Watson and Nelson (7) introduced the following boiling point
and gravity ratio, t e r m e d the
“characterization factor,” aa an indication of the chemical character
of a hydrocarbon and as a correlative factor:
where K
= characterization factor
= specific gravity
a t G O O F.
T E = atmosph:rjc b o i 1 i n g
point, Rankine
d
This factor is valuable in that it
is reasonably constant for chemically similar hydrocarbons. It is
useful as a quantitative means for
expressing the effect of variation in
chemical type and, in conjunction
R. L. SMITH AMI) K. M. WATSON
with one other property, serves as
a basis for correlating properties
U n i v e d Oil F’roducts Company, Riverside, Ill.
of pure componenL9 of widely varying c h e m i c R 1 types and boiling
points.
Application of the characterization factor was extended to
KOWLEDGE of the critical properties of hydrocarbon
mixtures by considering the boiling point of the whole mixmistures, particularly as a function of properties readily
ture to be the “molal average boiling point,” which was less
determinnble, is of great value for estimating the physithan the volumetric average by a correction which was a
cal snd thermal properties necessary for inteligent deeiqn of
petroleum refining equipment. Many of these properties are
function of the Engler distillation curve slope. It was
pointed out that the boiling point so obtained was actually
directly correlntable on the basis of the criticaI temperature
higher than the true molal average boiling point and was
and pressure by r pplication of the theorem of corresponding
established in an effort to obtain a single, simple average
states.
boiling point which would eliminate dependence on width of
The most easily determined properties definitive of a hydrocarbon are the specific gravity and the boiling point. Many
boiling range from the characterization factor and correlations
other properties, such as molecular weight, chemical charof other physical properties. However, more complete dnta
acter, cracking reaction velocity constants, hydrogen content
and apF.lcation of the principle of additivity of the properties
and heats of combustion, are correlatable as functions of these
of mixtures indicate that for wide boiling mixtures the Watsimple inspection data. However, petroleum problems alson-Nelson “molal average boiling point” represents an unmost always invoIve m.:stures of great numbers of components
satisfactory approximation to the proper variable for corwhich it is usually impracticable to treat separately, even to
relation of many properties.
the extent of determining individual bailing points and
If components of the same chemical character or charncterigravities. I n such cases it 2s custDmary to measure the
zation factor, IC, are mixed, the resultant blend should have
gravity of the entire mixture and to report the boiling point
the same K . From the above definition of K , since d varies
in the form of a batch distilla’ion curve. For the higher
linearly with volume per cent, ( T ~ ) 1 /must
3
also ~ a linearly
q
boiling substances, the distillation must be carried out a t
with volume per cent.
sufficiently lorn tempernturas to avoid decomposition, necessitating prcssures lower than atmospheric and correction to
the equivalent atmospheric pressure values by means of
where j v l = volume fraction of component 1 in mixture
vapor pressure relations. Since the true boiling point distillation is nat yet in general use by the industry, boiling point
Thus, the proper boiling point to use for calculation of li is:
curves obtained by the A. S . T. M. and h a l e r distillations will
be considered in the lollowing discussion and correlations.
T s b v . ) lfv*(To)‘’ I
fvdTo*)”* * * 1
-
Additive Properties of Mixtures
LIany properties of mistures niay be estimated by assuming
that the propertics of the individud components are directly
d d i t i v e . For example, such properties as the weight or
hydrogen content of a mixture are equal to the sums of the
corresponding properties of the componenta. Siqilnrly, it
mny be satisfactorily assumed that in liquid hydrocarbon
mixtures a t atmospheric temperature and pressure the volume
of the mixture is equal to the sum of the volumes of the componcnk, neglecting volume changes in mixing. On the basis
of these principles of additivity, it is possible to establish
conditions which must be met by correlations of various
properties that are to be mutually applicable to mixtures and
pure components.
+
+ -
4
This boiling point will be termed the “cubic avcrnge boiling
point.” For petroleum fractions i t m n y be obtained by subtracting from the volumetric average a correction cspresscd
n8 a function of Engler distillation curve slope and volumctric
average I~oilingpoint. This correction is prcsc&A grepliicnlly by the scrics of curves in Figure I , dtxignatd ~8 cubic
average.
If two componeots of diffcrcnt K and thc samc or different
i>oilingpointsare mixed,
INDUSTRIAL AND ENGINEERING CHEMISTRY
DECEMBER, 1937
+
or
Kdx. = fmKi
/wr&
where fwi = woight frnction of component 1
(8)
Therefore, the characterization factor, K, when based on
the correct cubic average boiling point, is additive with weight
fraction in all mixtures where changes in volume with mixing
(arenegligible.
It may be similarly demonstrated that A. P. I. gravity is also
d d i t i v e with weight fraction in all mixtures where volume
changes in mixing are negligible:
A. P.I.mix. = /mi X A. P. 1,i $- j w ? X A. P. 1.2
+ ... .
