ADDITIVE PHTSIC.4L PROPERTIES IN HYDROCARBON MIXTURES
859
(5) BARTELL, F.E., AND OSTERAOF, H. J.: J. Phys. Chem. 97,543 (1933). F. E.,AND WAITNEY, C. E.: J. Phya. Chem. 86, 3115 (1932). (6) BARTELL, (7) BUTLER, J. A. V., AND WIGHTMAN,A . : J. Chem. SOC. 1999, 2089. (8) DAVIS,J. K.:Thesis (unpublished), University of Michigan, 1940. E.A., AND ADAM,S . K.: Proc. Roy. SOC.(London) Al99, 218 (1933). (9) GUGGENHEIM, (10) H A ~ I N W. S , D., AND WAMPLER, R. W.: J. Am. Chem. SOC. 69, 850 (1931). (11) LEWIS,G. N.,AND RANDALL, M.:Thermodynamics. McGraw-Hill Book Co., New York (1923). (12) MACK,G.L.,AND BARTELL, F. E.: J. Am. Chem. SOC. 54,936 (1932). J. H., AND STAMM, A. J.: J. Am. Chem. SOC.46,1071, 2880 (1921). (13) MATHEWS,
ADDITIVE PHYSICAL PROPERTIES IN HYDROCARBOS MIXTURES R. M. DEASESLI’
AND
L. T. CARLETOS
Shell Decelopment Company, Emeryuille, California Received April 10, 1942
The idea of the additivity of physical properties is by now a familiar one. Various investigators have shoivn how in a pure paraffin, for example, a physical property of the whole molecule results from combining constant characteristic values for the individual atoms of carbon and hydrogen. For more complicated molecules, the structural units may be taken as rings or as groups, such as CH, or C=O, along with simple atoms. A convenient summary of the mass of work on such properties as parachor, fluidity, molecular refraction, etc., is that of Brode and Leermakers in Gilman’s Organic Chemistry (4). In reviewing various physical properties of a considerable number of hydrocarbons, the authors observed a behavior pattern of properties, additive as to elements of constitution, which proved capable of being generalized in the form of a single equation. This formulation of additivity has proved useful in the development of a new method of type analysis of hydrocarbon oils (I), using the property of molecular refraction. It is presented here in generalized form, in view of its broader applicability. The conditions which must be fulfilled by such an additive property of a hydrocarbon misture are as follows: ( 1 ) If f is an intensive physical property of an individual hydrocarbon i, of formula C‘,,H?,i+Tiand illi is its molecular weight, then for .Iffi to be addit i w (in this .sense) the following relation must hold good:
+
+
+
M i f i = ani bxi cRi dzi (1) in which Ri is the number of rings (presumed to be either five- or six-membered, these being the predominant types in petroleum); z i is the number of double bonds, either olefinic or aromatic; and a , b, c, and d are characteristic constants, values of “atomic” f for one CHg group, hydrogen atom, ring, and double bond, respectively ,
860
R. M. DEANESLY AND L. T. CARLETON
In the absence of triple bonds and unusual ring structures, n, x, R, and z are the only elements of constitution considered to deet the value of f for the molecule. Admittedly, the equation takes no account of kind and degree of branching, the method of joining the rings of polycyclic compounds, and other details of structure. In actual cases these have effects which are clearly secondary, and are for the time disregarded. (2) For a mixture of individual hydrocarbons 1, 2, . . , the property f must be a weight average of the values for the respective components, fl,fn, * , as defined by the equation f = wlfi
+ wlfi + .
