and, for non-po]ar compounds, to predict Íts value from the value of the acentric factor. It is interesting to remark that, although it is possible to use for each temperature the value which exactly reproduces the saturated liquid density, that would give a perfect reproduction of the critical density (the dream of equation-of-state builders!), but a poor critical isotherm at aII other points.