(4)
Application of Pure-Component Correlations
to Mixtures
If B correlation of any dependent vnrinblc is developed ns a
function of two definitive properties, it may be validly applied to both mixtures and pure components only if the
average definitive properties or the mixture are so defined
that the dependent variable for any mixture is the same as
that of a pure component having the Fame definitive properties as the average definitive properties of the mixture. This
principle may be used to determine directly the proper average
definitive properties for mixtures when pure-component dntn
arc available. When only data on mixtures are available,
the principle may be applied, by trial-and-error assumptions
of average property, to determine the correct average d e
finitive property and consequently establish a valid purecomponent correlntion if the range of variation in each misturc is small relative to the total range covered by the correlation.
As an npplicntion of the above principle, the correlation
of Iuu!cnulnr weight with boiling point and A. P. I. gravity
may be ccnsidered. The molecular weight of a mixture is
additive with the mole fractions of its components as is nlso
Consideration of the principles of additivity of mixtures shows that, for the application to mixtures of correlations of
properties of pure components with ~ p e cific gravity and boiling point, no one
type OS average bailing point is suitable
for all properties. The following different average boiling points have been
developed for application to correlation=
of the various physical properties: (a)
cubic average boiling point-characteri;cation factor, viscosities; (6)true molal
average boiling point-pseudocritical
temperature ; (c) mean average boiling
point-molecular weight, hydrogen content, heat of combustion, pseudocritical
pressure ; (4 weight average boiling
point-true critical temperaturee,
Using these average boiling points, new
correlations of true and pseudocritical
temperature and pressure datir, are presented which a m applicable both to pure
components and wide boiling mixtures.
1409
its true molal average boiling point. However, if inolecuhr
weights of pure components are plotted against boiling points
with lines uf confitrtik A. P. I. gravity, i t is found that the
molecular weight of a pure component, having an A. 1’. 1.
gravity and boiling point the same as the A. P. I. gravity and
true molal average boiling point of a mixture, is not the same
as the molecular weight of the mixture. Therefore, the true
molal average boiling point cannot be the proper definitive
property for applying the pure-component correlation to mixtures. Similar consideration shows that the cubic average
boiling point is also unsound as a correlating variable. However, an average boiling point which is the arithmetic average
of the true molal and cubic averages does form a sound bash
for correlntion of molecular weights. This average, which
will be termed the “mean avorngc boiling point” of n blend,
corresponds to n molecular weight on the pure-component
correlation, which is equal to the actunl molecular weight of
the mixture.
In Figure 1 nre plotted curves for obtaining true niolal
avernge boiling points of petroleum fractions from Enyler
distillation curve slopes and volumetric average boiling points.
Midway between thcse curves and those estnblishing the
cubic average boiling point are plotted similar corrections for
obtaining the mean average boiling point. When working
with mixtures of pure components or narrow cuts, fur which
the distillntion curve slope is meaningless, the menn average
boiling point is directly calculated as the arithmetic average
of the true molal and cubic average boiling points of the components.
A similar analysis may bc applied to the published correlation ( 8 ) of hydrogen contcnt with boiling point and specific gravity. Again i t is found thnt the mean average boiling
point is the proper dcfinitivc variable for npplicntion of this
purc-component correlation to mixtures and that correlntions
involving the characterization factor and cubic nverage or
true molal average boiling point nre not sound for misttires
of wide boiling range.
Examination of the mean average boiling point shows i t to
approximate closely the original Watson-Nelson “molal
average,” which was developed initially for molecular weight
correlations. In general, the previously published correlations employing the Watson-h’elson average boiling point
are satisfactory for the pure components and relrttively nnrrow cuts for which they were developed, but break down when
applied to mistures of extremely wide boiling rnnge, escept
where the proper correlating vnrinble is the mean Rvernge
boiling point ns defined above.
The viscosities of blends of petroleum fractions c n n w t be
cnlculatcci by directly additive methods, ns are molecular
weights and hydrogen contents. IIowcver, the recornmended
A. S. T. hL viscosity blending procedure, togetherwith th‘e unpublished results of a fuel oil blending investigation, hnve
been used for predicting empirically the rclntions between the
viscosities of components and their blends. These rclntions
indicate thnt the cubic average boiling point is the proper
vnrinblc for iisc in the correlation of viscosity with grnvity
nncl boiling point. Therefore, the published corrcln tioiis between charncterixntion factor and viscosity are sound and
indcpcndent of width of boiling rnngc when the chnrnctcrizntioti factor is based on the correct cubic Rvernge boiling point.
Irowever, rnensurcmcnt of viscosity nlono pcrniih ~stiiiiiition
of only thc cubic nverngo boiling point, and d a h r t w & be
avnilnble on the Engler distillntion curve slope to pcrinit cnlcirhtion of either menn or true molal average boiling poinb.
