J. Phys. Chem. 1982, 86, 654-658
654
Deuterium Isotope Effects on the Dipole Moment and Polarizability of HCI and NH, Chava Scher, Baruch Ravld, and E. Amltal Halevl’ Department of Chemistry, Technion-Israel Institute of Technoicgy, Haifa 32000, Israel (Received: July 24, 198 1, I n Final Form: September 14, 1981)
A previously described adaptation of the conventional Debye procedure for the direct determination of small dipole moment and polarizability differences between two polar gases is applied to the isotopic pairs DCl-HCl and ND3-NH3. The dipole moment difference obtained for the first isotopic pair, by using the Debye-Van Vleck equation for electric susceptibility,p(DC1) - p(HC1) = -0.0055 f 0.0002 D, is consistent with published spectroscopically determined values of poo(DC1)and poo(HC1),while that obtained by using the classical Debye equation is not. For the second pair, use of the DebyeVan Vleck equation, along with a correction for thermal population of vibrationally excited levels, is shown to be essential and yields M(ND),- p(NH3) = +0.0135 f cm3, 0.001 D and a(ND3)- a(”,) = -(2.2 f 1.7) X
Introduction A problem that engaged Simon Bauer’s attention at the inception of his immensely fruitful w e e r is that of defining unambiguously the relation between the polarization and polarizability of individual molecules and their manifestation as bulk dielectric properties.’ The investigation of isotopic polarity differences, which was carried out over an extended period in this laboratory, although motivated by other concerns, turned out to be directly relevant to just this problem. It is thus not inappropriate that this reported be dedicated, with esteem and affection, to Simon Bauer. Over 40 years ago, Bell and Coop2 suggested that the difference between experimentally determined dipole moments of a diatomic hydride and the corresponding deuteride could be estimated by PD - PH = ~ e l ( (-r r~ e ) 0 - (FH - re)o) AP = perA(r)o (1) in which p l is the dipole moment derivative at their common equilibrium bond length re. In order to justify eq 1, they argued that the quadratic and higher terms in the series expansion of the dipole moment function in powers of ( r - re), occasionally referred to as electrical anharmonicity, could be neglected. The expectation value of the bond length ( r ) , over any vibrational level, u, is invariably larger than re, more so for the normal than the deuterated molecule, which lies lower in the anharmonic potential well. It follows that A (r)@the difference in mean bond length over the lowest vibrational level, it negative. They determined the dipole moments of HCI and DC1 experimentally and, finding Ap to be positive, concluded that per must be negative. A few years later, Bell3applied similar reasoning to the isotope effect on molecular polarizability and proposed that A ( a ) = a,’A(r)o (2) Refractivity studies of several isotopic pairs were available; in all cases A a was negative, indicating that optical polarizability increases with the bond length. Subsequently, one of us4 extended Bell’s approach to polyatomic molecules, replacing eq 1 by AP = Pk’A(Sk)O (3) kCri
(1)H. 0.Jenkins and S. H. Bauer, J.Am. Chem. SOC., 58 2435 (1936). (2)R. P. Bell and I. Coop, Trans. Faraday SOC., 34, 1209 (1938). (3)R.P.Bell, Trans. Faraday Soc., 38, 422 (1942). (4)E.A. Halevi, Trans. Faraday SOC.,54, 1441 (1958). 0022-3654/82/2086-0654$01.25/0
Here, p i = (ap/L/aSk),is the partial derivative of the dipole moment function with respect to the kth internal symmetry coordinate, the sum being taken over all of the coordinates belonging to the totally symmetric representation, rl. The calculated value of p(ND3) - p(NH3), estimated from spectroscopically derived values of p i and (&)o for the totally symmetric modes of ammonia, agreed reasonably well with the experimental value Ap = 0.015 D, which Bell and Coop2had recalculated from the data of DeBruyne and Smyth? (Inclusionof a neglected factor of 2 in the equation relating the two anharmonicity constants (ref 4, bottom of p 1447), which should read Xk = xk/2(1- q), reduces the estimated value of Ap from 0.018 to 0.013 D.) This agreement with experiment was gratifying, since, if it were also necessary to include quadratic terms like 1/2pk’rA(sk2)o in eq 3, the need to know the curvature of the dipole moment function with respect to all of the symmetry coordinates, not just the totally symmetric ones, would have made the treatment impractical. Thus encouraged, we constructed an instrument with which the dielectric constant difference between a pair of isotopic gases could be measured at various temperatures.‘j The values of Ap and Aa could be obtained directly and compared with those calculated, respectively,with eq 3 and the analogous extension of eq 2: (4)
