M. H. Everdell
The University of Aston in Birmingham, U.K.
Some Remarks on the Thermodynamics of the Xylenes
It is well known that any one of the xylenes will, in the presence of an aluminum halide and hydrogen halide, rearrange to give some of each of the other two isomers. These reactions were studied first by Norris and Rubenstein,' more thoroughly by Norris and Vaala,= and Ohen by Pitzer and S c ~ t t . ~ The composition of the equilibrium mixture at 50°C according to Norris and Vaala is 16 * 10% ortho, 65 + 10% meta, 19 e 5% para, and according to Pitzer and Scott is 12 3% ortho, 71 * 5% meta, and 17 2% para. As will be shown, these figures are given broad support by thermodynamic calculations.
*
Thermodynamic Calculations
The values of the standard molar heats of formation and standard molar entropies of the liquid xylenes at 25°C and those of the standard molar entropies of the gaseous xylenes a t the same temperature shown in Table 1 are those reported in the National Bureau of Standards Circular C.461 of 1947. These values are derived from three papers; that by Pitzer and Scott, that by Prosen, Johnson, and Rossini,&and that by Taylor, Wagman, Williams, Pitzer, and Rossini." In principle these values permit the composition of the equilibrium mixture of the xylenes a t 50°C to be estimated. Thus for the reaction para-meta i the liquid phase at 25'C
=
2.66
The van't Hoff equation may then be used to estimate t,he value at 50°C and gives the value 2.74. Similarly for the reaction ortho-tmeta at 25OC
=
234
*
Table 1.
Thermodynamic Data for Xylenes
these values the composition of the equilibrium mixture at 50°C may be estimated5 to he 20 * 6% ortho, 58 1 10% meta, and 22 + 8% ortho. As was said earlier these figures give broad support to the experimental values reported by Pitzer and Scott and Norris and Vaala. Furthermore they show that since the values of AHo are extremely small the composition of the equilibrium mixture will be relatively insensitive to temperature.
so that
K*..
(very good agreement having been obtained between the calorimetric values and those obtained as the result of spectroscopic measurements and the formulas of statistical thermodynamics) so that any error in the values of (So,,, - SoD,,) or (So,, - SoOrth,) is probably very small indeed, the same cannot be said of the values for AH,. By whatever means a value for the heat of formation of a compound of this sort is obtained, a t some stage use has to be made of the experimental value for its heat of combustion. This for each of the xylenes is over lo8 cal and the error in its determination can hardly be less than 100 cal. In fact Prosen, Johnson, and Rossini estimate the uncertainty of their value for AH, for ortho-xyleae to be *0.26 kcal, that for meta-xylene to be e0.18 kcal, and that for para-xylene to be +0.24 lrcal. The value of AHo for the reaction para-meta should therefore be written more correctly as 237 + 420 cal, and that for the reaction ortho-tmeta as 234 440 cal. Using
The Entropies of the Xylenes
From the thermodynan~icviewpoint the most interesting fact that emerges from the calculations shown above is that the quantities ASo, i.e., (So,., - S o)., and (SomeU- Soorfh,,)are certainly as important
+ 406
so that
Kns
= 2.91 and
K,,
=
3.00
These figures correspond to an equilibrium mixture of composition 19.5% ortho, 58.5% meta, and 22.0% para. Unfortunately less reliance can he placed on some of the values quoted in Table 1 than would be desired. While the entropy values are probably of great accuracy 538
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Journal of Chemical Educofion
S P ~ ~K.zS.,~ AND ~ ,SCOTT,D. W., J. Am. C h . Soc., 65,803 (1943).
' PROSEN, E. J., JOHNSON, W. H., AND ROSSINI, F. D., J . Research, Nall. Bur. Standards, 36,455 (1946). TAYLOR, W. J., WAQMAN, D. D., WILLIAMS, M. G., PITZER, K. S., AND ROSSINI, F. D., J. Research, Natl. Bur. Standards, 37, 95 (1946).