Critical Temperatures
Several methods of correlation of the criticnl tcmperatures
of pure hydrocnrbons have been proposed, based on evtrapola-
1410
INDUSTRIAL AND ENGINEEII INC CCIEICI1Sl’ll Y
tioils oi iiietisurctiieiits of low-boiling materials togcthcr with
exi)erinicntsl dnta on petroleum fractions. In attempting to
apply tticsc correlations to mixtures, the problem is coniplictltcti by ttie lnck of Rdditivity of critical properties. It has
loxlg L)cen recognizcrl that tlic true critical pressure of a mixture mny be much grwter than the critical prcssure of a pure
component having tlic same definitivc properties, and ordinarily is grcnter thnn tlic criticnl prcssure of a11j‘ component
present in it. It hns been assuincd, liowcwr, that thc critical
temperature of mi.\turc wns the snnic ns that of n pure componetlt of the same gravity, i n conjunction with cithcr c h w
nctrrization fnctor, nioleculnr \wight, or molal n\‘cragc boiling point, lIo\revcrJ tlie recctit itivcstigntions of Ilocss (6)
tiitlicntc t h t the triic criticnl tciiipcrntuw c$ tlic mixture is
n l w higher tlinn tlint of n corrcsponding pure component
Iinvitig tlic w i i c grnvity niitl niolcculnr n’ciglit.
Trite c*riticd propcrtics of mixtures nrc of pnrticular w l u c
i n the cnlculntion uf 1)linsc rclatiotis ntitl viiporizntion cquiIibriritii dntn. IIoivc~.er,for tlic cq~rallyimportiint problerii
of t-npor volutiic niid ccnipwssibility cnl) .ilntirinq, it has long
bccii rccogtiized that u9c of tlic true criticril properties of a
mixtui.c for cnlcrilntiiig ~~crlr~cccl
tcmpcrnturw nnci pressures
retitlcrs tlir tticorcni of corrcspoiitli--s statcs inttpplicnble.
T’tiris the coiiiprcssi1,ility fnctor of n nlisture n t specified rc(luccd conditions is gcnctxlly gi*cntcr than thc corresponding
vnlue for n purc cotnponctit n t tlir wine reduccd conclitions.
T<ny (d), n.i R w b \ i l t of ciucful iiicnsuwiiicnts of thc 1’. Y.1’.
relntions of v:\por rriistiires, introtlucctl the ingcnious and
sitiiplifyinq corivcpt of tlic pscutlocriticnl point; he tlcfincd it
xs ttint tcrripcrntiirc niitl prcsattre which, used 114 criticnl propcrtics to cnlculntc rcrluccd prcssurc n n t l tcmpcrature, produce the snmc ictluccrl comprcssil>ility ftictor correlation :IS
tlirit of pure conipoiictits.
:\]thoiigli tlicrr is no t1icorctic:il ccrtiriiity of the csisteiice
of such n point, Iiny provcti its csistencc, within csj)cr.inicntnl
accuracy, for those inisturcs t l i n t lie studied; it will be RFsumed that cvery misturc lins a clcfinite pseudocritical point,
and it is the sniiic ns tlie criticnl point of n pure componcnt
having the same t1cfiiiitii.c pro,,crtics ns sonic type of average
definitive proprrtic~(Jf toe mixture.
Thc corrclntion prcscntcd by l i r ~ yplots pcudocritical
tcrnpeiaturc npinst rnolcculnr IrciKIit for lincs of constant
chnrnctcrizntion fnctor, K . 8cscrnI considerations indicate
that the estrnpolntccl portion i.: iticoncct, and that K and
rriolccukir tvcipht are probnbly not the proper corrclating
vnriaLIes. If, HS ICny Pbtetl, pseudocritical tempernture
is additive with inolc fraction, his constant K lines must be
straight since niolcculnr weiglit is also additive with mole
fraction, but tlic plot nctunlly s h o ~ sdeviations from linearity of triorc than 100” F . Also, since it has bccn shown in
the foregoing discussion t l i n t Zi is properly :L function of the
cubic n n d moieculnr weiKIi t of the riicnn nvcrngc boiling point,
it :ippcnrs thnt two different average boiling points hnvc bccn
iiscd i n the p~ecutlocriticnltcmpcrnturc corrclntion. Fiirtlicrmore, comLnn t-boiling-point 1 incs tiiroii g h the cxtrapoln tcd
portion of I<ny’s psciiflocriticnl tctripcrnturc plot how n decrcnsc to R rnirrirnum n n t l then r i risc i n the vnliics of pscutlocriticnl tcinpcrrtttire 119 I< incrcnses. l‘lic minimum vnlric
rind the rise :irc contrnrg to nvnilnlAo 1)rrrc coniponcnt dntn,
t h ~ x i ~this
h plioriornc~non in Siriy’s corrcintioti rnny Le only
spprircn?, d i i ~‘o ~ m w i l ~ iri:lrc.rirnvics
l(*
in lxding point-molccuInr n r i i g l i t twrrr! :it’ions.
1111 irirlirntiori tltnt I l o i l i r i K point i q I\ licttcr correlating
fnrtnr than rnolcciilrtr wciglit lim in the fnct tlirit liydrocnrh n isomcrs having tliv iciirrrt~r~iolccrilt~r
weight rrntl prncticnlly
the m m e chnrnctcrizntion fnctor mny ~ O S S C P Swidely diffcrcnt
h i l i n g poinh nnd criticnl trrriprrntiircP. ClonRcqriently
YOL. 29, ND. f?
critical tcmperaturcs of isomers clo not correlate well with
molecular wcight and K. They do, howevcr, with boiIing
point and A. 1’. I. gravity, though this gives no indication of
the proper averagc boiling point.