We were specifically interested in a comparison of alkyl and deuterioalkyl compounds, since the study of their polar properties was complementary to those of kinetic and thermodynamic secondary isotope effects, which were under way in our laboratory7and elsewhere.& The bulk of the relevant results could be rationalized in physical organic terms, if deuteration in an alkyl group increased its effective electropositivity but reduced its polarizability.7,8bThese two opposing effects could be related, via the anharmonicity of the totally symmetric stretching and ( 5 ) J. M. A. de Bruyne and C. P. Smyth, J. Am. Chem. SOC.,57,1203 (1935). (6)E.A. Halevi, E. N. Haran, and B. Ravid, Trans. Faraday SOC.,67, 44 (1971). (7) E. A. Halevi, M. Nussim, and A. Ron, J. Chem. SOC.,866 (1963); see also Y. Bary, H. Gilboa, and E. A. Halevi, J. Chem. SOC.,Perkin Trans. 2,938 (1979). (8)Reviewed by E. A. Halevi, “Progress in Physical Organic Chemistry”,Vol. 1, S.G. Cohen et al., Ed., Interscience,New York, 1963 (a) p 109 jf; (b) pp 114-23.
0 1982 American Chemical Society
Isotope Effects on Polar Properties of HCI and NH,
bending modes, to deuterium isotope effects on the effective Coulomb integral of the H3 pseudoatom in the formal H3=C triple bond and on the overlap integral of that bond’s A ~omponents.~ We were therefore eager to ascertain whether the perdeuterioalkyl halides, for example, would indeed be more polar but less polarizable than their protiocounterparts and, if so, whether the observed differences could be dissected into contributions from the different totally symmetric modes with the aid of eq 3 and 4. We were obliged, however, to begin with a reinvestigation of the isotopic pairs DC1-HC1 and ND3-NH,, because the earlier results had been challenged on theoretical grounds. It had just been conclusively shown,1° on the basis of relative intensities of the P and R branches of its infrared spectrum, that the dipole moment function of HC1 has a positive first derivative and a negligible second derivative. It follows that eq l is valid in good approximation, but that Ap must be negative. Somewhat later, Bartell,” taking the cross terms in the potential energy function of ammonia into consideration, estimated A( Sbnd)to be positiue, and concluded that ND3 was more nearly planar than NHB. It this is so, the positive experimental value of A p is incompatible with the reasonable assumption that NH3 becomes less polar as it approaches planarity, unless considerable weight is given to the quadratic term, neglected in eq 3, involving the square of the symmetric angular displacement and the curvature of the dipole moment with respect to it. By the time this research was under way the classical Debye procedure, of whhich ours is a differential adaptation, had been largely superseded, as far as the dipole moments of small molecules is concerned, by spectroscopic techniques. In particular, Stark effects on rotational spectra were being successfully applied to the evaluation of deuterium isotope effects on the dipole moments of numerous molecule^,^^-^* including several which were of direct interest to us, whereas selected isotopic pairs were being investigated with extremely high accuracy by molecular beam electric resonan~e.’~J~ For this reason, our experimental program was discontinued after completion of the two isotopic comparisons presented in this paper and a study of CD3F vs. CH3F, which will be published ~eparate1y.l~As will appear, they supply information, in each case of a different .kind, which is complementary to that obtainable from spectroscopic studies of the same isotopic pair. In particular, the analysis of the results presented in this paper provides insight into the interpretation of bulk dielectric properties in molecular terms.
Experimental Section Materials. The reference gases (HC1 and NH3) were Matheson Research Grade as were the calibrating gases (H, and Cod. Each of the deuterated gases (DC1and NDd were obtained from at least two suppliers (Merck, Canada; (9)E.A. Halevi and R. Pauncz, J. Chem. SOC.,1974 (1959);A. Ron, E. A. Halevi, and R. Pauncz, J. Chem. SOC.,630 (1960). (10)W.S.Benedict, R. C. Herman, G. E. Moore, and S. Silverman, J. Chem. Phys., 26, 1671 (1957). (11)L. S.Bartell, J. Chem. Phys.,38, 1827 (1963). (12)D.R. Lide, Jr., J. Chem. Phys., 33, 1519 (1960).
