THE JOURNAL OF
PHYSICAL CHEMISTRY (Registered in U. S. Patent Office)'
VOLUME61
(0Copyright, 1957, by the American Chemical Society)
NUMBER12
DECEMBER 27, 1957
THERMODYNAMIC PROPERTIES OF INTERNAL ROTATORS WITH SMALL MOMENTS OF INERTIA BY J. 0. HALFORD Department of Chemistry, University of Michigan, Ann Arbor, Michigan Received February 2 , 1967
A method of evaluating the partition function for internal rotation which applies integration only to systematic parts of the external one-dimensional rotator series is described. The method is used in verifying the equations and newly tabulated thermodynamic properties of Li and Pitzer and in determining the spread of partition functions which must be represented by a single entry in the general free energy table.
Li and Pitzerl recently have published tables of thermodynamic properties for symmetrical top internal rotators with lorn moments of inertia at low temperatures. The newly calculated properties provide an extension, without any decrease of implied precision, of the tables of Pitzer and Gwinn2 into regions of the pertinent variables where the original methods of Pitzer and Gwinn did not appear to be sufficiently detailed to yield the answers. The thermodynamic properties could not in fact be calculated in the region of the new extension from the theory developed by Pitzer and Gwinn. The greater detail of the energy levels derived by Koehler and Dennisona is required. Use of the Koehler-Dennison theory to determine the thermodynamic properties in the region beyond the limits of the Pitzer-Gwinn tables was first illustrated by the present writer4 with a methanol model. Above 200"K., it was demonstrated that the calculation did not have to be more detailed than that of Pitzer and Gwinn. The extra detail was required only in the supporting argument and not in the final calculation. Subsequent calculations with other models and a t lower temperatures have shown that the simplified Halford formulation is quite generally accurate. The model for which the energy levels have been derived consists of two opposed symmetrical coaxial rotating tops between which a force hinders internal rotation. The partition function for the two de(1) J. C. M. Li and K. s. Pitzer, THISJOURNAL, 60, 466 (1956). (2) K. 5. Pitzer a n d W. D. Gwinn, J . Chem. Phys., 10, 498 (1942). (3) J. S. Koehler and D. M. Dennison, Phys. Rev., 67, 1006 (1940). (4) J. 0. Halford, J . Chem. Phys.,18, 1051 (1950).
grees of freedom involved in rotation about the common axis is readily factored out of the over-all partition function in excellent approximation, but these two degrees of freedom remain coupled. Summation over both motions, however, yields a partition function which permits a formal separation into two parts, of which one is assigned to the external rotation, the other t o the internal rotation. I n the internal rotator energy level pattern a series of (usually) separated bands of levels occurs. Within each band, for each external quantum level K , there is one internal energy level in each of the several symmetry classes. For symmetry number n, the theory provides n equivalent periodic curves in each band on which the energy levels must lie. These curves are displaced laterally a t uniform intervals on a scale expressed in terms of u (Koehler and Dennison), p (Li and Pitzer), or K . Either of two K scales is valid. If C1 and Cz are the moments of inertia of the opposed tops of the internal rotator, and C = C1 Cz, u can be either f KCI/C or f KCZ/C. I n general, the levels of a single symmetry class mill be found systematically distributed over several of the periodic curves. The variable p has been defined by Li and Pitzer to contain 2n units per cycle and to measure the positive distance from a minimum point of each of the periodic curves. For any one periodic curve
+
p =
27rKC,/(nC)
+2~p/n
(1)
where /3 is an integer which is different a t any one K for each symmetry class. It is evident that the choice of K determines a p and a Q p for each symmetry class. Here Q p is a
1581
1582
9.0. HALFORD
Vol. 61
sum of exponentials a t constant p, or constant K and P, over the bands of energy levels. The overall partition function, for both degrees of freedom, internal and external, can therefore be expressed as
