Vol. 56
JOHN R. PLATT
328
PREDICTION OF ISOMERIC DIFFERENCES IN PARAFFIN PROPERTIES.’ BY JOHNR. PLATT Physics Department, University of Chicago, Chicago 37, Illinois Received December 8, 1061
Parameters pro osed for predicting paraffin properties are examined critically. The most important ones are: f, the sum of first C-C neighiors of every C-C bond in the molecule; p , the number of “steric pairs” or groups on carbon atoms three bonds apart, which is half the sum of second C-C neighbors; and w, the Wiener number, the sum Of the number of bondp between all pairs of carbon atoms in the skeleton. The variablesf and p determhie internal properties such as Ho, the heat of formation of the gas a t O’K.,and R, the molar refraction; p and w determine external properties such as L the standard heat of vaporization, B, the boiling point, and V , the molar volume. The heat of formation of the gas at 25d, Hla, Fequires all three, and the same promises to be true of most other properties at room temperature. Least squares equations are given for Ho and R and for the isomeric variance in H26, L, B and V . The mean deviations are close to the experimental errors, except for the boiling points, where additional or better external parameters are needed. The interaction terms of steric pairs-the p-coefficients-in H Oand R can bg obtained for different angles of twist about the intermediate single bond, from data on alkyl cyclanes. They follow a (1 cos e) law within experimental error, showing zero interaction at the 180” angle. The steric barrier height is about 1500 cal./mole at the 0 ” angle. First-neighbor alkyl substitution on a bond lowers its contribution to Ho, produces a red shift in its first ultraviolet absorption region, and increases R. Steric hindrance decreases R; a possible explanation is that steric strain causes differential lengthening of a C-C bond in high excited states, producing a shift of the high-frequency absorption to shorter wave lengths. The hypothesis is proposed that the Wiener number, w, is an index of the mean external contact area ?f the molecule, except for a correction for steric hindrance. This will account for the fact that w unstabilises an isomerJn !he gas phase (decreases the self-contact area and increases H26)b almost the same amount by which it stabilizes the liquid (mcreases the external contact area and L). The molar volume 6rmula contains a large negative steric term which seems to express the compression of steric pairs.
+
The isomeric paraffins are remarkably similar in stability, bond energies and force fields. Differences in their properties are largely of geometric origin. As ‘a result they are well suited for the isolation and study of geometric factors which must be of importance for all molecules. These properties were selected for study: H z ~the , heat of formation of the gas at, 25” R, the molecular refraction for the D lines at 20’ for the airsaturated hydrocarbon in the liquid state at one atmosphere V , the molecular volume under the same conditions B , the boiling point at, the same pressure
Later two more were added: Ho, the heat of formation of the gas at 0°K; L , the standard hcat of vaporization at 25
The most elaborate, systematic and successful formulations of these properties in the literature seem to be those of Taylor, Pignocco and Rossini,2 WienerlS and Platt.4 The Appendix defines the parameters in these formulations and gives the constants and mean deviations of some of the leastsquares solutions through the octanes (Tables I and 11). Wiener Formulas The great success of the Wiener 2-constant equations for all the properties is immediately evident. They are almost as accurate as 5- and 6-constant equations involving other parameters. Let us consider the Wiener variables. The importance and physical significance of one of his parameters, p , is easy to see. This is the number of pairs of carbon atoms three bonds apart in t,he hydrocarbon. It accounts alone for 80% of the isomeric variance in R and (1) Previously given in port s t The Ohio State University Sympob ium on Molecular Structure and Speotroscopy, Columbus, Ohio, Jane 11, 1947. (2) W. J. Taylor, J. M. Pignocco and F. D. Rossini, J . Reaearch Natl. Bur. Standard, 84,413 (1945). (3) H. Wiener, J . Am. Chem. SOC.,69, 17. 2636 (1947); J . C b m . Phys., 18, 766 (1947); THISJOURNAL, 52, 425. 1082 (1948). (4) J. R. Platt, J . Chem. Phye.. 15, 419 (1947).
for a large part of the variance in the other properties. It appeared in the TPR analysis as the factor kl used to relate their coefficients, and was responsible for the reversals of sign noted by Platt in his a2 coefficients. It is evidently an index describing the steric interference of the groups on carbon atoms three bonds apart during rotation about the intermediate single bond. Hereafter we shall call such pairs of groups “steric pairs.” Wiener supposed that p was due to an alternation in bond or atom interactions along a chain, comparable to the alternations found in conjugated systems. But the coefficients of p in the formulas are unreasonably large for such an interpretation, while the simple first-neighbor, secondneighbor, etc., interaction formula4 shows no further altefnation for pairs of carbons four bonds, five bonds, and SIX bonds apart. The analysis of the data on the cyclanes given below seems to show conclusively that p is an index of steric hindrance. The meaning of Wiener’s other parameter, w, the “path number,” which we might properly call the “Wiener number,” is harder to see. It is the sum of the numbers of bonds between all pairs of carbons in the molecular skeleton. It increases for large molecules as Na, where N is the number of carbon atoms in the paraffin. Evidently .p/;; is sort of mean molecular diameter; and Aw/N2, the term In the Wiener formulas, is approximately proportional to the increment in and represents the increment in mean diameter for a given isomeric change. The N a acts as a normalizing factor so as to mako the A w term for a given type of change almost independent of N . The success of the Wiener formulas for describing internal properties of molecules was at first surprising, because w seems physically unreasonable as a factor in such properties. Thus in going from a normal chain to a 3-branched isomer with branch lengths 11, Z2, Is, the change in w is the product - l & ~ . For a given N , this product continues to increase with increased lengths of the shorter branches, no matter how long they are, until all the branch lengths are Dependence on such a parameter in large molecules would imply very long-range forces indeed.