(13)J. S. Muenter and V. W. Laurie, J. Am. Chem. SOC.,86,3901 (1964);J. Chem. Phys., 45,855 (1966);J. Am. Chem. SOC.,88,2883(1966). (14)P. A. Steiner and W. Gordy, J. Mol. Speetrosc., 21,291 (1966). (15)L. Wharton, L.P. Gold, and W. Klemperer, J. Chem. Phys., 37, 2149 (1962);J. S. Muenter, M. Kaufman, and W. Klemperer, J. Chem. Phys., 48,3338 (1968). (16)E.W. Kaiser, J. Chem. Phys., 53,1686 (1970). (17)Z.D.Ganoth, DSc. Thesis, Technion-Israel Institute of Technology, Haifa, 1973.
The Journal of Physical Chemistry, Vol. 86, No. 5, 1982 655
Volk Radiochemicals; International Chemical and Nuclear). Before use, every gas sample was passed repeatedly through a “freeze-degas-thaw” static distillation cycle. Chemical purity was assumed when no extraneous peaks were detected in the IR spectrum. The isotopic purity (>98%) of DC1 was confirmed in the same way. The slightly lower deuterium content of ND3 was determined by mass spectrometric analysis of the D2/HD ratio in the hydrogen gas produced by decomposition over a glowing tungsten wire,’* and found to be 95.6% (Volk) and 97.3% (Merck). Measurement. All of the measurements were carried out with the instrument described7 or one identical with it in all important aspects. The procedure varied only trivially, when at all, from that employed in the comparison of HC1 and HBr.5 The instrumental parameter was determined with either C02or N2 and the instrument was “trimmed” with the reference gas to obtain bei,. The former remained substantially constant over the temperature range covered, generally 300-475 K; the latter was reevaluated at each temperature. The square of the frequency ratio, p = f1/f2, was plotted against the pressure, p , common to the gases in the two cells, and the difference between their ideal dielectric susceptibilities was determined from the corrected limiting slope:
in which ei = (ei - l)id$. The difference in total molar polarization (vide infra) between the two ideal gases at standard pressure, expressed in cm3, is APT = ( A e / 3 ) R T
(6)
The measurements were repeated at least once at each temperature, the light and heavy isotopic gases being alternated between the two cells of the instrument. The values of APT, either as defined in eq 6 or after correction as described below, were plotted against 1/T, the slope and intercept being calculated by the method of least squares. The former yielded the isotopic dipole moment difference Ap (= pD - p ~ and ) the latter, APE+A (= PE+A(D)- PE+A(H)) and (= 3hp~+~/4.rrN). Precision and Accuracy. At any given temperature, Ae, as calculated with eq 5 from the slope of the least-squares linear plot, was ordinarily reproducible within fl X lo*. In several experiments, the standard deviation increased appreciably with temperature, occasionally being as large as *6 X lo4. In order to ascertain whether this effect might be significant, these sets of data were replotted, each point being weighted by the reciprocal of the variance of Ae.19 In these cases, the weighted and unweighted plots were practically superimposable,so only the reesults of the latter are cited. The statistically computed errors in A p and APE+A were virtually independent of the functional form of the fitting equation. Since the quantity directly derived from experiment is A(p2),the method yields accurate values of Ap only for strongly polar molecules.6 This is true of our examples, and the random errors in ALL,as evaluated statistically from all of the data on the basis of any of the equations employed, were fO.001 D or less. The random error in APE+A was always within f0.05 cm3/mol, so the precision of Aa is f 2 X cm3/molecule or better. The (18) J. G. Jungera and H. S. Taylor, J.Am. Chem. SOC.,57,679(1935). (19)R. S.Burington and D.S. May, “Handbook of Probability and Statistics”, McGraw-Hill, New York, 1970.
656
Scher et al.
