E th yl C orporation, B aton R ou g e, La.
T
H E need for accurate estimates of the critical constants, to be used in conjunction with reduced equations o f state, has been pointed ou t b y Meissner and Redding (8). The correlation o f vapor pressure and latent heat data {10) likewise in
volves critical constants which in m ost cases must be estimated.
In general, the am ount of experimental critical data is insufficient to m eet the dem and o f useful engineering applications. H ence, there is a real need for reliable methods o f estimating critical con
stants.
Several m ethods o f estimating critical constants involve the use o f the parachor ( f , 2 ,4 , 8 ,1 2 ), b u t m ost o f them are o f limited application for predicting the critical constants o f a particular substance. T h e purpose o f this study, therefore, was to extend the range of application o f correlations between parachors and critical constants. Particular
emphasis is placed upon quanti
tative measures o f the reliability of the estimated values in order to provide a fair com parison with existing correlations. M an y of the existing correlations are not discussed in this paper, since Meissner and R edding (8) have already analyzed them su
c-cinctly. Gamson and W atson’s recent paper (3) is n ot given de
tailed consideration because it was published after com pletion of the present work. T h e correlations presented(S) are based entirely on normal paraffin data, which lim it their scope of application.
C R IT IC A L T E M P E R A T U R E
T w o useful relations between critical temperature Tc,normal boiling poin t Tb, and parachor [P ] were proposed b y Lewis {6, 7). These are o f the type:
T c- o [P ] + 6; T b - c[P ] + d (1) T c = e log [P] + f ; T b = g log [P] + h (2) E quations 1 apply to various groups o f chemically similar com
pounds having the same number o f atom s (e.g., C H ,I, C H jB r, C H jC l, and C H jF m ight consti
tute one such g rou p ); Equations 2 apply to various hom ologous series. The constants a h are different for each particular groupandeachparticularhom olo- gous series. The use of these correlations is lim ited to com -Various existing correlations o f the critical constants
are examined, and the reliability o f estimate is evaluated for several of them. Four relations in
volving the parachor are developed from which rela
tively accurate critical constants can be estimated, using the normal boiling points and calculated para
chors as auxiliary data. Statistical measures o f the reliability o f the estimated constants are presented.
T a b l e I. C r i t i c a l T e m p e r a t u r e
Groups of Compounds Covered by Equation“
Equation Groups of Com
No. Organic
3A Satd. & unsatd. hydrocarbons (acyclio and unsubstituted) 3B Aromática & cyclics (substituted
& unsubstituted)
3C Substituted aliphatics containing halogen* & S as functional groups
3D Aliphatic esters, ethers, acetals, oxides*
3E Aliphatio ketones, aldehydes, car-boxylio acids, & N compounds 3F Aliphatic alcohols & anhydrides
Inorganic , Paraffinic analogs, such
as silicanes
Inorganic halogen com
pounds
Some inorganic oxides
& 0 compounds such as SOi, Os, HsO Some inorganio N com
pounds such as Ns, NsO, NiOs
Inorganio “ anhydrides"
such as COs, H,
Equations Derived by Least ( Tc/Tb)„
Squares“ nk *
T c/ T b = 2.501 — 0.4176 log [P] 20 1.529 Range: 1.2 9-1 .7 0
T c/ T b - 2.640 - 0.4634 log [P] 28 1.510 Range: 1.38-1.65
Tc/Tb = 2.602 - 0.4449 log [P] 26 1.607 Range: 1 .3 7-1.78
T c/ T b = 2.544 - 0.4429 log [P] 38 1.467 Range: 1.35-1.64
T c/ T b - 2.301 - 0.3548 log [P] 23 1.515 Range: 1.41-1.69
T c/ T b - 1.783 - 0.1479 log [P] 14 1.441 Range: 1.41-1.52
Reliability*
Absolute Relative,
%
(2 S)d 0.034 0.082 0.075
0.036
0.067
0.033
2.2 5 .4 4 .7
2 .5
4 .4
2.3
“ Equations are based almost entirely on organic compounds, for whioh good values of the parachor can be calculated from atomic and struc
tural constants. Inorganic compounds to which equations were found to apply are indicated. For compounds falling in two or more groups, as chloroacetic acid 3G and 3E, the average of values obtained by using equations for each group is recommended. Compounds not covered by any of listed groups can be handled by the principle of chemical similarity. Reliability of estimate for such compounds is indeterminate and probably poorer than tabulated values. Indicated ranges are for experimental (Tc/Tb) values used in establishing equations.
b n = No. of experimental ( Tc/Tb) values used in determining equation.