' *
=
z wifi
(2)
in which w,, wp, + . are the respective weight fractions. If these equations are fulfilled exactly,, then for the mixture C,,HS,+2, having an average hydrogen content of y per cent and unsaturation (olefinic and aromatic, as weight of hydrogen per unit weight of sample) of h, the following equation holds exactly:
in which A , B, C, and D are constant for any given property. In practice, the conditions hold approximately for certain properties; hence the equation likewise holds only approximately. Nevertheless, the degree of accuracy is sufficient to provide a useful evaluation of h, and herein lies a definite application of the equation. Often unsaturation can be measured directly, by hydrogenation, only at great trouble; i t is relatively easy to calculate it from the readily measured quantities y (from combustion analysis), M (from cryoscopic gr ebullioscopic measurements), and f, the physical property under consideration. An accurate evaluation of h is the essential first step in any type analysis of oils (1). DERIVATION O F THE GENERAL EQUATION
Equation 3 is a consequence of conditions 1 and 2, and can be rigorously derived from them, for the variables in condition 1 are expressible by the following, purely constitutional equations: (2m
+ xi)1.008
= 14.026%
+ 1.008xt
whence
14.026yi - 2 xi = 1.008 Mi 12.010
ADDITIVE
PHYSICAL PROPERTIES IN HYDROCARBON MIXTURES
861
whence
14.026
hs M - m y - 2
=I--
2.016
24.020
M
.
Substituting these values in equation 1, Mifi = a ( 1 ) M i 12.010
+b
-2
1.008 12.010
M 14.026 1.008 ! I- 2 M 24.020
hi +C-C-M-C 2.016
hi + d 2.016 -- M ,
By introducing new constant coefficients, A , B , C,candD, this may be rearranged to Mifi = AMi
+ ByiMi + C + DhiMi
whence fi =A
+ Byi + C-M1i + Dhi
(4)
If equation 4 is multiplied by wi,and a summation is made over all the components, condition 2 gives: f =
z Wifi
while, from the definition of average properties, y
= z wiyi
and '
equation 3 results directly.
h = 2 wihi
862
R. M. DEANESLY AND L. T. CARLETON APPLICATION TO SPECIFIC PROPERTIES
Application of the equation to density may be considered first. It has been commonly observed that hydrocarbons mix without appreciable volume change; this behavior means that condition 2 is satisfied for reciprocal density, or specific volume. To test the applicability of condition 1 to specific volume, the authors investigated, among many individual hydrocarbons, the molecular volume changes accompanying the addition of CH2 groups, rings, and double bonds. The change of each type was found to be approximately constant. (This finding agreed with figures of Egloff and Kuder (3) and of Kurtz and Lipkin (6),who found nearly constant increments for different elements of structure.) A systematic trial of the equation on all available data appeared to be justified. For some time a critical review of the physical properties of hydrocarbons has been conducted in these laboratories. The procedure has been to transcribe all data from an intensive search of the chemical literature and, from careful consideration of these, to select best values of the important constants of each compound. Our own material has been supplemented by the similar compilations of Egloff and coworkers (2). This combined total afforded a good basis for correlation of any of the various properties considered. The ordering of these data for trial in the equation depended on our understanding of constitution. Since, as far as the equation is concerned, constitution is fully specified by n, R, and z, it seemed convenient to group the compounds by “series”,-collections of groups of isomers characterized by constant values of R and z. Only n was variable within each series. Progress of the grouping soon justified this arrangement ; specific volumes of each collection of isomers within a series were found to lie close together by comparison with the spread of values between