The Journal of Physical Chemistry, Vol. 86, No. 5, 1982
absolute accuracy of the values obtained for the various quantities depends on the validity of the equation from which they were derived. The exploration of this question constitutes the principal feature of this investigation. The imperfect isotopic purity of our ND3 samples required a further correction to 100% deuterium content. The purity was sufficiently high that we felt justified in adopting the theoretically dubious but practically innocuous expedient of multiplying the experimental values of Ap and A P E + A by 100/%D.
I
0.1
-0.2~ .0.3
~
a 0.5
0
1.0
1.5
2.0
2.5
3.0
25
30
jx103 [K-']
Results and Discussion HC1 us. DCI. In their pioneering study, Bell and Coop2 recognized that the classical Debye equation is inadequate for dealing with the temperature dependence of the electric susceptibility of gases as light as HC1 and DC1. Instead, they used the DebyeVan Vleck equation,20which reduces in the case of linear and symmetric-top molecules to b 0
05
IO
15
20
i x IO3 [K-I]
3
a+---
3kT
9k2P
(7)
in which P E + A is the "distortion polarization" and PoRis the "orientation Polarization". The third term, which contains the mass-dependent rotational constant, B, is, in effect, a correction to the Debye equation for the nonclassical behavior of the rotational partition function. The rotational constants, Be = h/89cIe, of HCl and DC1 were taken as 10.591 and 5.445 cm-l, respectively.21a Since the isotopic difference in p has a negligible effect on this term, because p 2 A B >> BA(p2),p could simply be set at Kaiser's16 value: pLoo= 1.1085 for HC1.
APT' = UE+A + UOR
AB (= BD- BH)is necessarily negative, so the effect of the quadratic term in (1/T) is to decrease Ap and raise Aa, thus correcting the tendency of the classical Debye plot to make DC1 appear to be more polar and less polarizable than it actually is, when compared to HC1. Figure 1plots 28 values of Ae, obtained for the isotopic pair DC1-HC1, in the manner described in detail for HCl vs. HBr, over the temperature range 293-440 K. Figure l a is the Debye plot, i.e., APT vs. 1/T, whereas Figure l b is the DebyeVan Vleck plot of A P T c vs. 1/ T. The slightly greater electric susceptibility of HC1 is evident in Figure la, the difference increasing slightly with increasing temperature. Figure l b shows that the Van Vleck "correction" increases the magnitude of APTc relative to A P T , but the "correction term", which is much larger than APT itself at the lower temperature, decreases rapidly as the temperature is raised, reversing the slope of the plot. Leastsquares regression analysis of the two plots yields the following: classical Debye plot (Figure la): Ap = 0.0022 f O.0Ol9D, APE+, = -0.14 f 0.07 cm3/mol, 1OZ6Aa= -5.7 f 2.9 cm3/molecule; Debye-Van Vleck plot (Figure lb): Ap = -0.0055 f O.0Ol9 D, A P E + , = 0.00 f 0.08 cm3/mol, 1026Aa= 0.0 f 3.0 cm3/molecule. (20)J. H. Van Vleck, Phys. Reu., 30, 46 (1927);'The Theory of Electric and Magnetic Susceptibilities", Oxford University Press, Oxford, 1932,p 198. (21)G. Herzberg, 'Molecular Spectra and Molecular Structure", Van Nostrand, New York: (a) Vol. I, 1950,p 534; (b) Vol. 11, 1945,p 437.
Figure 1. DCI vs. HCI: (circles)HCI in cell 1, DCI in cell 2; (squares) DCI in cell 1, HCI in Cell 2; (a) classical Debye plot; (b) DebyeVan-Vleck plot.