If a corrcct plot of pseudocritical tcmpcrature against boiling point snd gravity is developed, it is necessary that this
plot be applicable to piire components, for which the pseudocritical tcmperature is cqual to the true critical temperature.
This same corrclntion must be correct for true critical tem-
pcrntu~c-(Jf iiiistuics- of‘ tliic jirqict ty C:III be correlated 011 tlie
basis of only boiliiig poilit n i i d gravity, using tlic proper avernge boiling point. Conversely, n plot tlcveloped from true
critical tcnijiet*ntrircs rnust be correct for tlic pseudocritical
tetnperaturch, usiiifi p t q x r ni’cragc boiling poiuts. This
must follow from tlie fiict tlint n pure component is the littiitirig case of niistrire,q, for which the pseittlocriticnl snci the true
critical teitiperaturc corrclntiotis ttiust q r c c and for n.liicli nll
:ivcragc boiling pointf nrc itlcnticul.
13ccn11scof tlic scarcity of pscutlocriticnl data on \\Neboiling mivturcs, it is difficult to dctcrminc wit11 certainty
tlic proper nvernge boiling point for a correlation. IIowever,
tlie principles of valid pure-componcnt-niisture corrclntions,
~ p p l i c dto thosc iiiiuturcs studied by Kay tlint have a wide
ciiough range to bc significntit, intiicntc t h n t thc truc molal
nvernqe hoilitig point is ttie proper corrclnting vnrinblc for
pwwlocri tical tctnpcrntiircs. The rtieaii nvcrnge boiling
points or tnolccular weights of the inisturcs give, in all cnscs
wlicii usml with n piirc-component corrclntion, pseutlocriticnl
tcmIicmtiircs considcrshly higher t l i n n the csperirncntal
i ~ ~ l t t e s Tlic
.
true nioinl average boiling point used in the
same ninniirr gives ~ r i l t i e sonly sliglitly higher tlinn, or the
sntnc ns. t h cspcr~iitictitnl. liny’s corrclntion, although
ngrcc‘iiig iwll with his csj)ct*iiiiciitnl vtilrics, docs not q r c e
n d I Iritli prii~c-rciiii~,nticiitclntn i n the rcgiori lie inwstiffRtCd.
Roes:, ( 8 ) , hy :III iiigciiioi,., flow ::ictliod, nirirlc srcvtrnte
rlt.tcrmiiiatioti.9 of trite criticnl tetnpcrnturcs nntl jwcwires
of 3 p e n t nrinihcr of riiistiirrs; tiintry V-CI-C n t tcniprrnturcs
cotioitlcrnl)ly r i h o v ~tlir crncking rntiKc. Tinfortunntcly the
inspectioiis .of hi* sntnplcs probnbly Iirivc n lcsscr dcgrcc of
accuracy than Iiis criticnl tlntn, since tlic :iLtnosplicric EnRlcr
r;lL~iilritiori
w n ~t‘
tly 1)oilinji point data obtniiicri. For
tlir higher hoiling stoclis it is r v ~ t n i ntlint avcrnge boiling
pointd cnlculatd r r m thwc r\tmosphcric prcssurc distillntions
Rrc too low RS R result of decomposition. Efowevcr, by m a n s
of n corrclntiori obtnitrcci Ly comparing .I grcnt number of
vaciiuni nrd nttnosplirric prwurc distillntioas,nn npprosimatc
1.
DECEMREl\,1937
INDUS’I’RIAI, AND ENGINEERING CHEMiSTfZY
correction inny be obtained to bc atlded t o the atmospheric
pressure Engler distillations as,a function of the temperature
nnd the per cent distilled over. ROCSS’distillntion curves
were corrected in this manricr to npprosininte the results of
vacuwn dis?illation. Fortunately these corrections were
small in the majority of cases, m t l i n the worst significnnt
ciwc the corrcction nfycctcd the nvcrage boiling point by
25“ E’.
Iiocss corrcliztctl true critical tctiiperntures ns a function
of ~olurnetricaverngc I~oilingpoint nnd A . P. I. gravity. Thc
volurnetric avcrngc n.ns selcctcti bccnusc dntn 011 wideboiling
range blends of ~ . c l n t i \ ~ lnnrrow
y
cuts intlicnted thnt the
true criticnl temperature wtis morc ncarly proportional to
volume per cent than to rnolc per ceiit, nnd t h n t , using volumetric average boiling point as tlie corrclatirig fwtor, better
figreerncnt was obtnincd betwcen the espcrirnental critical
tciiipcmturcs on thcsc blends with those predicted from II
correlation of t h e tfntn on nnr! ~ n cuts.
.
Howcver, continuation of this snme type of reasoning favors the use of the \vciglit
average boiling point as opposccl to the ~oluinctrie,though
the ciiscrcpancies bctmecn the two are much less than between
the volumetric nncl the molal. &Ioreovcr, true critical tlatn
on nicthanopropnnc mixtures (6) and pentane-hcptnne mixturcs (3) show that, for this low-boiling region where Kay
found pseudocritical temperatuw t o be nearly linear with
tiiolc pcr cent, truc criticnl tcmpcrnturcs arc morc ncarly
linear with weight pci*ccnt thnn with liquid volunic per cent.