‘q-,
New Formulas I n an attempt t o duplicate the success of the Wiener formulas for internal properties without encountering this objection, other parameters were examined which it was hoped would be measures of the molecular compactness, similar to w, but which (6) I am indebted to Dr. J. E. Mayer for this observatiou.
-
March, 1952
PREDICTION OF ISOMERIC DIFFERENCES IN PAR~FFIN PROPERTIES
would represent interactions extending for no more than three or four bonds along the skeleton. By its nature, such a search cannot be straightforward. With the limited types of variations possible in paraffin isomers, every structural parameter is correlated with every other one.6 A spurious parameter may therefore substitute for a true one, because of this correlation, with no great loss in accuracy. It may even give a slight gain in accuracy if somehow, possibly by a physical connection, possibly by unfavorable random fluctuations, it is highly correlated with experimental errors in a limited set of data. The problem of finding the correct variables resembles the problem of finding the fundamental law of force in a mechanical system whose motions are limited and related by fixed constraints. Under these circumstances it seems best to choose our formulas by balancing several requirements. Accuracy.-In predicting the data, the formula should give minimum deviations, or deviations close to the experimental error. A variable is relatively inessential if it does not reduce the deviation when added to other variables. Simplicity.-The formula should have a minimum number of variables. Reasonableness.-Their effects should be physically understandable and the magnitudes physically reasonable. There should be no absurdities in the limit of large or small molecules. Stability.-If the variables are really fundamental, their effects should be independent of N , and their empirical coefficients in least-squares formulas should be almost unaffected by the presence or absence of other variables. This last requirement is peculiar to the examination of correlated parameters. The obvious variables for representing terminated interactions were tried: the sums of numbers of first-neighbors, second-neighbors, etc.'; the sums of squares of numbers of neighbors, etc.; the products of neighbors on one end of a bond with those on the other, etc.; and the number of gauche (or trans) pairs in rotational isomerism about every C-C bond (called h in the Appendix; both the most stable and the least stable rotational isomers were tried). The coefficients and mean deviations of some of the better least-squares formulas are shown in the Appendix. Upon balancing the criteria stated above, we find that none of these formulas is better than the Wiener formula, except in the case of one property. ,
'
(6) That is, skeletal parameters. The skeleton of a hydrocarbon completely defioes the structure. No examination of chain isomers can isolate variables which are theoretically more primitive. such as the contributions of H-atoms. of C-atoms, of C-H bonds, of C-C bonds, or their interactions, etc. These contributions are lumped indistinguishably into the coefficients of skeletal parameters. What linear combination of them is involved depends on one's taste in choosing his aeroorder theoretical approximation. The correlation of these Variables is complete. The correlation of different skeletal parameters is not complete and so a choice among them or an estimate of their independent contributions is still possible. (7) The Wiener number is in fact a particular neighbor-sum ( J . Cham. Phys., 15, 766 (1947)) but with the property that the coefficients of successive orders of neighbors, instead of approaching zero, increase linearly with increasing distance. This is another way of saying that it wuuld correspond to long-range forces if it were an internal property.
329
The comparison nevertheless led to significant conclusions which could hardly have been obtained otherwise. The property for which the Wiener formula was not best is Ho. The best variables in this case are the first-neighbor sum, j ; and the second-neighbor sum, which is twice the number of steric pairs, p. Call a formula with these variables a two-neighbor sum. This property, 'Ho, which is fitted without using the Wiener number, w, is as purely internal as any property could be. This suggested that w is not in fact an internal parameter but an gxternal one. This idea was supported by the fact that w is the principal and essential variable for describing the boiling point-a most external property. Whenever w is needed to describe presumably internal properties, it must mean that external variables are entering into the measurement of these properties to such an extent that w, which after all is correlated with f (and with the other neighbor sums and products), becomes a good substitute for f i n the equations. Thus if w is involved in H26 but not in Ho, "external" effects must contribute to the gaseous heat content in warming up from OOK. (Any one of the f, p , w coefficients in Hzs can be adjusted initially over a wide range without causing much loss in accuracy in a least-squares solution for the other two. By taking the f-coefficient the same as for Ho, which is arbitrary but plausible, the isomeric differences in heat content are described by the alteration in the p-coefficient from 0°K. to 25OC., and by the w-coefficient a t 25'). If w enters into R, perhaps there are external influences on R which are not completely represented by the simple density term in the classical formula. Actually in this case, a two-neighbor sum gives about the same accuracy as the Wiener formula and both seem to be pressing the limit of experimental error. Since we expect R to be internal, the f, p descriptionis adopted here. For the largely external properties, V , L and B , the Wiener formula is best. The adopted least-squares formulas for the properties are : Ho
= -12.450
- 4.074(N -
1)
-
+
0.490f 0.6881, zt 0.33 kcal./mole 0.0112f - 0 . 1 3 0 4 ~f R = 6.460 4.750(N - 1) 0.016 ml./mole AH25 = -0.500 Af 0.796 Ap 5.33 Aw/N2 f 0.38 kcal./mole AL = 0.210 Ap 5.086 Aw/N2 f 0.052 kcdjmole AB = 5.640 Ap 98.36 Aw/Nz f 0.55 AV = -2.256 Ap - 10.95 Aw/N* f 0.21 ml./mole
+
+
+ +
+ +
The formulas for Ho and R give the total value of the property, the first two coefficients being determined from the n-Cz to n-Cs series for H o and from n-C6 to n-Cs for R. The other two terms in these formulas, and the terms in the other formulas were determined from least-squares solutions for the isomeric variance in the c5 through Cs groups, and the mean deviations given are for these solutions. Formulas for the total value of the other properties must await more precise theoretical definition of w, since its contribution is known only for the isomeric variance. Its contribution to the CH2increment in
330
JOHN R. PLATT
the line of homologs is different for several different mays of computing w (which give the same Aw), and its physical meaning needs to be better understood before it can be intelligently used in formulas, The deviation of H26 for methane from the expected value given by linear extrapolation from the line of the higher n-paraffins was successfully predicted earlier by the six-constaiqt formula. The present simpler foymula for Ho gives the ethane deviation accurately, but misses the methane value by about 3 kcal./mole. The quadratic neighbor terms which'were dropped from the higher paraffins with no great loss in accuracy may be essential for describing the methane deviation. All the present formulas, except the one for boiling point, give mean deviations approaching the experimental error. It does not seem that the coefficients of additional parameters could be determined with any confidence. The boiling points are known to perhaps fifty times the accuracy of the formula, and a search for more elaborate external parameters to add to the Wiener formula for this property might be rewarding.
