74
TATSUO MIYAZA WA [CONTRIBUTION PROM
THE
AND
KENNETHS. PITZER
Vol. 81
DEPARTMENT O F CHEMISTRY, UNIVERSITY OP CALIFORNIA, BERKELEY]
Low Frequency Vibrations, Polarizability and Entropy of Carboxylic Acid Dimers' BY TATSUO MIYAZAWA AND KENNETH S. PITZER RECEIVEDJUNE 19, 1958
A normal coordinate treatment of the low frequency vibrations of formic acid and acetic acid dimers was made using the rigid monomer model. The potential function was expressed in terms of the H..O stretching and the 0-H. 43 and €1. .O=C-O bonding potential constants, which were calculated from the observed infrared frequencies and the entropy of dimerization. The H .O stretching force constant was found to be 0.33 X 1 0 6 dyne/cm., which is 5% of the H-0 stretching force constant. The low frequency vibrations are described in terms of the translational and rotational amplitudes and the fractional potential energy associated with each potential constant. The assumption of the rigid monomer model was examined and found to be satisfactory. The value of atomic polarization calculated on the assumption of independent rigid monomers with the monomer dipole moment is somewhat too small but plausible chauges in the dipole moment of each monomer unit yield agreement with the observed polarization.
A carboxylic acid monomer R-COOH has nine and the dimer (R-COOH)2 twenty-four degrees of vibrational freedom other than the internal vibrations of the R groups. As the result of an electron diffraction study the dimer molecule was found t o be planar2 with the point group (2%. Therefore those twenty-four vibrations of dimer molecules are grouped into nine a,, eight b,, four a, and three b,vibrations. Of these seven a,. seven b,,, two b, and two a, vibrations can be correlated with the corresponding internal vibrations of two monomer units, while the remaining two ag, one b,, one bg and two a, vibrations may approximately be described as the translational and rotational vibrations of two monomer units against each other. I n the present paper, the nature of the vibrations of the latter group will be discussed. Due t o their low frequencies, few measurements have been made of these vibrations of carboxylic acid dimers. Bonner and Kirby-Smiths observed a Raman line at 232 of formic acid dimer in the vapor phase. Infrared absorption of trifluoroacetic acid dimer was studied by Oetjen,4 but no strong band was observed in the region 90-250 crn.-l. However, Millikan and Pitzer6 observed two infrared bands of formic acid dimer a t 237 and 160 ern.-' and a band of acetic acid dimer a t 188 cm.-'. Normal coordinate treatments on these low frequency vibrations have been made by Halford6 and by Slutsky and Bauer.' In each of their treatments approximations were made which, although appropriate a t that time, should now be removed in order t o allow more realistic comparison with the recently obtained spectral frequencies. Low Frequency In-plane Vibrations.-The vibration frequencies and normal coordinates may be calculated by the G F matrix method of Wilson.8 I n ring molecules such as carboxylic acid dimer, however, the total number of internal coordinates exceeds that of the vibrational degrees of freedom, and it is preferable to express the inverse kinetic (G) and potential energy matrices (F) in terms of (1) This research was assisted by the American Petroleum Institute through Research Project 50. (2) P. W,Allen and L. E.Sutton, Acto Crysl., 3, 46 (1950). (3) L. Bonner and J. S. Kirby-Smith, Phys. R w . , 67, 1078 (1940). (4) R. A. Oetjen, quoted by N. Fuwn and M. L. Josien, J . O P f . SOC.A m . , 43, 1102 (1953). (5) I?. C. Millikan and E. S. Pitzer, THISJOURNAL. 80, 3615 (1958). (6) J. 0. Halford, J . Chetn. Phys., 14, 395 (1946). (7) L. Slutsky and S. H.Bauer, THIS JOURNAL, 76, 270 (1954). (8) E. B. Wilaon, Jr., J . Ckrm. Phyr., 7 . 1047 (1939); 0 , 76 (1941).