different series. Accordingly, specific volume was plotted against n for the compounds of each series, and a smooth locus of mean values drawn by inspection. This locus gave a “representative” specific volume for each n. Series treated in this manner were the following: (1) acyclic, CnH2,,+2 (paraffins) ; (9) monocyclic, C,HZ, (alkylcyclopentanes and alkylcyclohexanes) ; (3) monocyclic, CnHzn--l(alkylcyclopentenes and alkylcyclohexenes); (4) monocyclic, C,HZ,-~ (alkylbenzenes) ; (6) bicyclic, C,HZ,,-~ (all bicyclic naphthenes formed from five- and six-membered rings) ; and (6) bicyclic, C , , H Z ~ -(alkyl~~ naphthalenes). From the representative values so obtained, constants in equation 3 were evaluated by a process of successive approximations, to give v =
1 a = -0.321 din
1 + 0.1033~+ 12.00 + 0.0662h M
(5)
Here y is expressed as weight per cent, and h in grams of hydrogen per 100 g. of sample. Figure 1 shows the agreement of the equation with these values. As may be seen, it was good, and equation 5 was accordingly adopted for the limited range of molecular weights considered. The equation is based on an average increment of molecular volume of 16.33 per CHZgroup, a decrement of
ADDITIVE PHYSICAL PROPERTIES I N HYDROC.4RBON MIXTURES
863
6.84 for the formation of each olefinic or aromatic double bond (assuming the Kekul6 structure) with the loss of two hydrogen atoms, and a decrement of 20.18 for the formation of each ring with the loss of two hydrogen atoms. For purposes of practical use, the graphical representation of figure 2, in which y is plotted against v , seems most convenient. The loci of dserent homologous series (not drawn in) are a pencil of straight lines the center of which is the point y = 14.38, u = l.lG4. Progression along any one of these corresponds to adding CH2 groups to a hydrocarbon until it becomes almost completely paraffinic at infinite molecular weight. Lines of addition of hydrogen (broken) are straight and unique, as are the lines of constant M for saturates (full). The locus of 100 per cent paraffins is a straight line, and when a uniform progression 0 -
ACTUAL R E P R E Y I U I A I I V E V l L V r LOCVS Of T H T O l € T l W L VILUE?I
16
0.9
CnHzn-it
1
I
0 000
I
I
0004 0 008 ROCIPIOCOI Molnular Welphl
I
.
0012
FIG. 1. Agreement between actual and theoretical values of specific volume at 20°C for several hydrocarbon series.
of ring types is assumed, there are similar straight-line loci of 100 per cent naphthenes and 100 per cent aromatics; those shown on the figure correspond to the progressions cyclohexane, decalin, perhydrophenanthrene, etc., and benzene, naphthalene, phenanthrene, etc. Many other useful geometrical properties of such a plot may be demonstrated. This figure, like the previous one, typifies the graphical representation of equation 3 for all f's. Application of the equation to the so-called characterization factor of Watson and Nelson (8) Eas considered next. From examination of a large body of literature data, these authors concluded that, ". . . the ratio of the cube root of the absolute boiling point to the specific gravity is substantially constant for the paraffin hydrocarbons boiling between 100" and 700°F., if the averages of all the reported isomers are considered. When the boiling point is expressed in
864
B. hf, DEANESLY AND L. T . CARLETON
degrees Rankine and the specific gravity at 60°/600F., this ratio varies between 12.5 and 12.8 for the paraffins. For benzene the ratio is 9.8.” Similarly, when
07 I
1.LIb
a9
08
liC
I
120
110
IO I
Ibo
II
ob0
le
13 I
om
15
14 I
975
I
om
II 0.6s
Density. d$q‘
Fro. 2. Specific volc-e-constitution relationship for hydrocarbons
applied to narrow-boiling petroleum fractions, this ratio was found roughly to characterize constitution in terms of “paraffinicity” and “aromaticity”. Th,is characterization factor, denoted by the symbol K , is in extensive use (9).