&=ao+an+azra+ .... (4) The partition function of eq. 4 approaches complete accuracy in the limit and is independent of the choice of moment of inertia in the definition of p. The Li-Pitzer integration, a t the limit for high QEI = Qp(K)exp( - 6 Z P ) (2) K C,/C,, will give the result of eq. 4, provided that C,, and not C, as they recommend, is chosen for the for each symmetry class, where 6 is hP/(8g2CkT). Li and Pitzer have set up Q p as a general even definition of p and the integration is performed over all n periodic curves and subsequently norperiodic function of p, as malized. It appears, therefore, that there are Qp = U~COS mp (3) models corresponding to heavy alcohols and phem nols, for which the integration in terms of the They combine eq. 1, 2 and 3 and integrate for QEI. smaller moment C, is more accurate. At the opposite limit, with C,/C, approaching Since C1 in eq. 1 can be either moment of inertia, this ~villlead to two formally different results, of zero, it is readily seen by inspection that the integration with the p and Ii scales set by C, becomes which one must obviously be the more accurate. It is therefore necessary either to make a choice completely accurate. Thus a t both limits, for between the two moments of inertia or to show that C,/C, very small or very large, use of the smaller the difference between the two results is always moment gives the more accurate answer. When negligible under conditions of practical interest. the two moments are comparable in magnitude, it It should be emphasized that the two results can is not clear that either integration is accurate or always be obtained, regardless of the symmetry of which of the alternatives is more accurate. The the opposed rotating tops. The only difference use of C,, following Li and Pitzer, is likely to be introduced by the symmetry property is in the more accurate, when the moments are equal or number of individual integrations which must be nearly equal, because it shows differences between combined in order to cover a single symmetry class. symmetry classes which the use of C, is incapable When both tops are “strictly symmetrical,” Li of revealing. The question of accuracy a t intermediate C,/C, and Pitzer state that greater accuracy will be obtained with the smaller moment of inertia. The can be settled by comparing the integrated results theory, however, is thought to be accurate enough with direct numerical summation of the terms in for models like methanol and phenol in which one the definition equations. Unfortunately, Li and top is slightly asymmetric and has, in effect, unit Pitzer have not provided such numerical verificasymmetry number. In all cases of practical in- tions or their theoretical equivalent. A series of pertinent numerical comparisons has terest, the heavier top in such models is the “strictly symmetrical” one. For models of this type, Li therefore been carried out by the present writer. and Pitzer require the use of the moment of inertia For C,/C, from zero to unity, the recommandatioiis of Li and Pitzer yield highly accurate sums and of the symmetrical top in their integration. It develops, however, that this requirement leads thermodynamic properties, while the results obto a formally incorrect partition function when the tained by integration in terms of C , are unsatissymmetrical top is considerably heavier than the factory. The preference for C, undoubtedly conunsymmetrical one. If the nioments of inertia of tinues for some distance into the region of higher the symmetrical and unsymmetrical opposed tops C,/C,. Somewhere a t still higher levels of C,/C,, are, respectively, C, and C,, Li and Pitzer use C, the alternative integration must become formally in the evaluation of Q. When C,/C, is large, more accurate, but by this time the differences C,/C approaches unity, and the exponential factors between the two integrations have become negligiin the integrated equation approach zero, leaving ble in any practical ,sense. It is therefore concluded a. as the limiting value of Q a t high CJC, for all that the equations and recommendations of Li and symmetry classes, regardless of the symmetry Pitzer always lead to practically accurate results. Comparison of the integrals with the correct number. For any chosen temperature and level of ac- sums is facilitated, for a widely distributed series curacy, there will be a maximum K requiring in- of models, by dividing QEI of eq. 2 into a series