From thcsc consirlerntions it appears t h n t true critical tempcrnturc is the same function of weight avcrage boiling point
that pscudocriticnl tcnil~ernturcis of true molal nrcragc boiling point.
The weight avcrngc Lviling point is rentlily cdculntcti for
1411
blends of pure coinponcnb or narrow l~oilingciitfi. For
petroleum fractions with snrooth Engler distillation curves,
the weight avcragc boiling point may I)e obtnincd l ~ yadding
to the volumetric nvcrage I~oilingpoint :Icorrcctioil whicli i H a
function of Engler c1isti’;ition curve slope nnd \dutnctric
nverage boiling point. ‘~licsc corrcctioris :ire j)rcsented
graptiically i n Figurc 1.
The dntn of Itocss on true critical tctnpcrnturcs IVLLI’~ plotted ngainst nciglit averngc Ijoiling points together with the
pseudocritical tenij)ci nturc duta of Kay against true molal
average boiling points on the snnic plot with all available data
on pnrc hydrocarbons. Lines of const:irit A . 1’. I. gravity
were drawn through tliese points, resulting in the chart sliown
in Figure 2.1 Tnble I gives vnlucs from this clinrt in compnrison with sonic cspeririicn tally deteimincd true anti pseudocritical temperatures of niistures rind pure cornponcnts. The
average dcvistion of tlic correlation of all available data of
these three diffcrcnt types is * 4 . 5 O F . , with a niaxiinuni deviation of 31” F. Figure 2 is also in close agreement with a
chnrt publislietl by Watson ant1 Nelson (7), which was derived
hy mathematicnl estrapolation of the then-existing data on
pure compounds. The good agreement between tlie true
and pseudocritical temperature data, togetlier with ttiose on
pure compounds, gives further justification to the choice of
the weight average boiling point for true critical temperatures
nntl tlic truc rrioliil nvcragc boiling point for paeutlocriticaI
temperatures ns described above.
Critical Pressures
‘l’rue critical pressurcs of mixtures present R probleni of a
tliffcrcnt nature from any of tliose considered above in that,
for mixtures of any two componc~itsof npprcriably different
boiling points, the truc critical pressure nttnins a ninximuni
YRIUC tit sonic intcrrncdinte composition. Because of this
ninximum valuc, gyxtcr t h n n thc critical pressure of ei tlier
(wnponcnt, i t is cvidcnt thnt true critical j~ressurcscannot be
correlated as n function of only two average properties, such as
an avcrage boiling point and chnracterization factor or gravity.
However, Kay’s data on pseudocritical prcssures indicate
that no such niti~iniii~ri
oxists for them, and that the. pscudocritical prcssure is approximately proportional to mole fraction
i n binary mixtures.
If pseudocritical prcssure can bc corrrlated with two properties definitive of R hyclrocnrbon, such as boiling point and
Krnvity, it follows that this sarnc corrclntion must apply to
pure cornponcnts which are merely n limiting case of mixtures.
On this tsnsis B correlntion of pscutlocriticnl pressures m y be
h i l t up from three sources of data: criticnl prcssurcs of pure
components, rncnsurcd pseiirlocriticnl prrssurcs, and cst imntcd
critical pressures obtninetl by cstmpolnting vapor prcsmre
clatn to the critical tcmpernture. Unfortrinatcly thcsc! three
methods nre limitcd to materials of IOIV boiling point, since
true nnd pseudocriticnl prcssiircs of pure roniponents 11nw
not been mensrirctl for high-boiling substnnccs, :inct since
vapor pressure cstrnpolations i n the hirh boiling r n ~ i :ire
~r
uncertnin.
The Rocss tnensiircmctits of true criticnl ~ I ’ C S S I I I Wof rclntively hiah-l)oilirig-1,oint iinrrow wits coiistitutr tlir I w t
Ixisis for cstinintiiiK the critical ~,rcssurcsof Iiigli-l~oiliii~
compoiin(ls. IIowevcr, for thcsc dntn 21 rchtion l~cttvccrt t m c
tint1 psccuclocriticnl pressures must lie tlcvclol)cd for trniis1:rtion
of thc true critical dntn to pscudocriticnl.
1 T h e nuthors did not nt first intend to prcacnt their original. n o n i t w h a t
tloubllul cxtrnpolntions ol conatnlit A . P. t. Lincn lor thc light hydrocnrbon
KD.SCR, but the locntion of tlicw linca h u aince been rcri6ed by truc critical
tlnta of W. 13. Ksy on cthnnc-hcptnne rnixturoa (to k published in I N D F I T R T A L AND E n o i ~ r r n n ~CREUIOTRT
~a
undcr tho title ”Liquid-Vanor Phaw
I.:qrillibrium IIclntloiir in the Rthnnc-n-lleptnne Spteni”).
..1 . \ B L E
VOL.29, NO. 12
INDUSTRIAL AND ENGINEERING CHEMISTRY
1112
-- 1.