Cyclanes: Ho and R Anomalous molecules demand modifications in the equations. They provide an acid test for any physical iiiterpretation of the parameters. The methane and ethane deviations from the linear n-paraffin series are stringent tests of paraffin formulas. The anomalous cyclane properties are similarly hard to fit. Happily, it appears that the formulas given above for HOand
Vol. 56
R can be adapted easily and plausibly to the CB-cyclohexane and C7-cyclopentane isomeric deviations. The data may be analyzed in terms of a first-neighbor interaction, like that in the chain paraffins, plus a secondneighbor interaction which is a function of the relative angle of rotation of a steric pair about the middle C-C bond. When the middle C-C bond is a ring bond, the angles &re held close to certain equilibrium values, 60 and 180" for puckered cyclohexane with normal tetrahedral vaIence angles; and close to 0 and 120' on the average for approximately planar cyclopentane. If the intermediate C-C bond is adjacent to the ring, the angles are not fixed and we may have n "free" second-neighbor interaction similar to that in the chain paraffins. When the intermediate C-C bond is in the ring, examination of the data shows that the steric effect is the same whether the two interacting groups are both outside the ring, or one outside and one in the ring; this is true for both HOand R. By examining only isomeric variance for a fixed ring system, we avoid having to evaluate the steric interaction between different groups within the ring. We thus have three second-neighbor coefficients for the cyclohexanes: u2(60),az(180),and az(free); plus U I . Similarly for the cyclopentanes: u2(0),~ ~ ( 1 2 and 0 ) a2(free); plus ai. The four constants for each group can be determined independently, as seen in Table I11 in the Appendix. However, by modifying them slightly as shown in the "Adopted" set, they can be made more consistent with the alkane values with no great loss in accuracy for the cyclane predictions. The adopted formulas are: AH0 = -0.500A,f
1 .OOAPfree
AR = 0.015Af 0.130A~rres
+
I
+
O.OoAp180 =I=0.07 kcal./moIe for cyC6 fl.60Apo f 0.40Aptzo f 0.37 kral./mole for cyCj O.00Ap1~o f: 0.016
f 1.00 Apso
-0.24OApo - O.075Pi20 f 0.035 ml./mole for cyCj
In the R for the cyclopentanes the accuracy is appreciably worse than in the corresponding two-constant sum for the alkanes. This is mostly. because the methyl groups seem to have an abnormal interaction in 1,l-dimethylcyclopentane, possibly because of deviations of the ring bond angles from the tetrahedral values; and in cis-l,3-dimethylcyclopentane, possibly because they come in contact more than is usual for third-neighbor C-C bonds. Both effects can reasonably be associated with the well-known purkering of the Cgring. The second-neighbor interactions in Ha and R are.plotted as functions of angle in Fig. 1 . I n both cases they increase smoothly from m a r zero a t 180' to a maximum effect at 0'. The zero at 180" and the absence of dip near 90" show that no appreciable part of the interaction can be due to hyperconjugation, which would give maxima a t 0 and 180" and minima at go", approximating a cos0 function. The observed curve can be fitted within its accuracy by a (1 COS 0) function without higher order terms. The Free Interactions.-The free second-neighbor interaction in the alkanes is close to the mean of the fixed-angle interactions. For properties observed a t room temperature, such as R, this can be understood if an angle near 90" is a turning-point of the motion under hindered rotation,. SO that most of the steric pairs are found a maximum fraction of the time near this position. The free second-neighbor interaction in the ethyl cyclanes corresponds to a turningpoint nearer to 45". This may of course be normal for rotation about a bond which is doubly-FubBtituted at one end. This behavior of the free interactlon i s harder to understand for properties measured a t absolute zero, such as Ho. One would suppose that a steric pair would have deep POtential minima a t 60 and 180" from the hydrogen-atom repulsions on the central bond like the well-known minima in ethane and the substituted ethanes; and that a t absolute zero the pair would be frozen In one of these pos1t1om, preferably a t 180" where possible, in view of the higher energy required to reach 60" in Fig. 1. Rut such a model leads to three expectations which are not verified experimentally. First, there is no longer any reason for the mean angle of a "free" steric pair a t absolute zero to agree with its mean angle at room t#emperaturesince the first pair is frozen while the second is really almost free.
