The Value of the Critical Compressibility Factor for the Redlich-Kwong Equation of State of Gases Reino W. Hakala Governors State University, University Park, Illinois 60466 The Redlich-Kwong ( I ) equation,
We should solve eqn. (3)in another way to provide a check. Equation (3)is easily rearranged to b3
is reputedly the best two-parameter equation of state of real gases (2-5). The value of the compressibility factor at the critical point for this equation is, however, variously given in the literature as 0.328 and 0.333. These disparate values are due to the manner in which z, is determined, a way which encourages a numerical rather than an analytical solution. An alternative approach will be presented here by which i t is readily proved that z, = 113 exactly. This exact value will be compared with the entire set of the literature values of 2,. Lastly, a recent modification of the Redlich-Kwong equation will be described. Direct Approach If we use eqn. (1)to find (aP/aV)T,equate the result to zero a t the critical point, solve for a , substitute the expression for a into eqn. ( I ) to eliminate a, and solve for zC PcVJRTc,we obtain
-
+ 3b2V, + 3bVC2- V;? = 0
(blVJ3 + 3(blVJ2+ 3(blVJ - 1 = 0
(6a) (6b)
Let blVc = u - 1. Then eqn. (6b) simplifies to u3 - 2 = 0 which immediately provides the solution b = (2I"
- 1)Vc
(7)
As a check on our two results,
(3) (L)= ( b
I+I
+)
( I - 1)
V,
Substituting eqn. (7) into eqn. (2)provides us with
where We now need to evaluate V , in terms of b, so that we will be able to calculate z, using eqn. (2). If we find both ( d P l d V )and ~ (a2P/dV2)T,equate each of them to zero a t the critical point, solve respectively for RT,I(V, - b)2and R T J ( V , - bI3,take the ratio of the resulting expressions, and simplify algebraically, we get the cubic equation
which is also tedious to simplify algebraically, though less tedious than before. Algebraic simplification of eqn. (8) proceeds as follows:
This eauation has one real root which can be found bv Cardano's (Ferro'i) method ( 8 )(which issomewhat tedious,.ihough srraighrforward, to apply). The root of this equation is
Substitution of eqn. (4) into eqn. (2)yields
These details are provided here because not all of the steps are immediately obvious.
where
Since algebraic simplification of eqn. (5) would be very tedious, the value of z, will he found by means of calculation instead. A modern electronic calculator with a y x key and a sum-to-memory feature readily provides us with the result z, = 0.3333333 which strongly suggests, although i t does not rigorously prove, that z , = '13 exactly. In the old days before electronic calculators were available and slide rules were customarily used for this type of calculation, the value 2 , = 0.328 was calculated and published.
110
Journal of Chemical Education
Alternative Method We shall now present a different approach by which it is made clear, without any tedious algebraic simplification of either eqn. (5) or (81,that z, = 113 exactly. Indeed, this alternative method entirely avoids having to solve eqn. (3)or (6) directly. When we multiply both sides of eqn. (2)by the denominator of the right-hand side, collect terms in like powers of V,, and divide through by the coefficient of V,3, we find another equation in V,:
Because this result is equivalent to eqn. (3), we can equate corresponding coefficients:
In each case we find that z, =
exactly.
Value of a and Alternative Value of b Since we have obtained the solutions of eqns. (3) and (6), we can find an exact expression for a , which is generally given only approximately. We will need this expression later. The expression for a found by equating ( a P 1 a V ) ~to zero a t the critical point is
T o 10 sienificant dizits. the numerical factor in this result is equal to-l.28244076. ' We can find a in terms of P, instead of V , bv multi~lvina .. the right-hand side of eqn. (9cj(which is mope convenient for doing atithmetic than is eqn. (9h)) by RTc13PcVc, which equals unity in the context of the Redlich-Kwong equation, and simplifying. The result is By means of the same procedure, we can also find b in terms of PCinstead of V,: b = ((W- 1)/3)RTJP, = 0.08664034995RTJPC
(11)
A Recent Modification Giorgio Soave (9) has pointed out that the temperature function in the second term of the Redlich-Kwong equation (1) is not sufficiently accurate to be able to use this equation, together with mixing rules, for multicomponent vapor-liquid equilihrium calculations. Accordinalv. - . he modified eqn. (1) by replacing by R2TC2 a(T) = 0.42747PC
in which o is Pitzer's acentric factor. Equation (12) should be compared with alT112 as given by eqn. (10). (We have found that "0.42747" is an approximation for 1/(9(2'13 - 1)).Soave's modification results in a considerable iinorovement. ~ a r t i c ularl?. for substances with large valuri uf the acentric factor. 'I'his modiiication of the Hedlich-Kwonr eauntim. however. leads to the quadratic equation zC2- z J(2
- 2'13) + 1/(9(2'4(2113 - 1))= 0
of which the smaller root is z, = %,the same as for the unmodified R-K equation. C6mbarISOn with Ex~erimentalValues It is useful ron~mparethe Redlich-Kwong valueofz, with z , valws of'actual sul~stances.The somewhat more than 200 experimentally derermineil values ofz, ( 6 , ; ) rangr from 0.12 (for HFI tn0.47 ( N 2 0 n , .The mode of thiiset of values is0.27, representing 27% ofall of the organic and inorganic substances whose z , values have been calculated from measured critical constants. The narrow interval between 0.26 and 0.28 includes 61% of the values, 77% are included in the slightly wider interval 0.25-0.29, and 9070from 0.23 t o 0.31. (It is interesting to note in passing that a histogram of experimentally determined z , values is closely approximated by the normal distribution.) Thus, even though the Redlich-Kwong equation is outstandingly good elsewhere it does not provide a value of z, that is a t or close to the mode. However. it should he added that even if an analytic equation of state did provide ~ h modal r value of z,, e.a., the Dieterici equation, it c~,uldnot nmdel the region in the-neighborhood of the critical point accurately anyway since the critical point is singular. Literature Cited (I1 RedIiih.O.,snd K i i g . J.N. S..Chem. Re.,44,233 (194'3).
(21 Shah, K. K..and Thadoa, 0.. Ind. En8 Chrm., 57.30 (19651. (31 Biene.A..andBak.T.A..Acfo Chem.Seond.23.1783 119691 iai oit,~.'B.,'coate~,i R.,& H ~ IH. , T., J ~ .~.CH"w~Eouc..hs,~~~ , (1971). ( 5 ) Kerno, M. K., Thomoson. R. E., and Ziemne. D. J.J. CUEM. EDUC..52.802 (19751.
~
~
~.
~
dihber c;., ~l&land;OH. 1968, pp. 9%95: Spiegol, M. R.. "Mathemstbal Hsndbmk of Formulas and Tables,"Sehsum'a Outline Series, McCcsw~HillBook Co.. New York, 1968, p. 32. (9) Soaue.0..Chem.Ene. Sci..27,1197 (1972). ed., The Cheiical
Volume 62
Number 2
February 1985
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