The classical plot thus requires DC1 to be slightly more polar, if anything, than HC1; it is certainly inconsistent with Kaiser's16 precise, spectroscopicallydetermined, value for the dipole moment difference between the nonrotating molecules in their ground vibrational levels: poo(DC1)pLoo(HC1) = -0.0052 D. Furthermore, since atomic polarization ( P A ) has been shown on the basis of both classical22 and quantum mechanicalz3arguments, to be substantially mass independent, A a should be directly comparable to that derived from optical refractivity measurements," Le., A a = -0.55 X cm3/molecule or, in terms of molar electronic polarization, A P E = -0.014 cm3/mol. Instead, Figure l a yields what appears to be a statistically significant polarizability difference that is too large by a factor of ten. In contrast, the dipole moment difference obtained from Figure l b is in complete accord with that of Kaiser,16 which, as he pointed out, is fully consistent with eq 1. Moreover, the very small value expected for A a falls within our experimental uncertainty. Thus, half a century after Van Vleck's derivation of the theory of electric susceptibility from quantum mechanical principles, we have provided what appears to be the first direct experimental confirmation of two of its principal features. Insofar as an isotopic comparison can show, we have demonstrated the following: (a) The dipole moment derived from bulk susceptibility measurements is indeed pW, the expectation value of the nonrotating molecule in its vibrational ground state. (b) For light molecules, diatomic hydrides in particular, electrical susceptibility can only be correlated with the spectroscopically determined value of pm by means of the Debye-Van Vleck equation, the classical Debye equation being inadequate. ND3 us. NH3. de Bruyne and S m ~ t hplotting ,~ PTof each gas separately against 1/T in the range 273 5 T I 433 K, obtained Ap = +0.03 D and A P E + A = -0.9 cm3/mol. Bell and Coop2 recalculated these values from the temperature dependence of PTc,yielding AI = +0.015 and U E + A = -0.3 cm3/mol. (22)I. E.Coop and L. E. Sutton, J. Chem. SOC.,1269 (1938). (23)K.H.Illinger and C. P. Smyth, J . Chem. Phys., 32,787 (1960).
Isotope Effects on Polar Properties of HCI and NH,
,/
0.8
-0 6981
/
9
The resultant plot (Figure 2d) further reduces the size of Ap slightly and of A P E + * markedly. The average values from eight sets of measurements, each plotted as in Figure 2d and corrected for incomplete deuteration as described in the Experimental Section, were as follows: A p = +0.0135 f 0.001 D; APE+*= -0.056 f
,
1
3
2
I
I/Tx103[K-']
Figure 2. Sample set for ND, vs. NH,: (a) classical Debye plot; (b) Debye-Van Vleck plot; (c) Debye-Van Vleck plot, corrected for vibratlonal excltatlon. Kestimated with eq 10 (five iterations). (d) same as (c), but K calculated wlth eq 11.
Figure 2 depicts one of eight sets of measurements with ND, in one cell of our instrument and NH3 in the other, in the somewhat higher temperature range 313 I T I 478 K. Figure 2b, calculated with p H = 1.468" D and AB = -4.803 cm-',21b bears the same relation to Figure 2a as Bell and Coop's recalculation does to De Bruyne and Smyth's original analysis of their data. It diminishes an excessively large difference in polarizability and halves the dipole moment difference. (Substitution of the more recent25 value, 1.475 D, makes no significant change.) It was then realized that the frequencies of the symmetric bending mode of NH, and, particularly, of ND3 are sufficiently low that significantly vibrational excitation had to be taken into account at our temperatures, so as to convert A P T c and APTo, the isotopic difference in the expectation value of PT for a mole of classically rotating molecules in their lowest vibrational level. The corrected equation, derived in the Appendix, ie APT0 = AFJTC
-E[
VD
-
VH
1
T eXp(hvDC/kT) - 1 eXp(hVHC/kT) - 1 (9) At the time that this study was being carried out, the only practical way of estimating the isotope-independent parameter K was from the isotopic dipole moment difference
K-
TAPOR' ._
2(vD
- VH)
. ,
Since TAPORois the slope of the plot of APTovs. 1/T, the calculation had to be iterative. The slope of the plot of eq 8 was used to obtain a trial value of A P E + A o and initial values of *ORo for each temperature. The resultant trial values of A P T were fit to eq 8 to obtain new values of A P E + A o and @ORo. Convergence to well within our experimental error was achieved after five iterations. The (24)D. K. Coles, W. E. Good, J. K. Bragg, and A. H. Scharbaugh, Phys. Rev., 82,877 (1951). (25)F. Shimizu, J. Chem. Phys., 51, 2754 (1969).
(26)D. M. Dennison, Reus. Mod. Phys., 12, 175 (1940). (27)E.A. Halevi, E. N. Haran, and B. Ravid, Chem. Phys. Lett., 1, 475 (1967). (28)T. Yoshino and H. J. Bernstein, J.Mol. Spectrosc., 2,213(1958). (29)K. Kuchitsu, J. P. Guillory, and L. S. Bartell, J. Chem. Phys., 49, 2488 (1968). (30) Y. Morino, K. Kuchitau, and S. Yamamoto, Spectrochim. Acta, Part A, 24, 335 (1968).