+
k '
/ cf
-0.30 * Fig. l.-Angular
dependence of steric pair interactionr.
March, 1952
PREDICTION O F ISOMERIC
DIFFERENCES IN PARhFFIN
PROPERTIES
33 1
But they do agree almost within experimental error as shon:n on the regularities in R, and were indeed the initial in Fig. 1 for Ho and Hz6 and R. (The small increment in stimulus for the present study. the p-coefficient for Hg5 may be due to a tendency for a steric The transmission limits, which are determined by pair to go higher up the steric barrier a t room temperature; or i t may be due to a n effect discussed for the L coefficients the lowest-frequency absorption bands, were found below, the effect of steric hindrance a t room temperature in to depend principally on the number of first C-C reducing the probability of intramolecular van der Waals neighbors around the most-substituted C-C bond attractions.) Second, the success of a two-neighbor sum formula should in the molecule. This same factor seems largely be worse a t absolute zero than at room temperature if in the to determine the known first ionization potenfirst case the second-neighbors are not all alike and if a t the tial~,~~10 as though the most substituted bond conhigher temperature they are all alike, on the average. But tained the least tightly bound electrons in the the two-neighbor sum is better a t the low temperature (where the experimental errors are worse!) than t8hethree- molecule. The situation is parallel to that in subconstant formula for H26. On the other hand this may be stituted ethylenes where the spectra and the due to the inadequacy of w as a variable for describing the ionization potential both move to lower frequencies heat content. with increasing number of first neighbors of the C= Finally, if the steric pairs at absolute zero are fixed in the configurations described above as most probable, it should C bondslo The increase in R of the paraffins with increasing be possible to predict HO better by replacing p with a new variable, h, which is the sum of the number of 60' pairs values of the variable f corresponds to this shift to when the most stable configuration is taken about every longer wave lengths with number of first neighbors. C-C bond. This would assume that the 180' interaction could be taken to be zero, as found for the cyclanes. But a The shift in limit for every additional peighbor on least-squares formula for H Oin terms off and h was found the most subtitituted bond is about 30 A., or about to have a mean deviation of f 0 . 4 6 kcal./mole, 40% worse 1.5% in wave length. The increase in R per firstthan the formula with f and p . neighbor is about 0.2% of the increment for every The simplest way of accounting for these results is to suppose that the most stable configurations were not in fact ob- C-C link. No doubt R is mainly determined by tained in the low-temperature paraffin studies. The steric higher-frequency absorption bands which have pairs may be frozen into the 60 and 180" positions, but at large excited-state orbitals whose energy is relarandom, in a nonequilibrium distribution; so that p , tively insensitive to local isomeric variations of which is indifferent to configuration, is better than the ordered parameter, h. Such situations are familiar in the structure. The decrease in R with increased number of quenching of alloys, in the freezing-in of dislocations in metals and crystals, and even in solid paraffins themselves, steric pairs in the molecule represents a mean loss many of which are glasses and crystallize only with great of intensity or shift to higher frequency in the difficulty if at all. A paraffin gas molecule by itself may be absorption spectrum. The effect is several times a large enough aggregate to show such behavior also. H.-The height of the curve at 0' in Fig. 1 is about 1500 larger than the first-neighbor effect. The transcal./mole for each steric pair. This is comparable to the mission limit shows no signs of such behavior; heights of free-rotation potential barriers in paraffins and in fact, second-neighbors of the most-substituted substituted paraffins found by other methods. It is irnportant to remember that the curve for H in Fi 1 would be bond are associated with a small shift to lower only an approximate potential curve even if ttere were no frequencies rather than to higher. experimental error in the measurements, since each point on One attractive explanation of these effects is as it represents only a time-aveyage value over a zero-point follows. Consider the central C-C bond of a steric range of rotational configurations. I n the alkyl cyclopmt8anes, because of the zero-point puckering motion of the pair and the upper states produced by exciting an ring, the angles of a steric pair may range to f.60' on either electron from this bond. We may suppose that the side of the average value. addition of the neighbors to this bond in forming The first-neighbor term in the adopted formulas for H of the steric pair lowers the energy of each of these the alkanes and cyclanes would appear as the stabilization of a C-C bond by about 500 cal./mole on methyl substitution, states. Assume the energy reduction mould be if we considered the effects as being due simply to perturba- equal for all angles between the pair in the absence tion of the C-C skeleton. Such a picture is too simple. of steric effects. But the steric strain will cause This energy term is a sum of differential effects of substitu- lengthening of the central C-C bond, by approxition on C-H and C-C bonds; or on the C and H atoms, for those who prefer that formulation. For the prospective mately the same amount in the ground and first theorist, the best statement of the significance of the first- excited states where the excited electron is still neighbor term is probably phenomenological: that the whole nearby; but by much greater amounts in the higher system a t the end of a paraffin chain is "ideally" stabilized excited states where the sexcited electron is essenby about 1000 cal./mole on rearranging to the 2-methyl form; and that this is probably largely electronic or intemal tially removed and the bond becomes a weak oneenergy with small or negligible contributions from steric electron bond. Since the Franck-Condon principle effects. requires vertical excitation, differential lengthening The data on H26 for the cyclanes were not analyzed since of the bond in the upper state will produce a shift it was not clear how to compute w or an analog for it in these structures, or even Aw; and since the w term had already of the absorption to shorter wave lengths, which shown its importance in the alkane H25 formula at these may overshadow the lowering of the upper state temperatures. potential minimum, and which will cause a steric
Transmission Limits : R The molar refractivity, R, depends on the wave lengths and intensities of ultraviolet absorption in the molecule. Some observations on the absorptions of paraffins, as shown by their transmission limits, have been made by Klevens and Platt.8 Regularities found in these limits throw some light (8) H. B. Klevens and (1947).
