Normal Modes-of
Vibration and Relaxation
K. Razi Naqvi Department of Physics, University of Trondheim-AVH, N-7055 Dragvoll. Norway The similiarity between the kinetics of coupled first-order equilibria and normal modes of vibration has been discussed but , a satisfactory elementary treatby many authors (ld) ment does not seem to be available. Accordingly, I present a simple account of the topic here. Two Reaction Schemes Let us first consider the following reaction scheme:
Using Ai to denote the concentration of A;, we can write the pertinent rate equations as dA,ldt = -a@)
(2)
where a = kAl
- k'Az
(4)
We recall that a t equilibrium the concentrations of both reactants become, by defmition, time:independent; that is to say, one can write dA1ldt = 0 = dAzldt, where the tilde symbolizes the equilibrium value. I t is not difficult to juggle eqs 2 and 3 so as to get the two differential equations that follow: dAldt = 0 ( A I A, + A,) (5) The total concentration, A, remains consmt throughout, and a , a linear combination of A , and A?, must attain, like A,. a stationary value in the equilibrium state; thus daldt must vanish at equilibrium. Now, according to eq 6, dorldt = 0 implies a = 0 , and vice versa. Thus a is a measure of the departure from equilibrium of the system under consideration, and the relaxation constant ( k k') determines how fast or approaches the equilibrium value if a ( 0 ) Z 0. Next, we consider the following scheme,
+
**
k,
k2
kr
Bl+B,=B3
But eqs 13 and 14, being of the same form as eqs 2 and 3, can he treated in the same manner; this gives a new pair of equations in lieu of eqs 13 and 14:
Thanks to some elementary manipulations, we have contrived to replace the original three rate equations, eqs 8-10, by another set, consisting of eqs 15-17. Whereas B is the analogue of A, B- and 0 account for deviation from equilibrium oithe three-comoonent svstem in exactlv the same wav as does or for the two-componentsystem: one must have B- = 0 = Bat eouilibrium. The three-com~onentsvstem can thus be said tobossess two modes of relaxation, say mode a and mode b: if p(0) = 0 and B-(0) f 0 , the system moves toward equilibrium with a relaxation constant k l corresponding to mode a, the slower of the two modes; on the other hand, if B-(0) = 0 (i.e., B1 = Bg) and p(0) z 0 , approach to equilibrium will be more rapid, governed by the time constant (kl 2kz) of mode b. If neither B-(0) nor p(0) equals zero, both modes will come into play. It is worth observing that, in consequence of eq 14, 0 = 0 implies that, in mode a, Bz remains stationary throughout. We recapitulate: though the individual concentrations A; and B; do not exhibit mono-exponentialdecay, certain linear combinations of these variables (viz., a for eq 1, B1- Ba and p for eq I ) , which we found merely by juggling with the rate equations, not only obey the law of exponential decay but also serve as indicators of deviations from equilibrium. The solutions of the new, equivalent set of differential equations can he immediately written down; a little shuffling, too obvious to be detailed here, provides the expressions for the actual concentrations of the reactants, which are displayed below mainly for completeness. ~~~~~
~
~~
~~
~~
.
+
(7)
for which the rate equations read dB,/dt = -k,B,
+h a z
(8)
After defming the symbols,
-
Bi.B,*B,
and @
k,B+ - 2k&
(11)
Two Vibrating Systems We change course now and proceed to discuss longitudinal vibrations of two familiar mechanical systems: ( 1 ) two masses, m and m' connected by a spring of elastic constant K and ( 2 ) a central particle of mass M coupled (by identical springs of elastic constant K) to two outer particles, each of mass m. The equations of motion for the two-particle system are
(12)
one gets a new equation at each of 'the following steps: ( 1 ) adding eqs 8 and 10, ( 2 )substituting p on the 1.h.s. of eq 9, ( 3 ) subtracting eqs 8 and 10:
which imply that
+
(d2/dt2)(mx, m'x,) = 0
(23)
We begin o w manipulations by defming a new coordinate
x,
Volume 66
Number 9
September 1989
703
so that (m
+ mr)X = (mx, + m'x2); eq 23 now gives (rn
+ m')d = 0
or d = O (25) Multiplying eq 21 by m' and eq 22 by m, and subtracting the resulting equations, one gets, after minimal rearrangement, &- = -Kx(26) where P = mm'l(m + m') (27) and x- = ( x , - xZ) (28) The integration of eqs 25 and 26, a familiar exercise, shows that while X, one of the new coordiiates, describes rectilinear motion in accordance with the equation x = c,
+ c2t