658
The Journal of Physlcal Chemktry, Vol. 86, No. 5, 1982
Commerce, National Bureau of Standards, administered by the Technion Research and Development Foundation. We are indebted to Dr. E. N. Haran for ensuring the efficient performance of the apparatus which he designed, to Professors A. D. Buckingham and H. Friedmann for valuable comments on points of theory, and to Professor S. Kimel for critically reading the manuscript.
Appendix. Correction for Vibrational Excitation in NHa and NDS The total molar polarization is the weighted average of its expectation value over the occupied levels of the totally symmetric vibrational modes. Of these, vl, the symmetric stretching mode, is not appreciably excited at our temperatures, so only the effect of populating higher levels of v2, the symmetric bending mode, need be taken into account. If nu is the fractional number of moles in level u
Scher et at.
( 4 2 ) u= (Q2)u/m = 47r2v2c2(Q2),/k, = (u + f/2)huc/k, (A91 Here, we have used the basic relation 4?r2v2.=k/m and the standard expression for the mean square amplitude of a normal coordinate ( Q2), = (u + 1/2)h/47r2u~.32 The mean dipole moment, including the quadratic term, is given by
and, making use of (A3) and (A4) PORU
m
in which PTu is the value which P T would attain if all of the molecules were in state u. Such a situation can be realized only for u = 0, if the temperature is sufficiently low. PTo
= PE+Ao
+ PoRO
+ +
= POR~K(u f/2)~/T = PoRO K u v / T (All)
Equation A3 can now be rewritten
(A2)
PE+A is quite insensitive to angle bending,28so, to a good approximation
PT
= PE+Ao
+ PoRo + Ku - un, T up1
(A12)
Neglecting the slight effect of anharmonicity on the vibrational population, and defining 8 = exp(-huc/kT), we obtain m
nu = B"/CPu= P"(1- 8) 0
(A13)
m
For a polar molecule
PTO= PT- KVP( 1 - 8)CUfl"'
(A14
The sum in eq A14 is simply the binomial expansion of (1 - P ) - 2 , so Strictly speaking, pe in eq A4 should be replaced by the expectation value of the dipole moment over the lowest vibrational level of the symmetric stretching mode which is isotope sensitive. The effect of stretching anharmonicity on the dipole moment of ammonia is so small: however, that the mass-insensitive pe can be safely used instead for the purpose of this correction. Moreover, we regard v2 as if it were a pure bending mode, which depends solely on the angular coordinate 4, defined
4=
1 -[(a1
fi
- &e)
+ (a2-
Lye)
+ (a3 - a e ) l
(A5)
to which we assign the anharmonic potential energy function
u, = %k,42 - g,43
(A6)
Following Bartel1,ll we apply Ehrenfest's theorem31 ( d U,/d@) u = 0
(A7)
and obtain the familiar relation between the mean and mean-square amplitudes
( 4 )u = (3g,/k,) ( 4 2 ) u
(A8)
Since 4 is being regarded as a mass-reduced normal coordinate4 (4 = Q2m2-lf2,in which m2 is the "characteristic mass" for v 2 ) we can write, dropping the subscript 2 (31) P. Ehrenfest, 2.Phys., 45, 455 (1927).
and the correction to the isotopic difference is
The isotope-independent factor K defined in ( A l l ) is, in principle, obtainable from spectroscopic data, but is more simply regarded as an empirical parameter. If the dipole moment in the two lowest vibrational levels of either isotopic molecule is known, it follows from ( A l l ) K / T = (POR1 - P O R o ) / v (A171 Less reliably, it can be estimated from the experimentally determined isotope effect on PoRo,since, unless the shorter mean bond length of the heavy molecule displaces the effective potential curve for the bending coordinate" K/T ~A€'OR~/(U, - UH) (A181 The correction for vibrational excitation and that for nonclassical rotation are independent, and therefore additive, so APT can be replaced by APTc, producing eq 9 in the text. (32) E. B. Wilson, J. C. Decius, and P. D. Cross, "Molecular Vibrations", McGraw-Hill, New York, 1955, p 290.