and probably more important than the quantities AHo in determining the yield of the three isomers a t equilibrium. I t is therefore profitable to inquire how these differences arise, in other words to inquire into the reasons why the molar entropy of meta-xylene is greater than that of ortho and para. We shall in fact have to discuss the differencesin the molar entropies of the isomers as gases rather than as liquids because we wish to talk about the mechanical properties of the independent molecules. Our findings are however directly applicable to the reactions in the liquid phase because the values of ASo and AHoare almost the same in the liquid phase as they are for the gaseous reaction, and the reasons for the differences the same. It is well known that with some reservations (which need not concern us here) the entropy of an assembly of independent molecules a t a particular temperature and pressure may be supposed to be composed of the following distinct contributions: 50,-the translational entropy which depends only on the molecular weight Sm,-the electronic entropy Sw-the vibrational entropy &,,-the rotational entropy which depends on the rotation of the molecule as a whole about its oentar of mass and, in some cases, on the internal ratstion of one part of the ~ y molecule relative to the rest, and an the s ~ m e t number of the molecule
Since the object of this paper is to draw attention to the importance of the symmetry numbers of the xylenes and of internal rotations in determining the course of the reactions under consideration, both concepts call for further comment. The entropy of an assembly of molecules contains a contribution due to molecular rotation (whether this consists only of rotation of the molecule as a whole about its center of mass or of rotation of one part of the molecule relative to the rest) simply because such rotations increase the number of distinguishable coufigurations accessible to it. The quantity known as the symmetry number of a molecule is the number of otherwise diatinct orientations which are in fact indistinguishable due to the indistinguishability of like particles. The larger the symmetry number the smaller will be the number of distinguishable configurations accessible to the assembly and so the smaller will be its entropy. The symmetry number for molecules such as ortho or meta-dichlorobenzene, which possess a single twofold symmetry axis, is 2, while that for the molecule para-dichlorobenzene, which has three mutually perpendicular twofold axes is 4 (in the latter case only two of the twofold axes lead to additional orientations). The number of distinguishable configurations of ortho, meta, and para-xylene will however be greater because in addition to the rotation of the molecules as a whole each methyl group can rotate relative to the benzene ring, but since each hydrogen atom in the methyl group is indistinguishable from the other two, only one third of the total number of orientations are distinguishable. We must therefore multiply each of the figures 2 and 4 by 3 twice so that the symmetry numbers of ortho and meta-xylene are 18 and that of para-xylene 36. The contribution to the entropy of the xylenes if the internal rotation of the methyl groups is assumed to be completely free may be calculated from eqns. (4) and
(5) given below. Such internal rotation can however never be completely free but must be inhibited to some extent by the existence of potential energy barriers, and hindrance to free internal rotation will of course lead to a real entropy contribution which is smaller than that calculated from eqns. (4) and (5). Although today, techniques such as those of microwave spectroscopy make the direct estimation of such potential barriers possible for some simple molecules, the only procedure available in the case of the xylenes when their thermodynamic properties were first studiede was to calculate, using spectroscopic data and the formulas of statistical thermodynamics, the total entropy of the substance on the assumption that internal rotation was free, and compare the result with the value of the entropy obtained by calorimetric methods. If the two values agreed within the limits of experimental error it was assumed that the potential barriers were too low to have an appreciable effect. If, on the other hand, the statistical value was greater than the calorimetric value, the potential harriers were assumed to he high enough to restrict internal rotation to a considerable degree, and the corresponding loss of entropy assessed by the subtraction of one value from the other. Such calculations were first carried out on the xylenes by Pitzer and Scott and later revised slightly by Taylor, Wagman, Williams, Pitzer, and Rossini. They indicated that internal rotation in the case of meta and para-xylene is almost completely free, such restrictions which do exist leading to a loss of entropy of only about 0.3 cal deg-' mole-', but that internal rotation of the methyl groups in ortho-xylene is much more seriously inhibited, resulting in a loss of entropy of about 2 cal deg-'mole-'. The various contributions to the standard molar entropies of the three gaseous xylenes at 25'C and 1 atm pressure are shown in Table 2. The translational conTable 2. Contributions to the Molar Entropies, col deg-' mole-'
ortho
meta
para
tribution S(,,was calculated from the well known equation St.,..
