Jomes H. Swinehart University of Colifornio
Dovis 95616
I I 1
Relaxation Kinetics A n experiment for physical chemistry
Relaxation techniques, which include the perturbation of a chemical equilibrium by temperature, pressure, electric field, or concentration, have become an extremely valuable tool for making kinetics studies (I, 2). In general, these methods have been applied to fast reactions, but applications to slower reactions are possible and important. In many cases, the relaxation methods have significant experimental advantages over other techniques. The simplification in interpretation, particularly for complex reactions, can be enormous. The experiment designed here was chosen to illustrate relaxation methods by a kinetic study of the chromate-dichromate equilibrium (3). The experiment is applicable in a junior or senior physical chemistry laboratory. Theory
The relaxation methods rely on the fact that when a system in equilibrium or quasi-equilibrium is perturbed by a temperature, pressure, electric field, or concentration change, the rate of approach to the new equilibrium distribution imposed by the perturhation is exponential, dACi/dt = - A . The symbol ACi, is the difference in concentration of the ith species between time t and infinite time. The relaxation time, T, is the characteristic constant for the exponential change. Figure 1 contains an idealized relaxation curve indicating the variables discussed. The relaxation time is generally taken from a plot of in AC, versus time (see Fig. 1). I t should be noted that with an exponential curve the choice of zero time is immaterial. The units on the concentration axis need only be proportional to the concentration, absorbance, or percent transmission for the small changes observed. The relaxation time is related in a unique functional way to the rate constants, equilibrium constants, and concentrations depending on the mechanism for the reaction. In relaxation studies the rate constants and concentrations are those of the perturbed state. The number of relaxations occurring equal the number of equilibria present. For example, consider two equilibria to be present:
concentration of D has appreciably changed. Thus eqn. (1) can be treated as a separate equilibrium in the calculation of one of the relaxation times, 71. H o w ever, when changes in the concentration of D occur in eqn. (2), eqn. (1) is "coupled" to these changes through C and eqns. (1) and (2) must be considered together in a calculation of the second relaxation h e , 711.
Considering only eqn. (I), d[A]/dt = -k,2[A][B] If a small perturhation is applied and second order terms are neglected, dA [A]/dt = -klz([A]A [B] [B]A[A]) k?,A[C]. In this case A[A] = A[B] = -A[C] and dA[A]/dt = -[klz([A] [B]) kzl]AIA]. This equation satisfies the necessary exk12([A] [B]). ponential form and 1/71 = k2, Considering the "coupled" equilibria of eqns. (1) and
+ k2,[C].
+
+
+
+
dl[l)] - d l = h'sAIC]
and
- k,,A[D]
+
+
(3)
Several equations must bc satisfied A[A]
+ AIC] + l [ I ) ]
= O
A[A] = A[BI
Irl,
A+B=C km
C
=D
(1)
k*.
kl.
(2)
and the process in eqn. (1) equilibrates rapidly compared to that in eqn. (2) when a perturbation is imposed on the system. This means that changes in the concentrations of A, B, and C have occurred before the 524 / Jourwl o f Chemical Education
Figure 1. Top portion: idealized reloiotimcurve. Ci, I = 0, ond C,, t = refer to concentrations of the ith species ot zero time ond inRnite time. Lower portion: plot of In A C i verrur time (orbitrory unitrl.
,
Substitution of eqn. (5) into eqn. (6) and further substitution iot,oeqn. (4) followed by rearrangements yields
Snbstit,ution into eqn. (3) yields
Thus
The equilibria of interest in this experiment are: H + + In=HIn (7) H++ Cr04" e 1IICrO.c (8)
+
+
An indicator equilibrium, H + 111HIn, is used t o make colorimetric observation possible. If we assume that reactions (7) and (8) equilibrate npidly compared to reaction (9) and KI and KZ are the association constants for reactions (7) and (8), the relaxation time corresponding to reaction (9) coupled with the more rapid react,ions (7) and (8) can be calculated to be
dichromate solution (A) with about 3 ml of a more dilute chromate-dichromate solution containing indicator @) so that the total change in chromium concentration is of the order of 10-20%. Figure 2 illustrates a simple plastic "mixer" which can be used in conjunction with a 1-cmBeckman cell to facilitate the mixing operation. The "mixer" is a plastic rectangular cup with small holes in the bottom. A handle is attached to the center of the cup. The necessary amount of solution A is added to the cup and plunged into the Beckman type cell containing solution B. The holes in the plastic cup are small enough so that no solution passes through before mixing begins. Complete mixing is accomplished by several sweeps through the cell. The mixing time is about 2 sec. The sample cell is balanced against a reference cell containing solution B. Thus a diierence spectrum is always recorded. Observations are made a t 575 mp and 620 mp when the indicators are chlorophenol red and bromothymol blue, respectively. The resulting pH change from the perturbation as monitored by the indicator is used as a means of obtaining the necessary relaxation curve from which the relaxation time, 7 , is calculated. The concentrations of HCrO-, CrOdl-, and Crz072-are calculated from a knowledge of the final pH of the solution and the total chromium content. The pK of HCr04- and [Cr2072-]/[HCrO-1% are taken as 6.1 and 50 M-', respectively (8,4, 6). However, the values appropriate to the experimental conditions should be applied in the calculations. Table 1 presents a set of solutions which might be used in a student experiment. The solutions are made Table 1.
Typical Experiments, p = 0.1 (KNOB), t = 23°C
See the Appendix for a complete derivation. The experimental conditions can he chose11 (KeR >> 1 and [HzO] >> [CrZOlz-1) such that 1l.r reduces to 4kl [HCr04-] k,[HzO]. A plot of 1/r versus [HCrOa-] at any temperature will yield the rate constants kr and li,.
lo8 vol. A [HCrO,] l / r 010 0.05 0.10 0.05 0.10 0.20 0.10 0.20
+
The Experiment Spectrophotometric observation of the indicator can be made on the expanded scale of most spectrophotometers (e.g., 75-125% or 90-1 10% of a Beckman Rerording Spectrophotometer, or M . 1 absorbance scale
Figure 2. Side and top views of mixing device which is designed to Rt in o 1 -cm Beckrnon cell. A cornplete description of device is in the text.
0.60 0.55 2.1s 1 3.10 3.9% 5.71
0.034 0.03s 0.038 0.040 0.040 OOS1 0060 6.20 0.068
The indicator coneentratiou in solution B is 10-&M.
up with I(2Cr207and are 0.1 M in I