* Relative reliability (per cent) of an estimated value of {Tc/Tb) is defined as: 100 X 2S/{Tc/Tb), which is a measure of the percentage deviation of the value from the least squares line which will not be exceeded in about 95% of cases. {Tc/TB)m, the mean of experimental values, has been tabulated and used to calculate a reliability for the equation. This procedure was followed to give a common basis for comparison of the reliabilities of the equations. The total range of {Tc/Tb) values covered by the data used in establishing these equations is 1.29 to 1.78.
Calculated values which fall outside of this range will have poorer reliabilities than are determined by the defining equation above.
d S rs(rc/ r Bcaicd. - 7 v r s expti.)qv« , ~ _ .
--- g--- ; and 2S is a measure of accuracy to be expected m about 95% of cases.
* All fluoromethanes and moderately substituted fluoroethanes (as CiH,Fj) can be estimated with greater accuracy from Equation 3D. Equa
tion 3C gives results which are about 10% high. Highly substituted fluoroethanes (as CiCliFi) are handled with Equation 3C.
997
T a b l e II. P a r a c h o r C o r r e l a t i o n o f M u m f o r d a n d text or handbook. A portion is reproduced in Gilman’s "O rganic Chemistry, An A dvanced Treatise” , but the strain constants are not even mentioned although they were used in calculating some of the parachors. This crroJ has been called to the attention of Gilman and Leermakers b y George Calingaert and G. W . Thom son, o f E thyl Corporation.
* Value of hydrogen in com bination with other elements:
C 15.4 S 15.4 Cl 12.8
N 12.5 O 10.0 B r 16.4
6 R — hydrocarbon radical; X — negative group, Cl, C N , CO O R , O R , etc.; for X — Br, multiply strain constant by 1.5.
pounds which belong to groups or series for which sufficient data are available to determine the constants.
Equations can be derived from relations 1 and 2 to estimate T./Tb from [P] or log [P], respectively, for such groups and series. The following simpler equation
T J Tb = a - 6 log [P] (3) was found, however, to be of more general application since it re
quired the classification of over 140 compounds into only six groups, as shown in Table I. The six groups of compounds (3A to 3F) are described in Table I together with the derived equations and measures of the reliability of estimated Tc/Tb values.
Parachors as used throughout this paper were calculated from the atomic, structural, and strain constants of Mumford and Phillips (9) shown in Table II, unless otherwise noted. For some of the "simple” molecules (CO, Oi, Nj, etc.), where calculated- parachors differ appreciably from measured values, the measured
values were used. For compounds not covered by Mumford and Phillips constants, such as those of germanium, measured values or the atomic constants given by Sugden (12) or Lewis (~) were used. Examples of the use of Mumford and Phillips constants are given in Table II.
Comparison of the equations given in Table I with the Lewis Equations 1 and 2 shows that a considerable extension in range of application has been effected. The six equations are applicable to many times that number of homologous series, chemically similar groups, and miscellaneous compounds, most of which were outside the scope of Lewis’ relations because of the paucity of experimental critical data. However, it must be emphasized that the correlation presented is subject to the inherent limita
tions of any empirical relation.
Returning to Equations 1 and 2, the parachor can be eliminated from each pair of equations to give a relation of the form:
T„ = IcTb + j (4) Two of the empirical equations proposed by Meissner and Red
ding (8) have this form and can be readily tested. The method of least squares was used to obtain the best possible constants for the equations as applied to the particular data available, with the results shown in Equations 5 and 6. The constants so derived differ from those of Meissner and Redding, since these authors based their equations only on the critical data for hydrocarbons, thus assuming that the behavior of the hydrocarbons was repre
sentative of all compounds. This assumption is open to some question, and while experimental critical data are available foi relatively few compounds besides the hydrocarbons, it appeared desirable to consider all available data in calculating the con
For ninety-three compounds (halogen- and sulfur-free), other than aromatics and naphthenes, boiling above 235° K .:
Tc = 1 . 1 1 2 Tb + 131.8 ( 6 )
Reliability of Tc = 5.4%
Although the stated Tb limit of these equations is 235° K., the value selected by Meissner and Redding, they actually inter
sect at 225° K . There is a similar discrepancy in Meissner and Redding’s own equations which, however, is well within their limits of accuracy.