1 k ) l ~ I X oP01,uTS AND
TRUEAND
PSEUDOCRITICAL TEMPERATURES M D PnESSURES OF REPRESENTATIW MIXTURE6
AND
1454
-
PURECOMPONENTS
1472
1419
Datu11
--Kay
a
(20.02
(25%
1452
mole %
etliane
PIUS
PlUS
75%
73.98
1411))
mole %
n-hop-
tnne)
10.G
..
O
F.:
442
488
538
OU.I
7311
3.71
27 1
274
2a7
"ai
204
110
11.88
332
570
550
527
638
205
10.70
35.3
..
500
45.2
34.9
35.7
20.7
27.2
44.5
44.8
39.8
22s
245
205
254
280
45.3
535
508
....
oo2
?.4
(4.25)
a04
580
584
674
50 1
567
245
11.02
bil~
d BO
320
311
320
132
11..54
..
., ..
I .
50
bjs
(3.4)
4i7
300
370
385
152
11.23
(5.3)
i67
47 1
44 1
450
170
10.95
To, ' F.
'rrtie, ohavd.
590
041
003
088
787
507
935
809
004
783
T r u e , corrclntion
I'se ud 0 ,cor r r l i i ti o n
584
004
881
050
725
I'scudo K a y corrolirlion
000
910
030
670
740
Pacudo: obsvd.
.,
LTC/,OTC
(nhs.)
1 . 0 1 3 1 . 0 2 4 5 1.014
1 . 0 4 1.0405
tPe/yJ'c (ntw.)
1 , 1 1 3 1.213 1.122 1 . : ~ 1 . 4 3
P c J Ib./sq. in. n!,s :
rrue, obavri.
470
399
275
5.17
55G
Pseudo obsvd.
*.
Pseudo: culcd. ( f r o m
oba\'d. t l ' c )
427
329
245
400
389
Pseudo correlation
407
336
238
401
385
Pseudo: l i n y correlation
400
29i
230
384
350
408
288
850
641
'True c o r r c l a t ~ o n
454
247
554
000
448
437
T r u e : Roous c o r r e l n t i o n
..
..
..
..
..
870
8G3
810
530
..
..
..
..
I .
bio
30.0
..
..
i .a3
iiS
ii3
251
255
10.5
237
24 1
io7-ioa
11.55
410
305
388
160
11.0
704
702
700
740
..
403
445
489
11.91
830
833
770
830
..
1.36
1.0439
b01
..
403
..
405
360
A05
336
204
Relation between True and Pseudocritical
Pressures
Development of a correlation between true and pseudocritical pressures is particularly difficult with the data now
available because of the fact that no measurements of both
properties are reported for any mhture. IR Kay's work with
low-boiling mixtures, pseudocritical but not true critical
properties were measured. Roess, on the other hand, measured true but not pseudo properties.
Qualitative consideration of the variation of true critical
temperntures and prcssurcs of mixtures suggests t h a t there
4iould be a relation between the ratio of the true critical to
the pseudocritical temperature (tT,/pT,) and the ratio of
true critical to pseudocritical pressure ( P J p P , ) . Considering s binary mixture, these ratios are both equal to unity for
the two pure components and inciwse to maximum values
a t intermediate compositions.
An approximate quantitntive relation which would be espected to exist between these two ratios may be predicted
from the reduced compressibility factor diagram for pure substances. According to Kay's postulate of pseudocritical
propertieu, any compressibility factor isotherm of n vapor
mixture when plotted on R rcducccl bnsk will coincide with
h e isotherm of the same reduced vnlue on the comprwsibility factor plot for purc componcntn, if the rcduced propcrties of the mixture are based on its pseudocriticnl properties. Thus the pscudocritical tcmpcrntiire isotherm of the
mixtirre will coincide on the reduced compressibility factor
chart with a Y', = 1 line for pura componcnts, IIowever,
the true critical temperature isotherm of a mixture when plotted on a pseudoreduced basis will coincid:! with some p:ire
component compressibility-fnctor isotherm grcnter than T,=
1. A t some point on this isotherm occurs the trim critice!
188
22 1
200
278
u;4
(4.77)
1.0535
1.406
330
490
560
..
(4.83)
1.0415
..
02
332
316
487
403
1.378
..
288
274
397
440
318
245
a .
I .
* I
..
.,
..
..
4io
573
..
..
413
403
581
iio
258
258
258
258
258
114.13
12.55
31.2
..
......
P .
229
175
231
175
231
175
231
228
175
231
229
175
231
114.13 78.05 92.06
12.00 9.75 i o . i a
22o
229
..
..
..
tbi
555
580
551
bos
....
aoQ
4i5
362
$41
+64
iii
is5
477
365
ii9
iio
7Q4
740
ai1
iio
*...
t
420
. I
hi3
500
531
480
*.
*.
I ,
* I
20.0
..
....
410
. I
I.
,.
iia
mi
.*
ii7
377
387
I
452
97
97
07
113
07
129
97
81.4
72.1
13.12 1 2 . 0 4
ij3
71.4
. I
1 .
ii,
145
..
....
..
68.0
s;:7
..*.
304
ai0
....
iii
point. If the true critical pressure can be located on such a
pseudoreduced plot, the ratio of the true critical to the pseudocritical pressure will equal the pseudoreduced pressure at the
true critical point, nnd the ratio of true critical to pseudocritical temperature is equal to the pseudoreduced value of
the true critical isotherm.