appropriate symmetry coordinates which are orthogonal to redundant coordinates. Redundant coordinates of carboxylic acid dimer in the ag symmetry class are linear combination of the interbond angle coordinates only, while symmetry coordinates and redundant coordinates in the b, class are to be expressed as linear combinations of the interbond angle coordinates as well as of the bond stretching coordinates and the coefficients are quite complicated and impractical to calculate. Accordingly it is more practical to deal with each monomer unit as a rigid body. As a first approximation, then, the kinetic energy T' of the in-plane low frequency vibrations may be expressed as 2Ti
&f[(Ak~)' f ( A ~ B ) ) ]+ I z (;A2
+
LB')
(1)
where M is the mass of each monomer unit, R the distance between the center of mass of the dimer and that of a monomer unit, w the in-plane rotational displacement of an individual monomer unit around its own center of mass,, IZthe corresponding moment of inertia, and the subscripts A and B refer to the monomer units A and B, respectively (see Fig. 1). The internal symmetry coordinates in the ag symmetry class are and
SI = (ARA
+
s* = (W*
+
ARB)/^'/: WB)/2'/1
(2)
and those in the b, class are SO =
(WA
- W3)/2'/3
(3)
The coordinate ( A R A - A&3)/2'/'is redundant. The elements of the inverse kinetic energy matrix expressed in terms of these symmetry coordinates are Gii = 1/M Gs=l/Iz
and
Gij = O(for i # j )
(4)
The symmetry coordinate .$ yields non-zero angular momentum of 2'/1Iz&, which is to be canceled out with an over all rotation of the dimer (Iz':its moment of inertia) in the opposite sense with an angular velocity of 2'/~Iz$2/Iz'. The kinetic energy associated with S2is, then, equal to a ~ I ~ 3 and ~ ~ / 2 the corresponding element of G is G22 = 1/Iz& where az = 1 - (2I,/Iz'). If this correction is ignored, Gz2 = 1/Iz. It may be mentioned here that the correction for non-zero angular momentum is necessary only for the symmetric vibrations and not for the antisymmetric vibrations since the symmetry coordinates in the latter case yield zero angular momentum.
VIBRATIONS, POLARIZABILITY AND ENTROPY OF CARBOXYLIC ACIDDIMERS
Jan. 5, 1959
75
The potential function Vi for the in-plane vibrdtions is 2Vi = K [ ( A r l #
+ (Ar,)'] 4-HI'[(A~IW#4+ Hn'[(Acro~e)~ + (AcYsM)*](5)
(Aaw)']
06.- .- ..- .-H,
where r19 is the bond distance Ol-Hg, and a198 and a 0 1 9 are the interbond angles Og-Hg . 0 1 and Hg OI-CO, respectively (see Fig. 1). All the cross terms were neglected. The internal coordinates just given may be expressed in terms of the symmetry coordinates (see eq. 2 and 3). Then the elements of the potential energy matrix F are calculated as
-
F11 = FIz = F z ~= Fst =
+ +
+ + +
4K cosz 6 HI' Hz') sin' 6]/r10* 4 K R cos 6 sin 8 - HI' 4-H2')R cos 6 sin 6]/rle2 4 K R s sinz 8 HI' H2')R' cos* 6]/rlo2 4Ky2 XI^ Hz'xt8))/rle2
+
and Fz,
+
=z
Fir = 0
(6)
where R = ( X A ~ Y A ~ ) ' /and Z 6 = tan-' (YA/XA) and X A and YA give the position of the center of mass of the monomer unit A. Low Frequency Out-of-plane Vibrations.-The first approximation to the kinetic energy To of the out-of-plane low frequency vibrations may be expressed as 2T0 = IXX(X.4'
+
'XB')
- 2IXY('XAiA X B ~ -k )
+
//
Fig. 1.-Coordinates
+
$go)
(7)
of carboxylic acid dimer.
eol the
internal rotation angle around the Co=Ol bond. All the interbond angles are assumed to be 120' except for a(0-H 0) = 180'. The internal coordinates, expressed in terms of the symmetry coordinates, are substituted into eq. 12 to obtain the elements of the potential energy matrix
--
F.