The reliabilities in Table I show that the correlations proposed in this paper are considerably better than those for Equation 5 and slightly better than those for Equation 6. From the defini
tion presented in note (c) to Table I, the “ reliabilities” reported here are a measure of the maximum deviation, the average devia
tion being only about 40% as great in each case. The reason for adopting this more stringent definition of reliability is the paucity and doubtful value of much of the available critical data. The equations proposed by Meissner and Redding for aromatics and naphthenes boiling above 235° K. and for halogen and sulfur
Several relations between the critical volume, Vc, and parachor have been proposed. The simplest is due to Sugden (18):
Vc = [Pj/0.78 ( 7 )
and is based on his theoretical interpretation of the parachor.
The value of the “ constant” (0.78), however, has been found to vary by as much as ± 3 0 % . A purely empirical relation, having
a higher degree of accuracy, was proposed by Meissner and Red
ding:
V e = (0.377 [P] + 11.0)'/« (8) A quantitative measure of the reliability of this equation was not given by the authors.
A dimensional analysis o f the parachor (11) results in the equa
tion:
[P] = KVS/’ TS'* (9)
This relation, with K — 0.41, was first proposed by Ferguson (1), who subsequently (3) offered the following empirical relation which is not consistent with dimensional analysis:
[P] = kVS'Wc'/* (10)
Lautie (5) arrived at the equations
[P] = 0.681 VeU iTcl P (11) [P] = 0.316Fepe' " (12) by combining several theoretical and empirical equations which apply strictly only to normal liquids.
Since Equation 10 appears to have the most rational basis of all the proposed relations, it was first selected for further investi
gation. Rewriting Equation 10 in the form
V c = iqP]'-77V >.» (13) and empirically separating the available data into two groups, the following results were obtained:
1. For compounds having the functional groups — C = O,
— C s s N, — COOH and — OH, and one to three additional non
functional carbons:
V c = 3.34[P]‘ -7 7 V -s (13A) Reliability = 10.0%
2. For all other compounds:
V e = 2.92[P]1-77y>-» (13B) Reliability = 6.5%
Reliabilities were determined statistically as in Table I.
The reliability of the values of Vc estimated from Equations 13A and 13B is relatively high, and is not influenced appreciably by the accuracy of the T{ values used (e.g., an error of 5 % in T, will introduce an error of less than 2 % in Vc).
Lautie’s relation, Equation 12, was also studied further; re
writing it in the form
V c = C [ P ] / V - “ (14) and empirically separating the available data in three groups, the following results were obtained.
1. For aliphatic organic compounds containing two or less carbons in addition to the functional atom or group (e.g., CH«, CjHsOH, C2H5COOH), and all other compounds having T c <
450 ° K .:
Vc = 3 .5 8 (P ]/p c0-55 (14A) Reliability = 8.1%
2. For aliphatic organic compounds containing three or four carbons in addition to the functional atom or group, and all other compounds having Tc between 450 and 600° K .:
V c = 3.31 [P]/pc°-a (14B) Reliability = 5.2%
3. For aliphatic organic compounds containing more than four carbons in addition to the functional atom or group, and all other compounds having Tc > 600° K . (excluding H jO ):
Vc = 3 .1 0 [P ]/V -“ (14C) Reliability = 7.5%
The reliability of Vc values estimated from Equations 14A, 14B, and 14C is about the same as that of estimates made from Equations 13A and 13B. The latter equations are more useful because experimental values of Tc are more numerous than ex
perimental values of pc, and Tc can be estimated much more accurately than pe. When appropriate data are available, rela
tions 14A, 14B, and 14C can be of service by offering an inde
pendent check on values of Ve estimated from 13A and 13B.
In order to make a quantitative comparison between the equations developed above and those of Sugden and of Meissner and Redding, the available data were fitted by the method of least squares to equations of the types proposed by these authors.
Assuming Sugden’s relation to have the form Vc — A [P] + B, it was found that
V e = 1.447[P] - 20.09 (15) Reliability = 18.1%
Similarly, assuming that Meissner and Redding’s relation had the form Vc - (P [PJ + E ) l-n , it was found that
Vc = (0.3591 [P] + 14.00) >■“ (16) Reliability = 15.7%
The constants in Equation 16 differ from those in Meissner and Redding’s paper since the latter were based only on hydrocarbon data.
Inspection of the above results shows that the correlations developed in this paper (Equations 13 and 14) are more reliable than those of the type proposed by Sugden or Meissner and Red
ding, based on the data available at present. They also show that Meissner and Redding’s correlation (Equation 16) is not appreciably more accurate than the simpler modified Sugden relation (Equation 15). Further investigation of equations of the type:
Vc = (F[P] + G)" (17) where n varies from 1.0 to 1.25, indicated that the reliability of this type of correlation is not very sensitive to the value of n used.