The above relations are too familiar to need elaboration. The variables x- and x., usually termed normal coordinates (6).should really be called natural coordinates, as suggested by Desloge (7). It is evident that X, x- = ( X I - x3), and x, = (XI+ x3 - 2x2) are analogous to B, B- = (B1 - B3),,and B = k l ( B ~ +Ba) - 2kzB2, respectively; w12 = Klm and wz2 (Klm) (2KIM) are also somehow related to XI kl and X2 = kl + 2k2, respectively, but the actual kinship remains to be established (see below). A Second Look at the Rate Equations
If we recall that the coordinates xi appearing in the equations of motion are all distances measured from the respective positions of equilibrium, it becomes natural to ask whether the similarity between the mechanical and chemical systems could not be made still closer by working, not with the concentration variables Ci (C = A or B), but with A; = Ci - Ci, so that the equations of mass conservation, viz., eqs 5 and 16, now become
(29)
the other new coordinate x- executes simple harmonic motion, x- = ~ ~ (expf+iwt) 0 ) (wZ= KIP) (30) Retrospection reveals that X and A play similar roles, and so do x- and a;likewise, w2 = (Klm) + (Klm') appears to be the vibrational counterpart of the relaxation constant k k'. The equations of motion for the three-particle system are well known (6):
+
I t turns out that the rate equations remain valid if Ci is replaced by A; (for this substitution amounts to adding zero to both sides of each equation). If one now takes a second look at eq 7, and introduces the following symbols, ,onecan easily discern the following features, which are reminiscent of eqs 42 and 43: Mode a(A = 0 = x): Al+&=0=4 (42') Mode b(A = 0 = A-):
Addition and subtraction of equations takes one from the last three equations to the following set:
x = [rn(x, + x3) + Mxz]I(2m+ M)
d=o rnf- = -Kx-
x-=x1-x3
(34) (35)
Multiplying eq 36 by M and eq 32 by 2m, and suhtracting the equations so obtained, one arrives, after dividing throughout by 2m + M, a t the following result: m,E, = -Kzn (37) where xn=(x,-Zx2+x3) (38) and rn. = mMl(2m + M) (39) We can now assert, without solving any equations, that the three-particle system is characterized by a zero-frequency translational mode and two modes of vibrations, say mode 1 ' and mode 2, with characteristic frequencies wi (i = 1.2): and If the system is oscillating in a particular mode, one must set to zero the coordiiates for the other two modes; using this recipe and the definitions given in eqs 35,36 and 38, one now gets the following description of three modes: mode 0 (x- = 0 = x.): xI = zZ= x3 (41) mode 1(X = 0 = x.): mode 2 (X = 0 = x-):
704
x1 + x, = 0 = x, x, = -(2Mlrn)xZ= z3
Journal of Chemical Education
- -
+
(42)
(43)
A, = -A&
= A3
(43')
Complex Decay Constant and Complex Frequency It remains for us now to answer the question: Is w the companion of the decay constant k, or is it w2? On comparing (46) dZx/dt2= -u2x with the corresponding rate equation dCldt = -kC (47) one will be inclined to regard wZ as the analogue of k. Indeed, if one approaches the problem through the traditional route, the resulting secular equations, in which w2 and k play equivalent roles, will preclude any other interpretation. The contrary opinion, that w and k make an analogic pair, can also be supported, mutatis mutandis, by the following reasoning. Since eq 46 is consistent with dxldt = 5iwx (48) and eq 47 implies that (49) d2Cldt2= kZC one must concede, on comparing eq 46 to eq 49 and eq 47 to eq 48, that fiw is the analogue of k and, conversely, fik is analogous tow. To quote Widom (4): "From the point of view of the theory of vibrations, a relaxation is a vibration with an imaginary frequency, whereas, from the point of view of the theory of relaxation, a vibration is a relaxation with an imaginary relaxation rate." The foregoing arguments, though it has been convenient to develop them in connection with problems involving exponential decay or oscillatory motion, will manifestly apply to systems in which both phenomena coexist. For example, if we write (50) dzldt = -Xz and take
z=z+iy
and X=k+iw
(51)
the real part of z,Re z = x , comes out to he
simplest way to allow for the decay of a "stationary" state of a quantum system is to assign the state a complex energy (8, 9). Literature Cned
Thus, exponential decay ( w = O), undamped oscillation (k = O), and damped oscillatory motion (k + 0 + w ) can all he encapsulated in a single first-order linear differential equation, viz., eq 5001 its analogue, dzldt = fiDz with 61 = w + ik. Students who have come across this argument are less likely to be mystified if they are told at a later stage that the
1. Jmt, W.Z.Noturfrfrrfch.1947,la. 15S163. 2. Mstaen,P.A.:Fmnklia, J.L.J.Am.ChpmSac. 1950,12,33373341. 3. Wei. J.: Prater.C. D.Ad". Cofolrsi* 1962.23. m 3 9 2 .
Volume 66
Number 9
Se~tember1989
705
