James P. Birk and S. Kay Gunter Arizona State University Tempe 85281
Water Dipping Kinetics A physical analog for chemical kinetics
General chemistry students are commonly overwhelmed bv the abstract nature of the concents involved in discussions oi the rates of chemical reactionsand of dynamic chemical equilihrium. I t is helpful to have physical analogs to these chemical systems which can both provide visualization of these abstract concepts and generate data comparable to that collected for chemical systems (1-2). Several useful analogs have been developed. The cooling of hot water follows firstorder kinetics and is noteworthy for the simplicity of the experimental apparatus ( I ) . Capillary flow, as investigated by Davenport (2), requires more specialized equipment but is exceptionally versatile in demonstrating a wide variety of kinetic behavior, including the establishment of a state of equilibrium in a reversible reaction. However, i t does not illustrate the dynamic nature of equilihrium. The simultaneous transfer of water back and forth between two containers has often been advocated for the demonstration of the attainment of a state of dynamic equilihrium ( 3 5 ) . The steady use of two heakers of different sizes (i.e., different relative rate constants) as dippers illustrates vividly the relationships between rate, capacity, and volume of water remaining in the containers. The analog easily hears qualitative extension to consideration of the individual processes involved-the change in rate with time, the equality of rates a t equilibrium, the dependence of the rate on the amount of reactant (water) and on the rate constant (capacity of beaker), and the activation energy. All of these can easily he demonstrated and discussed during the time required to reach a state of equilibrium. It is necessary to carefully define the rate as the volume transferred per dip rather than as the frequency with which dips are made. Although various aspects of this demonstration have been discussed before (3-5), no attempt has been made to develop this physical analog into a quantitative experiment. We describe here a series of quantitative experiments on water-dipping kinetics which have great potential as useful physical analogs since they combine the desirable features of versatility (2) and simplicity of equipment (I). The required equipment consists of a container with sides perpendicular to the bottom, a dipper of cubic or rectangular prism shape, and a graduate cylinder to measure the volume of water transferred in each dip. Plastic trays with dimensions' of 25.5 X 19.0 X 6.5 cm, which are normally used in our general chemistry labs to hold small acid bottles, were used as containers. Dippers were constructed by cutting an appropriate section from the bottom end of milk cartons (half-gallon and half-pint sizes were convenient) or plastic chemical bottles (less desirable since comers are usually rounded). The dipping procedure had to he performed reproducibly: The dipper was placed in contact with the container bottom in an appropriate configuration (Fig. 1) and allowed to fill naturally, then tilted toward its hack edge in a smooth motion. Care had to he taken not to use a scooping motion which generally gave inferior results. It was desirable to correct for the volume of water remainine at the end of the Drocess since the finite thickness of the sire of the dipper limited the volume which could he removed. The value of this infinity volume (analogous to infinity absorbance, pressure, etc. in chemical systems) was calculated by taking the difference in areas of the container bottom and the portion of the dipper in contact with it, and multiplying by the thickness of the dipper wall. This gave
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Figure 1. The configuration of dippers in a container necessav to simulate various kinetics: (A) zero-order, (6) first-order, and (C)secanddrder. The open face of the dipper is at the top, front, and front, respectively.
2000
1000
v-v. 500
200 0
10
20
30
40
DIPS Figure 2. Plots of volume (in mi) and of ( V number of dips fw a first-order process.
V.)
(ona lagarithmic scale)versus
excellent results, as will he seen below, although deviations were ohsewed towards the end of the process since surface tension restricted the free flow of water when the water level approached the thickness of the dipper wall. Although the infinity volume was small, 30-120 ml with our apparatus, failure to correct for it led to deviations earlier in the process. Zero-Order KineticsZ ko
A-B
If the dipper is oriented with its hottom touching the hottom of the container and if the level of water is higher than the sides of the dipper (Fig. lA), the same volume o h a t e r will he transferred by each dip, until it is no longer pwsihle to fill the dinner. .. Using a 9.5 X 9.5-cm milk carton cut off to a h e i ~ hoft 1.0 cm, a plot of the zero-order kinetic equation, (V - V,) = (Vo - V,) - k o (dins). eave an excellent straieht line with a
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'The dimensions of containers and dippers are always presented as length x width x height when the open face is on top. ZInthe representation of a process by an equation, each letter stands for a distinct container of water and each rate constant represents a distinct dipper. Volume 54, Number 9, September 1977 1 557
v
2
v-v. 1000
1
0
10
20
30
40
DIPS
Fiaure 4. Plots of volume (in mll and of I V - \Im)-' versus number of dim for a seconddrder process.
first-wder rate mstant (in dips-') for variars dippws v e m Figure 3. Plot of me area (in cmZ)of the side of the dipper.