ADDITIVE
PHYSICAL
PROPERTIES IN HYDROCARBON MIXTURES
865
Practical petroleum fractions usually contain large numbers of hydrocarbons covering a considerable boiling range. To generalize the application of the characterization factor, a suitable mean boiling point must be defined. This mean must fulfill, as nearly as possible, the condition of constancy for a wide variation of boiling range when specific gravity and average molecular constitution are constant. Smith and Watson (7) showed the “cubic average boiling point” to be a satisfactory mean. They defined this, for a mixture of 7~ component hydrocarbons, by the expression Cubic average boiling point =
Vi (absolute Fahrenheit boiling point):”
1 a
in which Vs is the volume fraction of the ithcomponent. (In practice, cubic average boiling point dif€ers by a small amount from volume average boiling point, as obtained from an A.S.T.M.’ distillation curve. Conversion from the latter may be made graphically.) Besides its suitability in fulfilling the condition stated above, use of this function has the additional advantage that the characterization fLctor calculated from it exactly satisfies condition 2, that is, K = Zw,Ki. Moreover, the present authors found evidence of conformity with condition 1 ; the experimental value of K for cyclohexene, for example, approximated that predicted from values for benzene and cyclohexane. A trial of the general equation 3 was therefore made. Following the method applied to specific volume, representative values of K were selected for a number of series of pure hydrocarbons. The constants A , B, C, and D were evaluated from these figures by a method of successive approximations, the method being applied repeatedly until the constants remained unchanged. Their final values yielded the equation
K = 1.12
+ 0 . 7 8 6 ~- 118 + 0.500h
in which y is expressed as weight per cent, and h in grams of hydrogen absorbable per 100 g. of sample. The fit of this final equation to the representative values is shown on the plot of K against reciprocal molecular weight in figure 3. From this plot, the equation appears to be fairly accurate in the region C, to CIS, and its adoption appears to be entirely justified. Figure 4 shows a plot of y against K corresponding to the plot of y against u of figure 2 and exhibiting the same general properties as that figure. A special case to which the equation applies is that of the Lorenz-Lorentz specific refraction. This function is defined by
1
A. S. T. M. = American Society for Testing Materials.
8GG
I?. M. DE.UESLY AKD L. 1'. CAHLETON
in which t L is refractive index (conventionally a t 20°C., for the sodium D line), and d is density, d",ob. In an earlier paper (l), the present authors showed the specific yefraction of petroleum oils to be related to constitution by the equation T
=
0.2084
kh + 0.008421y f __ 201.6
(7)
in which, as usual, y is expressed in weight per cent, and h in grams per 100 g. The absence of a term in l / M will be noted; this corresponds to a value of zero for the coefficient c in equation 1 . I t will further be observed that a specific niunrrical value has not been given to the coefficient of h. The reason is that k , 0 -
ACTUAL REPRESENTATIVE VALUE LOCUS OF THEORETICAL VILUES
0
12.5-
12.0-
E
d
5
110-
2
10.5-
10.0-
9s-
0000
0004 0 ow R ~ C I P I O C MoIwubr ~I hbht
1
0012
FIG.3. Agreement between actual and theoretical values of the characterization factor for several hydrocarbon series.
the value of J l r for one double bond, actually shows regular variation with structure, having a range of values of, roughly, from 1.4 to 2.8. Figure 5 illustrates a plot of y against 1' for an assigned constant value for k of 1.7, and is characteristic of plots for all constant values. Again, the broken lines represent addition of hydrogen, while the light lines, A M , B M , C M , and DM are loci of different homologous series,-the monoolefins, alkylbenzenes, alkylnaphthalenes, and alkylanthracenes (or phenanthrenes), respectively. It will be noted that all saturates fall on a single locus rather than on two separate loci, as in the other figures. However, assignment to k of a single, constant value limits the applicability of equation 3 to a narrow range of mixtures. In endeavoring to overcome this
'
ADDITIVE
PHYSICAL PROPERTIES IN HYDROCARBON MIXTLJRES
8G7
difficulty, the authors succeeded in relating k to an additional physical property, and were thus able to eliminate it hetween the two equations, and to express h