of clusion in the summation. This Ii is proportional sums which are individually readily evaluated. A If p is defined considerable simplification is effected whenever, in to C‘/2, or, a t high C,/C,, to C,’/z. in terms of C,, there will be, for symmetry number terms of the smaller moment of inertia, the number n, nC,/C, individual Q K or Q p values per cycle of of levels per cycle in each band is an integer. The each of the n periodic curves. It follows that the correct partition function is quickly and simply number of cycles covered, up to the maximum K , evaluated under all conditions where the symmetry is KC,/(nC,), or with IC,,,, proportional to Cs1/2, number approximation is accurate. When difthe coverage of the cycle becomes inversely propor- ferences between symmetry classes are negligible, the formal difference between the general partitional to Cs1/7. As a result, wheii C,/C, becomes indefinitely tion function of Li and Pitzer and the correct one is large, the fractional coverage of the cycles becomes clearly shown. The differences between ao, the zero, and every Q p (IC) in eq. 2 assumes the value partition function upon which Li and Pitzer have corresponding to K = 0 or p = 2n p/n. This, in constructed their tables of thermodynamic properturn, leads to the general result, as C,/C, ap- ties, and the partition functions obtained here are negligible for the internal rotator with a simple proaches infinity, that
i
i
THERMODYNAMIC PROPERTIES OF INTERNAL ROTATORS
Dec., 1957
1583
sinusoidal potential function. They may become determine, in terms of the am of eq. 3, the limiting appreciable, however, for other potential func- values of the partition function which a single entry in the free energy table represents, and thus to tions. A model will be considered which consists, like evaluate the limits within which general tabulation a phenol, of a top of symmetry iiuinber 2 rotating is permissible. against a slightly asyninietric top of moment of inIt is reasonable and, if general tabulation is perertia C1. When C1 and Cz are equal, there are four missible, necessary to assume that the a, decrease levels per cycle (nC/CJ and p is always an integral in magnitude with increasing m. The largest demultiple of ~ / 2 . Only three numerically different viation of Q from ‘ao will then occur for symmetry values of QP can appear, namely, Qo, Q T pand QT. number 2 a t infinite C,/C,, where, according to eq. For either symmetry class, each exp(-66K2) is 3, the internal partition function is multiplied by one of these three Qp values. Q = uo uz 0 4 . . . = (Qo Q r ) / 2 (8) Equation 2 for one symmetry class of this model takes the form The opposite limit is obtained for a model having symmetry number 2 with C, = 2C,. For this .....I QEI = Qo [l + 2 exp( -166) + 2 exp( model Q is a0 a3 . . . for one symmetry class + Qa,z [2 exp( - 6 ) 2 exp( -98) 2 esp( -256) and a0 - a3 . . . for the other. The limit is . . . . .J therefore a0 f a3 . . . , whichever deviates more + Q,[2 exp( -46) + 2 exp( -3G6) 2 exp( -100 6) widely from a0 according to the sign of a3. .....I (5) Since, according to Li and Pitzer, a2 is small and For the other symmetry class the Qo and QT are a3 is negligible, the practical limits on the general exchanged. partition function are a0 and a0 a2. General tabIn either case ulation is therefore permissible only within limits QEI = ( T / S ) ’ / ~ ( Q O 2QTiz Qa)/4 (6) for which a2 is negligible. In other words, Qp is for which leads to precisely the expression recom- practical purposes a simple sinusoidal function of mended by Li and Pitzer as an approxiniation to p , as originally verified by Halford for a methanol the general internal partition function QI (E&). model. If this mere not true, highly precise genIn terms of the coefficients of eq. 3, the internal eral tables of thermodynamic properties would not partition function from eq. 6 reduces to be possible. Q ~o +a4 ... (7) The expression of eq. 6 and 7, used by Li and The factoring operation illustrated with eq. 5, G Pitzer to calculate Q, is, with u4 negligible, a very and 7 can be applied for any symmetry number close approximation to their integrated general whenever nC/Cl is an integer. Integrat,ion is ap- partition function ao. This is one limit of the pracplied only to the undisturbed one-dimensional rota- tical range of partition functions