J. A. Creighton University of Kent Canterbury, CT2 7NH United Kingdom
I
A Vibrating Molecular Model
An important part of an elementary course in vibrational spectroscopy is to help students to visualize the motions involved in the various vibrational modes of simple molecules. The modes of a linear svmmetric AB9 molecule. such as C07 can be simulated by a &echanical mohel, and apart from the value of such a model in disdavine the modes. the fact that the vibrations ascribed to t'he"m&ule may be seen to be oerformed also hv a macroscooic svstem increases their credibility for some students. Such H vibrating molecular model is described here which is both inexoensive and easy to build. In addition to demonstrating the idternal vibrations of a COa molecule, including the circular or elliptical paths described by the atoms in the degenerate bending mode, the model also shows the external modes of the molecule trapped in a cage, the phenomenon of resonance (with the driving motor), and (sometimes) bond dissociation. The, mudel cvn. 1.5. The modes are then well tionallv such that m spaced, and occur in the order of frequencies u.,,, > us, > 6 which is commonly found for molecules. The equations of motion of the model, including a worked example, are given below. and mav be found h e l ~ f ufor l o~timizinp -a oarticular . model. For an elementary course demonstration where the model is used to simulate the modes of a COz molecule, the motor
I Thinner bandsmayalso be used provided their unstreWled length is also less, so that their force constants and the tension in the model are not significantly diminished. Thus bands which when unstretched were of length 13 cm and of total cross-sectional area 2.5 mm2 were also found to he suitable.
620 1 Journal of Chemical Education
IS-
--
I
Figure 1. The vibrating molecular modei
Figure 2. The vibration frequencies of a mlecular model ploned againsl Ihe ratio m2/m1. where m, and m2are me masses of an outer ball and of the central ball, and m, = 10 g, f = 10 Nrn-', t = 3 Nand r = 0.5 rn (see text).
may be started a t maximum speed and the speed reduced slowlv so that each mode is activated in turn. The laree din ameter and low momentum of the polystyrene balls ensure that the vibrations are quickly air-damped when not in resonance with the motor (a model constructed with more massive weiehts and sorines was found to "dnve" the motor and to inte-rfere withits Geed control). With the masses, force constrant, and tension similar to the values given above, the frequencies of the vibrations (see Fig. 2) are sufficiently low
for the details of the motions to be easily appreciated, but if a substantially slower display is required this may be achieved by illuminating the model with a strobosco~iclight source. -At n more advanced level the mndel may he used to dem)..A onstrate the modes of a triatomic molecule of point group - [ trapped in a cage. The three internal vibrations transform as Z,+ (symmetric stretching), Z.+ (asymmetric stretching) and n, (bending), and there are in addition three external vibrations, consisting of two modes belonging to Xu+ and nu in which the molecule as a whole translates, respectively, along and perpendicular to the molecular axis, and one IIg libration. To comply with the symmetry properties of these irreducible representations the Z,+ and & +vihrations, which are symmetric with resoect to all the reflection o~erationsc,,,must involve motions of the atoms only along the molecul& symmetrv axis. In the degenerate n, and nuvihrations however, where the motions may he resolied intotwo components, one of'u,hich is symmetric tin the plane) and the other of which is antisymm&ric (out of with respect to any a, operation (the total character for a, is thus zero), the motions must be perpendicular to the molecular axis for small vibrational amplitudes. These requirements are borne out hy the model. Finally the model may be used as the basis for an example in vibrational mechanics. in which the eauations relatine the force constants to the frequencies ma; be derived h; the I t is simplest standard GF matrix method used for molec~les.