F44 =: 4(HiN4- ~ H z ' / ~ ) Y ~ / ~ I Q ~ FG = ~ ( H I ' X I Y C~ H Z ~ X C V ~ ~ ) / ~ I Q ~ F,s = 4(HlWX1' 4H2"mz/3)/r~2 Fta = Fss 0 FM = HI' 4H2'/3)R2/r~9z (13)
-
+
=i
,
IYY($A'
0\3I
+
Up to this point we have treated each monomer where x and $ are the rotational displacement of a unit as a rigid body. This approximation is less monomer unit around axes through the center of mass of the monomer and parallel to the X- and P- satisfactory for the out-of-plane than for the inaxes, respectively, and I X ~IYY , and IXY are the plane vibrations because of the relatively low frecorresponding moments of inertia and product of quency and force constant associated with the 0-H inertia. The internal symmetry coordinates in the torsional motion. Consequently the validity of the rigid monomer model was investigated by a a, symmetry class are complete calculation for the out-of-plane motions. SI a ( X A -k XB)/2'/' The mathematical details are given in Appendix I and and the results are discussed below. SS = ($'A 'kB)/2'/z (8) Entropy of Low Frequency Vibrations.-Since and those in the b, class are only a few of the low vibrational frequencies have SB = I - ( X A - X B ) sin 6 (+.4 - + B ) C O S ~ ] / ~ ' / Z (9) been observed in spectra, it is necessary to use the The coordinate ( X A - X R ) COS 6 ($A - $B) sin 6 entropy data to assist in determining potential is redundant. The elements of the inverse kinetic constants. The dissociation equilibria of carboxylic energy matrix expressed in terms of these symmetry acid dimers have been measured by many workers. codrdinates are Recently Taylor and Brutonlo measured dimerizaG.4 = I Y Y / ( I X X I Y Y -IxY~) tion equilibria of formic acid and acetic acid over Gib = I X Y / ( I X X I Y Y - IxY') the temperature range 50-150' a t relatively low Ga I X X / ( I X X I YY Ixx') (10) pressures. The entropies of the low frequency viGee.* = ( I y y si112 6 - 2 1 x y sin 6 cos 6 + brations of the two acids were calculated from the Ixx cos) 6)/(1XXIYY - I X P ) measured values of the entropy of dimerization. and The results of these calculations are shown in Table G4e = Gm = 0 I. The symmetry coordinate SSyields non-zero anguThe translational and rotational entropies of lar momentum and the apparent G ~ Bmust * be cor- formic acid monomer were calculated by the use of rected as in the case of SZ. In this case the deriva- the bond distances and interbond angles detertion of the correction is much more complex;g mined by a microwave measurement.'l The valthe result is ues for the dimer are given below. The vibrational entropies of the monomer and dimer of formic acid were calculated by the use of the frequencies obwhere Ixx', etc., are the moments and product of served by Millikan and Pitzer.6012 A similar calcuinertia of the dimer. lation was made for acetic acid where the change in The potential function V o used for the out-of- vibrational entropy on dimerization was assumed plane vibrations is to be the same as in the case of formic acid. The 2V0 = Hi'l(A.Ples)z 3- (ABaa)*l Hz'[A.Bol)z bond distances and angles of acetic acid monomer
+ +
+
+
+
( A B S E ) ~ ](12)
where
is the interbond angle Os-Hg
. O1 and
(9) This formula was obtained by induction and verified numerically by appropriate reduction of the exact C matrix of Appendix I.
(IO) M. D. Taylor, THISJOURNAL, 78, 315 (1951); M. D. Taylor and J. Bruton. ibid.. 74. 4151 (1952). (11) R . G.Lerner, B. P. Dailey and J. P. Friend. J . Chem. Phys., 26. 680 (1957). (12) R. C. Millikan and K. S. Pitzer, ibid..27, 1305 (1957).
76
TATSUO ~ I I Y U A WXSD . A KENNETH S. PITZER TABLE I
ENTROPY O F DIMERIZATION AT 373°K. ( I X CAL./DEG. AIOLE)
---
Monomer
Entropl--------? Dimer Dimerization
(A) Formic acid Translation rotatioil 6U.23 67.04 - 53.42 High frequency vibration 1 81 2.46 - 1.16 Low frequency vibration 18.60 +18.60 Thermal -3.5.98" (B) .Icetic acid Translation rotatioil 63, 72 69.50 - 5 i 94 High frequency vibratiuri - 1 ,It? Low frequency vibration ... 22.26 t22.26 Thermal -36.84" Kef. 10. Assumed to be the same as in the case of formic acid.
+
+
mere assumed to be the same as the corresponding quantities in the dimer. Entropy of Dimerization by Classical Mechanics. --At the temperatures of interest for carboxylic acids in the vapor state the low frequency vibrations may be treated by classical statistical mechanics in good approximation. For example at 373°K. the error of the classical calculation of the entropy of the six low frequency- vibrations is only 0.1 cal./deg. mole for formic acid, and it would be less for any other carboxylic acid. The classical formulation of the entropy of dimerization explicitly cancels many extraneous factors such as masses and moments of inertia and shows the essential relationship to the force constants of the bonds connecting the monomer units. The dimensions of the monomer are assunied to remain unchanged in the dimer; this is not exactly true in the case of formic acid, where data are available, but is a useful approximation. The classical expression for the entropy of a viliratioiial mode $, = K [I + I ~ i ( k T / h c v i ) ] (14) is used together with the conventional equation for trarislational and rotatioiial entropy. Also the product of the vibration frequencies is obtained irum the pro(1ticL oi the determinants of G and F. as ATtcr extei1sii.e caiicclliition tlic e11tro1)y ol' tliiiieriLation is
+
A? = MV' I