The possibility presents itself that the true critical pressure on a pseudoreduced basis may be a function of the shape
of the true-critical-tcmperature compressibility factor isotherm. For pure components the critical pressure is located a t
the point of inflection of the critical temperature isotherm
nrid is also at the point of intersection of extrnpolations of the
concave downward portions of the negative and positive
slope sections of the isotherm. I n attenipting to locate
critical pressures on higher pseudoreduced temperature isotlierms, the point of inflection of the curve mas discarded ns a
criterion since i t soon reaches a nearly constant value of
pseudoreduced pressure. The values of pscudoreduced pressure, obtained a t points of intersection of estrapolrrtions of
two p u t s of each pseudoreduced temperature line, mag be
plotted against pseudoreduced temperature. If the pseudoreduced pressure locnted in this manner actuslly represents
the pseudoreduced pressure of the true critical point, then
this resultant plot is a plot of the ratio ( P J p P J vs. (tTe/
pTc). This plot is shown by points on Icigurc 3. IIowever,
further evidence is neccssnry before this assumption can bo
ncccpted.
A dircct cnlculntion of the vnlue of the above ratios nt one
point is possible from the true criticnl dnta on pcntnnc-hcptnne mixtures. By intcrpolntion of these data, the true
critical temperature of a mixture having thc same true mold
avernee boiling point RO hexnne and tho true critical pressurn
of 8 mixture having the snme molcculnr weight RB hcxnnc are
detorminnble. 40Q*iming that true m o l d n v c r a e boiling
INDUSTRIAL AND ENGINEERING CHEMISTRY
DECEMBER, 1937
1413
[m]
d log tP,
may be plotted against (tT,/pT,), and the best line drawn,
Since (tP,/pPc) = 1.0 at (tT,/pT,) = 1.0, integration of
the differential curve from 1.0 gives (tPc/pPo)as a function
of (tT,/pT,). Points with the same pseudocritical pressure
are located a t the point of intersection of lines representing
the mixtures on a plot of A. P. I. gravity against the proper
average boiling point. A plot of the relation so calculated
using true molal average boiling point as the variable for
pseudocritical temperatures and mean average for pseudocritical pressures is shown in F&pr~.&This line coincides
almost exactly with points derermined by the various other
methods. The use of either the true molal or mean boiling
point as the correlating variable for both pseudocritical constants gives relations that differ widely from one another and
from any of the above relations. Thus further evidence is
adduced as to the validity of the relation as presented and in
favor of the average boiling points selected. For values of
(tT,/pT,) greater than those investigated by Roess, extrapolation of the relation becomes necessary. In Figure 3
two extrapolations are presented as broken lines. The higher
line is calculated from extrapolated pentane vapor pressures
and the lower, which is the one recommended for use, results
FIQURE3
point and molecular weight are, respectively,
the proper correlating variables for pseudocritical temperature and pressure, the pseudocritical properties of this mixture will then be
equal to the critical properties of hexane. As
shown in Figure 3, the point calculated in this
manner falls almost exactly on the curve calculated from the compressibility factor curves, as
described above.
Another possible method evolves from consideration of the extrapolation of a vapor pressure curve beyond the critical. It might be
supposed that, for any temperature above the
critical Qf the pure component, the extrapolated
vapor pressure represents the pressure of complete liquefaction of an imaginary liquid for a
pure component, or the critical pressure of a
mixture of the same average properties with the
given temperature as its true critical temperature. If true, a plot of P, vs. T , of the extrapolated vapor pressure curve is, again, also a
plot of (tPo/pPo)vs. (tTc/pTc). Points calculated from the Brown-Coats (2) extrapolation
of the vapor pressure of n-pentane are shown in
Figure 3.
Application of the principles of valid purecomponent-mixture correlations to pure component data and the pseudocritical pressure of
Kay's wider boiling mixtures indicates that
molecular weight, or the mean average boiling
point, is the proper correlating variable for
pseudocritical pressures.
From Drecise data on mixtures of the tvDe
studied 6y Roess, it is possible to calculate" 6he true relation
between the ratios (tPc/pPc)and (tTc/pTc)if the proper
average boiling points are known. Since
log (tPc/pPc>= log tP.
- log pPe
log tP, may be plotted against (tT,/pT,) for various constant
values of pP,, the slope of these constant pP, lines may be
calculated,
FIQURE
4
from a direct extrapolation of the function as calculated from
Roess'data.
It may be that the relation between (tPJpP,) and (tT,/
pT,) is not the same for all types of mixtures. The assumption that it is the same is tantamount to any one of the following statements, and verification of any one of these statements would validate the assumption: (a) The compressibility factor must be the same at the true critical point for
VOL. 29, NO. 12
INDUSTRIAL AND ENGINEERING CHEMISTRY
1414
all mixtures with the same value of ( t T c / p T c ) ;(b) the compressibility factor isotherms for the true critical temperature,
when plotted against log P, must be exactly similar a t the
true critical point for all mixtures with the same value of
(tT,/pT,); and ( c ) the true critical temperature isotherms
on the true reduced basis must coincide for all mixtures with
the same value of (tT,/pT,). All of these statements seem
reasonable. They are similar in nature to the assumption of
the theorem of corresponding states and of the existence of
pseudocritical properties as defined by Kay.
from the same figure, using the true molal average boiling
point. The ratio of these two critical temperatures on a n
. absolute scale is then used to obtain (tPc/pPo)from Figure 3.