covered by the az,cortor function where the accuracy is readily deter- general tabulation. The other limit, a0 responds to eq. 8, which is the original expression mined . With C1 constant and C2 increasing, the fraction verified by Halford for methanol. Since, for acof the external series multiplying a single Q p de- curate general tables, one limit should be as good as creases, but the magnitude of this fraction remains the other, it is evident that the calculation denearly constant as a decreasing fraction of an in- scribed by Li and Pitzer is unnecessarily detailed. creasing total. At the same time, with nC/C1 in- I n support of this point, it has been found, for two creasing, more individual Q p are brought into play different models a t 1/Qf = 1.000 and V / ( R T ) = in a manner which decreases the absolute error of 3.436, that the simpler form of eq. 8, together with the factoring operation and rapidly decreases the the corresponding simple forms for the derived relative error. In spite of the a t first suspicious sums, gives thermodynamic properties in accurate increase in subdivision of the rotator series, with agreement with interpolated values from the iiem the apparent possibility of increasing error of inte- tables. The accuracy of the equations of Li and Pitzer gration, the illustrated factoring method actually grows rapidly more accurate as Cz increases a t con- was verified for a model with symmetry number 3 for which C, was 1.0 X c.g.s. units while C, stant C1, n and T . At 1/Qf= 1.000 and V / ( R T ) At very low temperatures, particularly when was 11.0 X C2/C, (or C,/C,) is not large, the pertinent sub- = 3.436, V is 859 cal./mole and T is 125.84”K. divisions of the one-dimensional rotator series will For this model, for which the individual symmet,cy deviate appreciably from the corresponding inte- classes have properties widely different from the grals. When this occurs, only a few exponential general tabulations, the results obtained by the terms will contribute appreciably to each sub- methods of Li and Pitzer are in accurate agreement division, so that direct summation mill not be very with the correct ones based upon direct numerical detailed. The illustrated procedure has proved summations. useful in verifying the integration results of Li and In the verification just described, an interesting Pitaer. feature emerged which has not been previously The new factoring operation illustrated here, brought out. The heat capacity is well outside the with the indicated parts of the one-dimensional range delimited by calculations from the statistical rotator series evaluated by integration, should be sums a t p = 0 and p = T. This can happen with accurate for all models to which the new tables of the heat and heat capacity functions but not with Li and Pitzer are applicable. It should therefore the free energy. The expression for Q”, which in be possible, if the extreme models can be found, to the chosen example controls the heat capacity, con-
+
+ + +
+
+ +
+
+
+
+ +
+ + + +
+
+
+
1584
R. M. DELL
tains contributions from the amplitudes of all three of the periodic curves for &, Q’ and &” against p. It is a rather large al’ which causes the heat capacity to fall outside the range in which it was a t first expected to appear. It has been stated that the methods discussed and referred to here separate the problems of the external and internal rotations. It is true that the properties and factors assignable to the external rotation in the limit for an infinite barrier are separated out. The rest of the thermodynamic property as calculated for both degrees of freedom is then assigned to the internal rotation and included in a general table. It should be kept in mind,
Vol. 61
however, that the separation is arbitrary as well as convenient. All solutions of the problem sum up for both degrees of freedom and subsequently divide the property between them in an arbitrary but convenient manner. Actually, when two degrees of freedom are coupled as in the internal rotation problem, there is no device by which a physically meaningful separation can be made. The character of the “separation” effected here is well emphasized by the column of positive free energies obtained by Li and Pitzer. For a truly separated independent degree of freedom a positive free energy would represent statistical nonsense.