J. R. Platt, J . A m . Chem.
Soc
, 69, 3055
decrease in the contribution of this bond to R. Many other explanations could be put forward. This interesting second term in R seems worthy of a careful theoretical examination. The Wiener Number: L,B and V Having examined the meaning of the first and second-neighbor parameters, we are in a better (9) W. C. Price, Chem. Rem., 41, 257 (1947). R. E. Honig, J4 Chem. Phys,, 16, 105 (1948).
(10)
.
332
JOHN R. PLATT
position to understahd the effects assigned to the mysterious and important external parameter, w. The behavior of the w coefficients suggests as a working hypothesis: that the quantity +ii, which is a sort of mean distance between the carbon atoms in a molecule, is an approximate inverse measure of the probability of one part of the moleculebeing attracted toanother part by van der Waals forces, or an inverse measure of the “mean self-contact area.” Since the total possible contact areasomething like the exposed surface of the hydrogen atoms-is approximately constant in isomers, .the sum of the self-contact area and the external contact area must be constant. If w is an inverse measure of the former it must be a direct measure of the latter. This notion gains support from the near-equality of the w-terins in HzS and in L a t 25’. Within a set of isomers, a larger mean molecular diameter reduces the internal contact area and decreases the intramolecular van der Waals attractions of the gaseous molecules, by about the same amount by which it increases the external contact area and increases the intermolecular van der Waals attractions in the liquid. The failure of w to contribute to HOcan be understood if we suppose that at absolute zero the molecules are frozen in configurations determined by the hindered-rotation potential barriers about each C-C bond, and that these configurations offer little opportunity for contact beyond the secondneighbor contacts already described. The mean area of the random long-range internal contacts described by w must, be small or zero. What then is the role of the p-term in L? For a given w, this steric term increases the intermolecular binding in the liquid. Plausibly we may suppose that steric hindrance makes some of the internal contacts more awkward and improbable, decreases the internal contact area, and increases the external. The increment in the p-term between Ha and Hzs may be due to this loss of van der Waals stability in t,he more sterically hindered isomers. The increment is of the same order as the steric term in L,but smaller. Because of the “surface irregularities” produced by steric effects, there is in this case no.reason to expect the loss of internal contact area to balance exactly the external gain. In brief, it is proposed here that the steric and Wiener terms in the adopted formula for L together describe approximately the isomeric variance in the mean intermolecular contact area of molecules in the liquid phase. The dependence of the intermolecular contact area on w and p may be calculable for molecules with partially-hindered rotation by statistical mechanical methods. About the boiling-point formula all that need be said is that the coefficients of the two terms in B are roughly proportional to the coefficients in L. As for V ,we see that an increase of intermolecular binding in the liquid as shown by larger L is accompanied by a decrease of molecular volume. A stronger total force between molecules produces a closer average approach. Or we can say that a compact molecule uses its volume inefficiently and contains holes; it will have a small external contact
Vol. 56
area and small L; while the opposite will be true for an extended molecule. Or both effects may be present. If the variance in volume were largely external, due to variations in the mean space between molecules, we should expect the V coefficients to be proportional to the L coefficients. If the variance were largely internal, then for a given external contact area as determined by L, we should expect an increase in steric hindrance to produce less efficient packing and more holes “within” the molecule, with a positive contribution to the secondneighbor V-coefficient. Actually, this coefficient is much larger (more negative) than the w-coefficient in V , relative to their values in L. In short, steric hindrance reduces the internal volume. No doubt this added p-effect is due to the familiar interpenetration of steric pairs to distances of approach smaller than the usual van der Waals radii. This penetration is necessary to account for the mean repulsive potential in Fig. 1. The apparent compression of volume per steric pair is about 1.8 ml./mole, about 12% of the total volume contribution per added CH2 group.
Indicated Studies The variables adopted here will undoubtedly be the simplest ones to use in making empirical correlations of other properties of the paraffins. Of especial interest would be further studies on free energy and entropy of formation, surface tension, octane number and so on. The type of reasoning used here can probably be extended in a straightforward way to alkenes, alkynes and alkyl aromatics; the disappearance of free rotation and the possible appearance of directional intermolecular forces will add interesting but difficult variables to the hnalysis. The treatment of substituted hydrocarbons will have its own delights and complications. Of the terms discussed here, the f-terms in H and R and the two terms in L are those which seem most likely to succumb easily t o theoretical prediction. The Wiener number, with its N2 denominator, especially needs theoretical examination and justification, with both theoretical and empirical attention to the question whet,her it is itself a fundamental variable or only a good approximation t o some more significant quantity. I n the boiling points, the search for an additional or a better external parameter should continue. l1
Appendix In Table I are given the values of parameters which seem likely to be useful in future analyses of other properties of (11) NOTEADDED IN PROOF.-Reference should be made to J. K. Brown and N. Sheppard, J . Chem. Phya., 19, 976 (1951), and to D. W. Scott, J. P. McCullough. K. D. Williamson and Guy Waddington. J . A m . Chem. Soc., 7 3 , 1707 (1951), on low-temperature rotational disorder in methslbutanes. Also to K. S. Pitzer. Disc. Far. Soc.. 10, 66 (1951), and to J. G. Aston, ibid., 10, 73 (1960, on the n-butane steric barrier. Also to J. L. Lauer, Thesis, University of Pennsylvania, 1947, and J . Chem. Phys., 16, 612 (1948), on the shift of the “cheracteristic wave lengths” of methylcyclohexanes to higher frequencies with steric hindrance. Also to A. R. Ubbelohde and J. C. McCouhrev. Ddnc. Far. SOC..10, 94 (1951), on coiling and self-contact of npara5ns.