.Dale
868
R. M. DEANESLY AND L. T. CARLETON
FIG.5. Specific refraction-oonstitution relationship for hydrocarbons: k = 1.70
and the new equation was
It is apparent that k may be eliminated between equations 7 and 8, giving an expression for h as a function of R, y, and S. This treatment of specific refraction suggests that the form of equation 3 for
869
-4DDITIVE PHYSICAL PROPERTIES I N HYDROCARBON MIXTURES
other physical properties might be made more accurate by expressing the coefficients as functions of additional physical properties rather than m constants. This possibility has not yet been investigated. UNSATURATION O F ACTUAL OILS BY THE VARIOUS EQUATIONS
To test the consistency of direct experimental measurements of hydrogen absorbable with values calculated from the changes of the various physical constants of oils in hydrogenation, the data of Kreulen ( 5 ) on a number of gas oils were considered. To our knowledge these are the only data in the literature complete enough to afford simultaneous evaluations of h from changes of y and M on hydrogenation, and from the densities, characterization factors, specific refractions, and specific dispersions of the original samples. Table 1 lists values of h obtained by the different methods. While the superiority of the determinations from specific refraction and specific dispersion (by the use of a TABLE 1 Calculation of unsaturation from various physical propertiea of a series of gas oils A OIL
Y
From I, Y, Y
From K y M
4.7 1.4 1.6 1.4 1.5 1.4 1.6 1.4 1.5 1.1 1.1
4.6 1.3
(equation 5 )
1 2 3 4 6 5 11 .7 8 9 10
180 215 244 192 227 216 251 217 236 196 233
(equation’6)
1.4 1.6 1.6 1.55 1.25 1.4
From v , S, y , M
4.6 1.05 1.3 1.2 1.3 1.3 1.6 1.2 1.4 1 .o
1.0
3y hydrogenation
4.65 1.1 1.5 1.3 1.2 1.4 1.6 1.1 1.6 1.1 1.2
nomograph for simultaneously solving equations 7 and 8) is not so evident as it would be for heavier oils, it is noticeable that these values show the smallest extremes of variation from the hydrogenation values. SUMMARY
The additivity of a “molecular” physical property of a hydrocarbon mixture is defined; from the definition a general equation relating the simple property to hydrogen content, molecular weight, and unsaturation is derived. This equation is of value in expressing the unsaturation of a petroleum oil in terms of readily measurable quantities. The equation is applied to specific volume, to the Watson and Nelson characterization factor, and to specific refraction. REFEREXCES (1) DEANESLY, R. M.,AND CARLETON, L. T . : Ind. Eng. Chem., Anal. Ed. 14, 220 (1942). (2) EGLOFF,G.: Physical Constants of Hydrocarbons. Reinhold Publishing Corporation, New York: Vol. I (1939); Vol. I1 (1940).
8iO
PAUL J. FLORY
(3) EGLOFF, G., AND KUDER,R. C.: J. Phys. Chem. 46, 281 (1942). (4) GILMAN,H. (editor): Organic Chemzstry, A n Advanced Treatise, Chap. 20. John Wiley and Sons, Inc., S e w York (1938). (5) KREULEN, D. J. W.: J. Inst. Petroleum Tech. 23, 253 (1937). (6) KURTZ,S. S., JR., AND LIPKIN, M. R.: Paper read before the Division of Petroleum Chemistry at the 98th Meeting of the American Chemical Society, \vhich was held at Boston, Massachusetts, September, 1939. (7) SMITH, R. L., . ~ N D WATSON,K. M.: Ind. Eng. Chem. 29, IC08 (1937). (8) WATSON,K.M., AND XELSON, E. F . : Ind. Eng. Chem. 26, 880 (1933). (9) WATSON,II. >I., XELSON, E. F., AND MURPHY,G. B.: Ind. Eng. Chern. 27, 1460 (1935).
VISCOSITIES OF POLYESTER SOLUTIOSS APPLICATION OF
THE
MELTS:ISCOSITY-;\IOLECUL.\R WEIGHTRELATIOSSHIP TO SOLUTIOKS
PAUL J. FLORY Esso Laboratories, Chemical Division, Standard Oil Decelopment Company, Elizabeth, New Jersey Received June 18, 1948
Recently ( 5 ) the author demonstrated the existence of the simple relationship log q = A
+ Czr
(1)
or
between the viscosity q of a molten linear polyester and its weight average chain length 2,“. A , a, and C are constants and the weight average chain length is defined by
z,= z w,z,
(2)
where w r is the weight fraction of species x having a chain length (number of chain atoms) 2,. No significant deviations from equation 1 were observed over a fortyfold range in 2, and a 3000-fold range in viscosity. The constant C is nearly independent of temperature, and A varies approximately as 1 / T , at least for a range of some 50’C. above the melting points of the polyesters. Hence, to include the dependence of 9 on absolute temperature as well as on 2, log q = D
+ BIT + Cz;”
or
where D ,B , and dare new constants.
(3)