The pseudocritical pressure is read from Figure 4 or 5 and
multiplied by this ratio to determine the true critical pres/' sure.
The correlations and methods presented here have the advantage over previous correlations of being based on and applicable to both pseudocritical and true critical properties for
all widths of boiling range and any type of mixture. The
correlation of true critical pressures presented by Roess does
not include the limiting case of the pseudocritical and is, in
any case, limited only to those mixtures with a uniform distillation curve where slope is significant. Moreover, values
read from his correlation deviate rather widely from his
experimental values.
A tabulation of some representative data, with the corrected
distillations of Roess, is presented in Table I, along with comparisons of values given by different correlations. Deviations from all the available experimental data of the various
correlations are as follows :
--Roeas-Max.
Av.
130
126.7
tPo, lk./sq. in.
P T O , F.
pPo, Ib./sq. in.
a
...
b
..
...
...
Above butane.
--Kay--Max.
Av.
29
36
...
*vi
---AuthorsMax.
Av.
94
19.0
15
13.9
110.2
a6.5
{igi]
Below butane.
Pure components are included with the pseudos.
Vapor Pressure Relations
Pseudocritical Pressure
For high-boiling materials the Brown-Coats and all other
vapor pressure extrapolations become uncertain and cannot
be used with confidence to determine pseudocritical pressure
from pseudocritical temperature. Comparison of the pseudocritical pressures of Figure 4 with those estimated from the
Brown-Coats chart (2) indicate that the extrapolated pressures are too low for the higher boiling materials a t high temperatures and pressures on the chart. For substances with
a characterization factor of about 12.0, critical pressures from
Figure 4 agree almost exactly with those of Kay for the highboiling materials. Critical pressures for aromatic substances
predicted from Figure 4 are higher than those of Kay in the
high-boiling region; they are closer to, but still higher than,
those predicted from the Brown-Coats chart.
Recently data were obtained (1) on the vapor pressures of
n-dodecane from very low pressures to the critical point.
These data indicate that the va-aor Dressure lines
150n160nComponent CHI
C B H ~ C ~ H P ClHs C8Hs CaHs
C4Hs CdHs CdHla C,Hio
On the Brown-Coats chart are'
straight Or
A. P. I.
440
167
213
213
138
145
104
QQ
114
llo
horizontal, but should slope first upward a t very
s p . gr.
o 247 o 473 o 41 0.41 0 526 0 511 0.600 0.613 0.576 0.585
low pressures with increasing pressure and temperature, and then downward to the critical, in
I n using Figure 5 , the A. P. I. gravity of a mixture must be
agreement with predictions of the present critical pressure
calculated as the weight average of these same values for the
correlation.
components. It may be that Kay's method of using the
Literature Cited
molal average of the criticals of the pure components, when
known, is as satisfactory as any Other for substances lighter
(1) Beale and Docksey, J. Inst. Petroleum Tech., 21,860 (1935).
than pentane.
(2) Brown and Coats, Univ. of Mich., Dept. Eng. Research, Circ.
There is some indefinite indication from Roess' data that
a t constant (tT,/pT,), (tPc/pPc) may increase somewhat
with increasing boiling point. However, in the absence of
any definitely invalidating evidence, the single-line relation
was used to calculate pseudocritical from Roess' true critical
pressures, and the correlation of the former with pure component data and Kay's pseudocritical pressures is presented
in Figures 4 and 5 as a function of mean average boiling point
and A. P. I. gravity, or molecular weight and A. P. I. gravity
for the lighter components. Molecular weight was not used
for the higher boiling substances, since such a correlation
would depend on another. The three sets of data correlate
consistently within themselves and with one another.
The correlation for the lighter components was based on
the following extrapolated values of A. P. I. and specific
gravities a t 60 F. :
O
0
Series 2 (1928).
Calculation of Complete Critical Data
By means of Figures 1 to 5 the critical prop,ezties, both
true and-pseudo, of any mixture are readily calculated. The
true molal, m_e_an,and weight average boiling- points and
A**' T*-Favity are first determined. The true
perature is then read from Figure 2, using the weight average
boiling
-- - point. The pseudocritical temperature is then read
_.*-
I-.
(3) Cummings, Stones, and Volante, IND.ENG.CREM.,
25,728 (1933).
(4) Kay, Ibid., 28, 1014 (1936).
E: ;
~
~
~
~
~
~
~
~
~
~
~
n
~ 214
~ (1934).
~ ~ ~
(7) Watson and Nelson, Ibid.,25, 880 (1933).
(8) Watson, Nelson, and Murphy, Ibid., 27, 1460 (1935).
R ~ c p , ~ vAugust
~n
10, 1937. Presented before the Division of Petroleum
Chemistry a t the 94th Meeting of the American Chemical Society, Rochester.
N. Y . , September 6 t o io, 1937.
~
~