PREDICTION OF ISOMERIC DIFFERENCES IN PARAFFIN PROPERTIES
March, 1952
333
Isomer
-1 1
-2
-3
2
4
4
2-m-3
6
2
6 8
5 5
6 -1
4 4
3 2
12
5
10
2 -1
-1
6 6
10 18
-1
1 22-m-3
2
12 1
-5 2-m-4
4
2
4 6 1
'4 36
-2
3
13
-1
-2
-3
9 0.06250 20 18 0.08000 16 0.16000
/
2-m-5
6 6
8 10
3-m-5
6
10
-6
6 6
4
4
14 22
8
2
24
1
4
1 22-m-4
6
14
23-m-4
6
12 2
7 7
10 12
3-m-6
7
12
3-e-5
7
12
15 32
8 1
I8 26
10
6
28
6
30
8
16
48
14
12
20
24-m-5
14
38
4 2 8
12
2
2 33-m-5
7
16
223-ni-4
7
18 '
-8 2-m-7
8 8
12 14
3-m-7
8
14
4-m-7
8
14
3-e-6
8
14
2
4
17 60
2
-3
8 8
22 30
12
8
32
10
32
10
30
14
22-m-6
8
10
18
52
8
23-m-6
8
16 16
25-m-6
16
33-m-6
18
34-m-6
16
24
42
8
1-1
2 12
2
10 1
8
38
14
8
56
8
44
2 16 3
32
28 5
4 22
2 34
3 32
8
17
36 11
8
24
22 8
3
3
7
9
1
10
22
32
40
7
3
10
2
2
.
3 30
6
15
2 24m-6
4
G
3
22
30
1
4
3
5 34
1
24
26
1 10
2
1
1
5
1
1
1G 18
24
I
0
8
4 12
8
30
18 20
1 1
28 6
21
10
7 12
2G
-2
12
26 3
8
.52
1
G 20
1
3
14
26
34
(i
3
10
-1
8
24 5
15
2
3
3
6 0
1
18
24
3
23-m-5
1
5
2
4 12 14
20
1 12
1 22-m-5
3
4
1
16
14 16
1
2
I
9
6 6
3
12
16
-2
8 8
14 2
12
-2
3
-7 2-m-6
5
44
G
1
14
-1
1
8 8
10 12
8 38
9
11
35 32 0.08333 31 0.11111 28 0.19444 29 0.16666 56 52 0.08163 50 0.12245 48 0.16326
40 0.20408 4G 0,20408 48 O,lG32G 44 0.24400 42 0.28571 84 79 0.07812 76 0 12500 75 0.14062 72 0.18750
71 0.20312 70 0:21875 71 0.20312 74 ,O. 15625 67 0,26562 68 0.25000
1
JOHNR. PLATT TABLE I (Coolltinued) Isomer
//t A/
2-m-3-e5
16
3-m-3-e-5
18
/a/
-
3 20
10
233-m-5
20
234-m-5
18
60 2
3
ax
40 11
15 90
4
CONSTANTS ANU
14
23
18
6
6 46
52
8
3
24
30
68 -2
4
16
12
19
4
18 4
2233-m-4
22
12
10
12 42
66 -1
4
10 42
19
6
3
AfIZ
11
-1
16
20
/ldi
38
00
ci
4
.
/ll/lA/II
44 1
3 18
4 224-m-5
fa/: ah
2
16 2
223-m-5
iu
-4
54 34
18
/adi W a z 38 11 48 16 42 13 18 1 50 17 40 12 54 19
ai
aiz
ale
all
0.544
aa
14.05
.344
- .19
.342
(f and h ) H25,kcal./mole -0.668 -0,500 -1.658 -1.570 -1.118
0.597 ,301 .398 - ,335 - .545 - .160
19.42 0.165 ,120 .lo1
9.201 ,235 ,210
0.011 .070 ,002
.0055 .0243 .0239 .0285
-
-
,0015
-0.0036 - .€lo30 - ,0041
0.003i . OOiO ,0038
0.0011 .0020
-
.0010
0.050 QPR-G-const ) 5.086
.0238 .OS92
4.001 98.36
100.76 -1.43
-1.112 0.76 ' 0.304
0.120 1.03 0.004
-0.139 0.14 0.030
92.51 (TPR-6-const)
a
-0.943 -1.173 -0.797 -0.505 -1.056
Fitted to older API data.