=
In M
+ 25.99 cd deg-'mole-I
(1)
where M is the molecular weight. The contribution is of course the same for all three isomers. The electronic is negligible. Thevibrational contribucontribution Scz, tion Sillis that calculated by Pitzer and Scott from spectroscopic data. The quantity S(4,, is the entropy contribution due to rotation of the molecule as a whole about its center of mass, ignoring the symmetry factor, it is calculated from the equation See references in footnotes 3 and 5 , Volume 44, Number 9, September 1967
/
539
The Pora
where I,,,is the rotational partition functiou of ihe molecule and is given by l.he equation
whew D is the prodnct of the three pril~cipalmon~eutsof inertia of the molecule. The quantitySclb1is the conlrihution due to the interual rotation of the two methyl groups relative t o the henseue ring, assuming such rotnt,iou to be completely free and, as before, ignoring the symmetry factor. It, is culculat,ed from t,he eequet,iou
whrre the partitiou funrtion for internal rolation is give11 by the equat,im
where I IS the reduced moment of inertia for internal mt,at.iou. The values of D and I are taken from footnote .5. Siro1 is the quant,ity by which theentropy is reduced hecause of the symmetry factor r a n d is calcnlaied from the formula AS(,^^ = 1f I l l r
(0)
t,hc symmetry factor having the value 18 for orb110 aud metit and 36 for para. S(4dl is the estimated quautity by which the entropy is reduced due to the potent,ial barriers hindering internal rotation. The standard molar entropy is given by t,he equation So = So) Sir1 &a Sir.> S m - S w - S ( m (7) There is of course some uucertainty in the value of So aud hence because Socan be evaluated only either from calorimetric measurements and so is subject to experimental errors, or from statistical formulas which include Slld1.The calorimetric values of the standard molar entropies of the gaseous xylenes are 84.20 i 0.25 cal deg-' for ortho-xylene, 85.64 0.35 for meta and 84.31 * 0.25 for para. These figures would provide 1.97 + 0.25,0.18 * 0.35, and 0.24 0.25 for the t.hree values of SW). For various reasons the "best" values for Sowere chosen as 84.31, 85.49, and 84.23, aud the values of S(nal changed t,o t,hose shown in Table 2 for the sake of internal connist,ency.'
+
+
+
+
* *
540
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Journal of Chemical Education
-t
Meta Readion
We are now in a posltion to identify the part played by each of these contributions to the value AS0 for the reaction para-meta in the gas phase, so as to account for the value A S 0 = So,.+, - So = 85.49 - 84.23 = 1.26 ral deg-I. I t follows from Table 2 that AS,,, AS,,, ASim AS9 = A R w AS,,:
,.,.
+
+
+
+
ASl4.1 =
0
+ 0 - 0.29 + 0.15 + 0.03 + 1.38 - 0.01
- ASi4dl
so t,hat the difference hetxeen the molar entrepies of mets aud para-xylene is more than accounted for by the ter~n In other wt~rdsthe entropy difference is more than accounted for hy the greater degree of symmet,ry of the para molecule. Since we have already shown thst the entropy term appears to play the larger part in the determinatiou of the equilibrium point in the parit meta reaction we now see that the fact that, meta-xylene is preferred t,o paw is due mainly to the difference in symmetry of the two molecules concerned, t,he less symmetrical isomer being the one more readily formed.
-
TheIOrtho -+ Meta Reocfion
We now have to accouut for thevalue A S o = So,., So,,,,,,,= 85.49 - 84.31 = 1.18 cal deg-I. AS" = AS,,, + ASrll + AS,,,, AS,,., ASilb) ASiw - AS(m
+
=
0 - 0
+
- 0.50 + ll.l:, + 0 + 0 + l.,53
so that the differe~~ce bet,ween the molar entropies of met,a aud ortho-xylene is due mainly to the contributiou AS(ldl. I t follows that the fact that meta-xylene is preferred to ortho is due, to a very large degree, to the relat,ively restricted rotation of the methyl groups in the nrtho position. The readiness with which one compound rather than anot,her is formed during a chemical reaction is of course determined by many factors which stem from the geometry of t,he molecules coucerned. The relative symmetry of the molecules nud the potential barriers inhibitiug internal rotation are only two factors among ma.111. and their effects are usually obscured by the greater importance of the others. The reactions st,udied here are of particular interest because in one case the symmetry factor aud in the other case the potential barrier factor are of sufficient importance to determine the course of t,he reactions concerned.