THE ADSORPTION OF GASES ON GERMANIUM POWDER BY R. M. DELL’ Admiralty Research Laboratory, Teddington, Middlesex, England Received March 21, 1967
A study has been made of the adsorption of hydrogen, carbon monoxide and oxygen on germanium powder prepared by reduction of pure germanium dioxide. At 25”, no chemisorption of H) or GO was observed on either reduced or oxidized germanium surfaces. Oxygen, however, was adsorbed readily a t this temperature, the uptake obeying a logarithmic rate law. The extent of oxygen adsorption appears to be independent of the semi-conductor type of germanium but to vary with the detailed preparative procedure, indicating a dependence upon the crystal structure of the germanium surface. with a resistivity of about 15 ohm-cm. Four different Introduction samples were used ranging in specific surface area from 0.45A knowledge of the adsorptive properties of 1.37 sq. m./g. according to the previous thermal treatment. germanium for various common gases is of interest Surface areas were determined by krypton adsorption a t from two points of view. First, the presence of a -195”. For quantitative comparison with the york of chemisorbed film of gas will, in general, change both Green, Kafalas and Robinson, a value of 19.4 A.2 was for the cross-sectional area of the adsorbed krypton the surface barrier height and the surface recom- chosen atom.6 Adsorption measurements were carried out in a bination velocity of holes and electrons. The elec- constant volume high-vacuum system of conventional detrical properties of germanium are thereby modi- sign, a multi-stage, high speed, mercury diffusion pump of fied. Secondly, recent work on the chemisorption the type described by Gray’ being employed. The gercontained in a silica vessel, was protected from of gases on metallic oxides has shown this to be, in manium, mercury vapor by a cold trap kept permanently a t -195”. many cases, an electron transfer process, and that a Between adsorption experiments the germanium was heated relationship exists between the semi-conductor and in hydrogen, usually at 650°, for periods of 21 to 100 hours the adsorptive properties of a solid. Since the and evacuated a t the same temperature for 10-24 hours. oxygen adsorptions, the amount of hydrogen taken up semi-conductor characteristics of germanium have After corresponded quantitatively to the removal of oxygen as been so well established, its surface chemistry is water. At higher temperatures of reduction, the amount worthy of study. of hydrogen used was rather less than the theoretical beThe present paper reports some studies of the cause some GeO evaporated to a cool part of the reaction interaction of oxygen, hydrogen and carbon monox- tube before reduction could be effected. ide with reduced germanium powder. Since this Results work was undertaken, a number of papers have A series of experiments was carried out to deterbeen published dealing with the adsorption of gases mine whether hydrogen is adsorbed on reduced on germanium films,2 filaments3 and crushed single germanium At 25”, no adsorption could ~ r y s t a l s . ~A comparison of these results with be measured,powder. the limit being 5 X 1O’O those for germanium powder reveals several in- molecules/cm.2 a t 5 x of detection mm. pressure and 2 X teresting features. 1013 molecules/cm.2 a t 0.2-0.4 mm. pressure. At Experimental - 195”, however, slight adsorption occurred (just Germanium waR prepared by reduction of Johnsondetectable at 0.2 mm. pressure) ; this appeared to Matthey “Spectroscopically Standardized” germanium di- be pressure sensitive and was probably physical oxide in pure hydrogen a t 600-650°.6 The stated impurities were copper and calcium, and a fused sample was n-type adsorption, No hydrogen uptake was found (< 6 X 10l2 molecules/cm.2) a t either 25 or 100” on (1) Houdry Process Corporation, Marcus Hook, Pa. germanium surfaces which had been pre-saturated (2) K. Tamaru, THISJOURNAL, 61, 647 (1957). with oxygen a t 25’ and 0.5-1 mm. pressure. (3) J. T. Law and E. E. Francois. Ann. N . Y . Acad. Sei., 68, 925 Similar experiments were performed with car(1954); J. T . Law, THISJOURNAL, 59, 543 (1955). (4) M. Green, J. A. Kafalas and P. H. Robinson, “Semiconductor bon monoxide. At - 195O, reversible, pressure deSurface Physics,” University of Pennsylvania, 1957, p. 349. (5) L. M. Dennis, K. M. Tressler and F. E. Hance, J . A m . Chem. Soc., 46, 2033 (1923).
(6) A. J. Rosenberg, ibid., 78, 2929 (1956). (7) T. J. Gray, Disc. Faraday Soc., 8, 331 (1950).