M.D. 1.38
0.44 .33 .33 .46 2.23 0.37 .44
I
-10.95
-14.063 .160
-0.169 .I20 - ,073
-
.015 .013 ,011" .012 ,013 0.712 .052 .097 .043 .55
2.38 0.53 1.54 0.67 .42 .55 2.14
V , ml./mole -1.128
0.358 -0.110 1.160 1.090 0.355
6
11.8
tl, "C. 1.198 2.943 0.165 -2.34 2.368
4
.016
2.820
-3.258 0.295 -8.476 -8.23 -1.639
'
6
0.163 .015
L, kcal,/molo
-
3
-0.325 -0.170
0.105
-0.172 .0385
6
7.31
,0652
- ,0680 - ,0749 - ,0741 - .073G
6
.38 .29 .19 25
R, ml./mole -0.0702 0.0112
*
'
5.33 -0.140
4
TABLE I1 MEANI)mVIATIOKS OF LEAST-SQUARES FORMULAS
Ho, kcal./mole -0.499 -0.505
h
W-AW/N'
67 0,26562 64 0.31250 63 0.32812 66 0.28125 62 0.34375 65 0.29687 58 0.40625
-0,001 ,100
-0.004 - .035
.016
.008
-
-
0.21
.38 .21 .28
'
.23" -10.90 (TPR-6-const)
.19 .16"
,
335
PREDICTION OF ISOMERIC DIFFERENCES I N P , ~ R . ~ F FPROPERTIES IN
March, 1952
TABLE 111
ANGULARDEPENUBNCE O F STERIC PAIRINTERACTIONS A. =/$A
Isomer
free
00
Parameters -AH0
'/$
obs.
180;
1200
x
A R X 1000
100 pred.
obs.
ped.
..
..
Ca-Cyclohexanes e-c6 11-m-c6 12-m-c6 cis(ep) !rans(ee) 13-m-c6 cis(ee) trans(ep) 14-m-c6 cis( ep) trans(ee)
..
e-c5 11-m-c5 12-m-c5
..
..
2
-2
cis
1 1
-2 -2
1 1
-2 -2
2
-2
2
1 1
-2 -2
3 1
1 1
-2 -2
2
1
-2 -2
1
trans 13-m-c5 CiS
trans
2 2
27
0
1 197
0 200
- 173 106
- 190
308 112
300 100
-
28 1 13
290 - 30
14 293
- 30 290 16
100 300 M.D. = 7
2
1
.. 200
114 305
2
..
.. 199
..
..
..
2
261
320
2 3
$58 231
60 180
2 2
157 220 21 1 220 M.D. = 37
-
130
..
..
-
80
170
99 108
- 100 65
134 185
140 140 35
B. Angular Coefficients Coeff.
CS
a2
(free)
a2
(0")
-0 I20 0.45 .76
a1
ap (60') a2 (120") a2 (180 ") Assumed.
-
Ho, kcal./mole Observed c6
-0.43 0.62 .5G
.10 .06
Observed
R , nil;/iiiole
Adopted
C5
c6
Adopted
-0.50" 0.50b .80 .50 .20
-0.049 - ,108 - ,142
0.020 - ,098
0.015" - .065* - ,120
-
- ,085 - ,038
,080
- ,038
. ooca
- ,012
.OO"
Assumed i n C6's.
hydrocarbons. They are listed for all the alltanc isomers t,hrougli thc octanes. .V is thc number of carbon atonis. f; is Zfai where fa, is the number of first-neighbor C-C a
bonds adjacent to the ath C-C, fa? is the number of secondneighbor C-C bonds, or bonds one bond removed from the a t h C-C, etc.; the sum being taken over all the C-C bonds in the molecule. f is ?I. i,f2 IS p , tho nuniber of st.erir pairs in t.hc mole cult^. When p replaces f 2 in a forniula, its coefficient is numerically twice as large. f i j is Bfaifaj. a The offset columns give half the isomerir incimicnt in cach parameter over the value for the n-paraffin of thc same 1%'. The offsetfi column gives Ap. vu is the sum of the numbers of bonds between all pairs of carbon atoms in the molecule. The offset column gives 1 / N 2 times the isomeric increment, which is the quantity actually employed in t,he Wiener formulas. h is the total number of steric pairs at an aiiglc of 60" t,o cach other around thcir central C-C bond, under the condition that each pair is allowed to take only the angles 60 09 180°,with the 180" position around every C-C bond being filled first. This variabh is most likely t,o be useful for describing low-temperature properties. Thcre arc some useful sum rules for the f's. Sincc Tfaj=N-2 J
aid ?;?;,faj"fak
= (:V -
212
j k
it follo\ts that &f, = ZZ.faj = ( N - l)(N - 2) j
ja
and x?;fjk
j k
= ~ ~ ~ f a j = f a(AY k a j k
I)(lv
- 2)'
The third of t.hese relations is useful for checking the deterniinat,ions of the f's. The j ' s ca,n be expressed in terms of c3, the number of tert.iary carbons; c4, the number of quaternary carbons; C ~ I , the number of tertiaries next to an end; etc. Thc first few formulas are fl = 2(N +fi = 2(N
- 2 + c3 + 3cd - 3 + 2C3 + 6 ~ 4- CZI - 2~41
+ 2C34 + 'hO + 4 N + 12c3 + 42c4 - 2c3, - 4c41 + 2C33 + + 8~44 = + .I?+ Gc3 + 24~4 = 10 - 4N + + 2f1 + fa + Gc, - 2Cql + 2C33 + 6~3.1+ 1 6 ~ ~ 3 3
jII= -10
4~34
fi
.fit
fi
The highcr J's, ,fJ and ,f12 and bryond, cannot bc expressed completely i n terms of c's with only one or two subscripts.