~ to use a cartesian coordinate representation for the model, and although it is more usual for molecules to use a coordinate system based on bond length and interhond angles, an example worked in cartesian coordinates is of value for making clear the relationship of the G matrix to the kinetic energy, which is less easily grasped in a bond length-bond angle coordinate representation. As a result of their orthogonality the parallel and perpendicular motions can be treated seoaratelv. and the oarallel motions are here considered first. he changes in length of the honds linking an outer ball 1 to the frame and to the central ball 2 are 2 , and 2 2 - z l , respectively, where 2, denotes a loneitudinal disnhcement of ball i from its eauilibrium ~osition. The total pbtential energy VL for lon$tudinal motion is therefore given as a function of the force constant f by ~
~
['" 1'
+ - Z I P + - 22)2+ = 2f(z,2 + + - 2223)
2VL = f[212
(22
222
(23
2321
(1)
2 3 2 - 2,ZZ
The F matrix, defined by 2 V = X F X , where X is a column matrix of cartesian displacements zi, is therefore
-f
2f
-
The corresponding G matrix, defined by 2 T = X G - ' X , is a diagonal matrix of the inverse masses, as follows simply from the kinetic energy expression ~ T= L rn,ilz+ rn2iz2
+ rn3is2
The matrices F and G may be transformed in the usual way2 to the matrices Y and? of a symmetry coordinate representation, using Y = UFU and 9 = UGU with
Thus
r 2f
0
01
-
and 9 = G . The frequencies ui are given2 by the roots hi = 4 r V i of the determinantal equation 199- XEI = 0. One thus obtains A, = 4r2u2,, = 2flrnl
(2)
for the Z,+ mode, and for the two &+modes the equation for the roots A2 and ha reduces to rnlrn2AZ- 2f(rnl
+ mz)A + 2f2 = 0
(3)
where A2 = ~ T ~ v ~ , , , ,and hg = 4r2[vtrsns(Zu+)I2. The potential energy for transverse motion is given by
+
+
~ V= T 2t(x12 zz2 .xs2 - ~ 1 x 2~zzdlr
(4)
where t is the tension and r is the bond length, as may be verified by comparing - a V ~ I a x i with the transverse force experienced by ball i as a result of displacements X I ,x s and xg. It may he seen by comparing (4) with (1)that replacement off by tlr in the F matrix for longitudinal motion gives the F matrix for transverse motion, while it is easily shown that the G and U matrices for the two types of motion are identical. It thus follows that the roots A j of the secular equation for transverse motion are tlfr times those for longitudinal motion. Thus for the II, mode ulib.
=~ ~ ~ ~ ( t l f r ) ' ~ ~
(5)
and similarly for the two nu modes
,.,.
6 = ua.,m(tlfr)1'2
ut,..(II,)
=
[U
(6)
(&+~)](tlfr)1~2
(7)
Students may verify these equations experimentally by comparing the observed frequencies with the frequencies calculated from the force constant and tension obtained from static loading measurements. For the model constructed by the author, one of the hands was hung vertically and stretched to a length of about 40 cm by suspending from it a weight of 300 g. Increasing the weight to 400 g caused an additional stretching of 6.9 cm. The force constant was thus 14.2 Nm-'. The masses ml and m2 were 10.0 and 17.2 g. Thus from eqn. (2) u,,, = 8.5 Hz, and from the values of A2 and Aa obtained = 9.9 Hz and uw,,(Z,+) = 3.9 Hz. from eqn. (3) one finds v., The tension in the model was determined by measuring the transverse displacement rAm2lt of the central hall which results from increasing its mass by Am*. For the author's model the bond length r was 42.5 cm. and a transverse load of 50 g caused a displacement of 7.0 cm in the central ball. The tension was thus 2.98 N a n d from eqns. (5)-(7) one finds ulib, = 6.0 Hz, 6 = 6.9 Hz and Y, (nu)= 2.8 Hz. These values were all within 10%of the observed frequencies of the model. It is interesting to note that in the early development of the studv of molecular vihrations, quite elaborate vibrating mod& were constructed to help with the interpretation of the Raman spectra of molecules such as carbon tetrachloride and henzene,hnd later as analog devices for the calculation of molecular force constant^.^
,..,
2 Wilson, E. B., Decius, J. C., and Cross, P. C., "Molecular Vibrations," McCraw-Hill, New York, 1955. Ketterine. C. F.. Shutts. L. W.. and Andrew. D. H.. Phvs. Reu.,
36,531 (1936.
MacDougall, D. P., and Wilson, E. B., J. Chem. Phys., 5, 940 (1937).
volume 54, Number
10, October 1977 1 621