C R IT IC A L P R E S S U R E
The estimation of critical pressures, pc, is generally made by means of equations which include T e and Vc. Meissner and Redding proposed:
pc = 20.8Tc/(Vc - 8) 118)
Wohl (IS, 14) proposed:
Pc = 21.8Tc/Vc (19)
The reliability of estimate is usually poorer than for Tc or V, alone. Equation 12, involving the parachor and Vc, was pro
posed by Lautie and has been developed in this paper as a method for estimating Vc. Rearranging Equation 12 into the form
Pc = (C [P ]/ V cY (20)
gives a relation for estimating pe (the empirical groupings and values of C are the same as for Equations 14A, 14B, and 14C).
The reliability of estimate for pc is relatively poor (about 20 to 30% ) because of the fourth power involved. This reliability is of the same order as that obtained by other generalized methods for estimating pc, and the relation is useful because of its broad scope.
In an attempt to obtain a more reliable relation for estimating P c the following line of attack was tried.
In general (11 ): log p/pe — m — n T c/T At the normal boiling point: p = 1, and T = Tb Therefore —log pe — m — n T J T » Combining with Equation 3: lo g p c = —m + n(a — b log [P])
Or log pc - o ' — b’ log [P] t211
Ta b l e III. Cr i t i c a l Pr e s s u r e
Reliability*
Equa Abso Rela
tion Groups of Compounds Equations Derived by Least (Log lute tive,
No. Covered by Equation“ Squares“ n» Pe)» (25)* %
21A Satd. & unsatd. hydrocarbons (acyclic, unsubstituted), ex
ing biphenyls & condensed ring systems
2ID Aliphatic acids, alcohols, &
anhydrides
“ Application of these equations to compounds not specifically covered in listed groups is accompanied by a decreased reliability of estimate (exact magnitude not predictable).
b S and n have same significance as in Table I; (log pc) m is similar to (T c /T s )».
“ Reliabilities for pe were calculated from values of 2,S and (log Vc) » ■ Since the plus and minus devia
tions were unequal (in transposing from logarithms to natural numbers), the larger deviation was used to give a more conservative estimate of reliability. The tabulated values were calculated from the expression:
100 X [(antilog [2S + (log pe) „ I/antilog [(log pe) „ ] ) - 1]
highest for the most limited groups. Therefore, the best re
sults should be obtained when a single homologous series or group of chemically similar com
pounds is fitted by Equation 21. Where sufficient data are available, this procedure is rec
ommended, but owing to the limited data on p,, the less ac
curate general relations have greater application.
Similarly, Meissner and Red
ding’s equation gave: the reliability of estimate is generally poorer. This result is not unexpected, since Equation 21 combines several rough approxi
mations. It is also evident that the reliabili' of estimate is
Substance [P]° Tb Exptl. Estd.* ation& Exptl. Estd.e taion Exptl. Estd.d ation
Bromine 132.1« 332 575 550 - 4 . 3 135.4 154 + 14 valuable for application to substances not included in the specific groups listed in Table III. It should be emphasized that Wohl’s relation will probably give poorer reliabilities than
are indicated above when ex- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ perimental values of Vc and Tc are not available and it is neces
sary to use estimated values.
6 Calculated from Equations 3 A to 3F : % deviation — 100 X (estd. — exptl.)/ex p tl.
“ Calculated from Equations 13A and 13B, using estimated Tc values.
“ Calculated from Equation 20, using experimental Vc values where avaitable, or 21A to 21F.
* Experimental parachors. check of the proposed equations, they were applied to several compounds for which critical data were available but which had not been used in establish
ing the equations. The results are given in Table IV. Devia
tions of the calculated constants from the experimental values are in excellent agreement with the reliabilities of estimate de
termined from statistical con
siderations. In view of the essentially empirical nature of the proposed relations, the sub
stantiation afforded by this test provides some measure of con
fidence in their utility.
I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y A C K N O W L E D G M E N T
The author wishes to acknowledge gratefully the constructive criticism of this paper by 0 . E. Kurt, G. F. Kirby, Jr., and G. W.
Thomson, all of Ethyl Corporation; and the able assistance of Justine S. Herzog in compiling the data.
N O M E N C L A T U R E [P] = parachor
Tc — critical temperature, ° K.
Tb = boiling point, ° K.
Vc = critical volume, cc./mole p = pressure, atmospheres pc — critical pressure, atmospheres A, B, C, . . . K ; a, b, c. ..k = constants S = standard error of estimate