~
336
P. TORKINGTON
Wiener showed that w is a neighbor-sum 1) @jfi w =: ‘/2N(N
-
+
J
The isomer designations in Table I are obvious simplifications, such that-4 means n-butane; 2-m-3-e-5 means 2methyl-3-ethylpentane; and 2233-m-4 means 2,2,3,3tetramethylbutane; etc. For each property G of Table 11, each row gives the coefficients determined for a least-squares fitting of a formula of the type CZsA.fd f a l ~ A f i 14- aizAfi2 f a t ~ A f ~ 2 awAw/N2 to the observed isomeric deviations, AG, obtained from tho
AG
alAfi f a2A.f~
+
API Tables. Some terms in this formula were omitted in each solution; the entries in each row correspond to the terms kept. The last column shows the average deviation (neglecting sign) between tthe observed values and the values predicted by the resulting formula for the property. The first entry in this column for each property shows the rdw isomeric variance, z . e . , the mean deviation of the data on this same group of isomers from the formula AG = 0. The amount by which this figure is reduced by using the other formulas is a measure of their merit. All the CS-CSisomer data were used in the analysis, omitting the 2,2-dimethylpropane and 2,2,3,3-tetramethylbutane, since some of the data for these compounds were taken under non-standard conditions. The data used in Tables I1 and I11 were from the Selected Values of Properties of Hydrocarbons, American Petroleum Institutre Research
VOl. 56
Project 44 at the National Bureau of Standards, as follows, by property, table numbers, and latest dates of revision (except as noted in Table 11): Ha: Iw, 2/49; 2 ~ 11/46; , 3 ~ 10/44; , 6w, 4/49; 7w, 4/47 Hzj: l p , 4/45; 2p, 11/46; 3p, 4/45 R and V : lb, 6/48; 2b and 3b, 12/48; 6b and 7b, 6/49 L: lq, 29 and 3q, 5/44 B: l a , 6/48; 2a and 3a, 12/48 The coefficients of thef, h solution for HOin Table I1 were not tabulated since this solution was unsuccessful. Mean deviations are also given for the T P R least-squares solutions for several properties. These authors used different parameters, c3, c4, cZ3, C Z ~ ,c 3 3 and c34 to fit essentially the same data used here. Part A of Table JII gives the isomeric deviations in the parameters f l and f z (angular) and the properties H Oand R for the Ce-cyclohexanes, taking ethylcyclohexane as reference compound; and for the Cr-cyclopentanes, taking ethylcyclopentane as reference compound. Part B shows the corresponding coefficients, a1 and a:: (angular) necessary to fit the observed values of Part A . The “Observed” colunins give fhe coefficients obtained by simple subtraction among the data in Part A. The “Adopted” columns give the best values of the angular coefficients when some of the others are fixed as indicated so as to be more consistent with the alkane values. The predictions from these “adopted” sets of coefficients are shown in part A, with t,heir mean deviations. The “Observed” and “Adopted” sets are plotted in Fig. 1 as functions of angle, after having been multiplied by two to convert them to p-coefficients.
POSSIBLE MOTIONS OF VIBRATING MOLECULES AND THEIR INTERPRETATION I N TERMS OF MOLECULAR STRUCTURE
.
BY I?. TORKINGTON British Rayon Research Association, Barton Dock Road, Urmston, Nr Manchester, England Received December 86, 1061
The possible motions of a system of vibrating particles in a normal mode are studied briefly in the general case, and it is indicated how in a given configuration they may be expected to depend on the potential function. The case of the symmetrical vibratioris of the symmetrical triatomic molecule is treated in detail, taking water, sulfur dioxide and chlorine monoxide as three typical systems. The problem is reduced to one in a single coordinate, and formulae obtained for the shape of the potential well representing the vibrating system, for all allowed solutions of the force constants. The contours of the potential well are found to be concentric ellipses of the same eccentricity, this latter and the orientation of the well being functions of the force constants. The directions of the vectors representing the two normal modes are related to each other and to the axes of the elliptical contours.
I. Introduction ,The study of molecular force fields has been considerably hampered by what perhaps can best be described as a feeling of discouragement which arises a t quite an early stage of an investigation. This possibly applies t o other fields of work as well, of course, but in the present instance one can at least be quite explicit as to the underlying cause. It is merely that one cannot in general be certain enough that a set of force constants chosen, after the usual fitting with the vibration frequencies, as a best set, is near enough to the real set to justify a, complete normal coordinate analysis, with its attendant additional information-atomic displacements and potential energy distributions. After all, the calculation of force constants is not the bea11 and end-all of a theoretical spectroscopist’s life-he wants bo use them to inves1ig:Lte the normal “life” of his molecule-the way it nioves i n its everyday exislence while twiislating in Le-
tween reacting. There are two obvious courses of action in proceeding beyond the choosing of a ‘!best set” of force constants, and its interpretatioii in terms of individual bond strengths, bond angle rigidities, oribital coupling and interatomic repulsions. The first is the direct calculation of the potential function by quantum mechanical methods. Until the exact significance of localised and over-all molecular orbitals’ is properly understood-why apparently one has to use one set for one type of phenomena, the other for a different type-this appears to be beyond the bounds of possibility even in simple triatomic systems; the problem is rather different from that of obtaining force constants of the correct order of magnitude as a demonstration of quantum-mechanical method. It might be noted in passing that the improvement of the simple Heitler-London treatmelit of the ( I I I,eiiiilrrJ-.luiit.s l r i i c l P o l h , F‘rtraday Roc. Discussion o n IlydrovnrIIUU.,
1861.
.