First-Order Kinetics
If one side of the dipper is placed on the bottom of the container (Fie. 1B). . . . the volume of water removed in a dio should he proportional to the height, and thus the volume, df water remaining in the container, ~ r o v i d e dthe water level is below the top of the dipper. h his is the requirement for first-order kinetics, and indeed, as shown in Figure 2, a plot of V versus dips gives a typical exponential curve. If a plot is made of the first-order function, In (V - V,) = In (Vo - V,) - kt (dips), good straight lines are ohtained for transfer of >90% of the original volume of water. For a typical dipper, a 9.5 X 9.5 X 9.5-cm section of milk carton, the value of k l is 0.159 f 0.002 dips-'. As might he expected, the value of k l varies in a linear way with the area of the dipper face which lies on the bottom during filling (Fig. 3). Under our conditions, - ~ This result is imk,lA = (1.82 f 0.09) X 10-3 ~ m dips-'. portant for applications to more complex systems since i t allows the selection of desired relative values of two rate constants by use of appropriately sized dippers. Second-Order Kinetics kl
A-B
The simulation of second-order kinetics can be attained in a simple fashion by water dipping, in contrast to the complex apparatus required with capillary flow (2). If the dipper is laced in the water container as shown in Fieure 1C. then i t can be shown mathematically that the volume of water removed is nrooortional to the sauare of the height . . - of water in the container as long as the water level lies in the lower half of the diooer. The rorrect ulnrement of the diooer can he simplified by marking guidelines on the containeibottom and wall. As shown in Figure 4, the volume does decrease in a second-order fashion and a plot of the equation ( V - V,Y1 = (Vo - V,)-' + kz (dips) does give astraight line. For a 9.5 X 9.5 X 9.5-cm milk carton, the slope of this plot is kz = 5.08 X 10-5 ml-' dips-'. Reversible First-Order Processes
AQB kb
Using separate dippers with two containers and dipping in the forward and reverse directions simultaneously, the results shown in Figure 5 were obtained. The rate constants, measured independently for the two dippers, were k. = 0.159 558 1 Journal of Chemical Education
Figure 5. Plot of volume (in ml) far each species versus number of dips lor the reversible system. A FI 6.
dips-' (9.5 X 9.5 X 9.5-cm milk carton) and kb = 0.067 dips-' (7.0 X 7.0 X 5.0-cm milk carton). For reversihle first-order processes, a plot of In (V - V,) versus dips should he linear kb) (6).The value of k, kb ohtained with a slope of -(k, from such a plot was 0.218 dips-', which compares favorably with the value of 0.226 dips-' obtained by addition of the independently measured values of k. and kb. The equilibrium constant for the process can he calculated as K = area of dipper Alarea of dipper B = 2.6, or as K = k,/kb = 2.4, or as K = VBIVA, = 2.9. The agreement between the various values of K is satisfactory.
+
+
Parallel First-Order Processes A%B A%C
The formation of two different products from a single reactant in parallel processes is shown in Figure 6. Again a plot of In (V - V , ) versus dips for the volume of A should give a straight line with slope of -(k.b + k.,) (7). Such a plot gives k,, = 0.233 dips-', while the independently a value of k.b measured values of k,a = 0.159 dips-' and k,, = 0.067 dips-' sum up to 0.226 dips-', in good agreement. The distribution of products should he in the same roportion as the relative rate constants (7). We obtain v%$= 2.8, while k,blk,, = 2.4, again a satisfactory agreement.
+
- -
DIPS
Figure 6. Plot of volume (in ml) for each species versus number of dips fwthe parallel processes, A Band A C.
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F i w r e 7. Plot of volume [in ml) for each rpeeias Venus number of dips for the consecutive processes, A B C ~
~
Consecutive First-Order Processes
Results obtained with k,b = 0.159 dips-' and kb, = 0.067 dips-' are shown in Figure 7. Complete analysis of this data but a plot of In (V - V , ) versus dips for the is not simple (8), volume of A gave a value for k,b = 0.158 dips-', in excellent agreement with that measured independently. The maximum volume of B is attained a t -9.5-10 dips, while the theoretical maximum, calculated from the two rate constants and the equation in ( k b e / k o b ) / ( k b c- k o b ) (8).would occur a t 9.4 dips. This system could be extended to any number of consecutive first-order processes by addition of the appropriate number of containers and dippers. Michaelis-Menten Kinetics
ASB-c kk
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Figure 8. Plot of volume (in ml) f a each species vwsus number of dips f a lhe Michaelis-Menten system. A a 8 C.
kk
Simulation of this system, which is extremely important in biochemical systems, can he carried out with three containers and three dippers. The results shown in Figure 8 were obtained with k,b = k h = 0.159 dips-' and kb, = 0.067 dips-'. The onset of the linear steady state decay of A at the time that B reaches a maximum is a twical feature of Michaelis-Menten kinetics. The theoretical shapes of these plots and the mechanistic features corresoondinz to the various r e. ~ o n have s been discussed by ~ i ~ ~ i n s - ( 9-) . Other Systems The demonstrated versatilitv of water-diooine in simulatine the kinetics of chemical reactions and the &eIl;?nt agreement between the observed and oredicted behavior in the svstems described lrads us to helieve that water-dipping ct~uldhe used to obtain simulated data for all systems involving Z P ~ O - ,first-.
or serond-order prncesses in any reversible or irreversible and conserutive or parallel romhinations. The quantitative application of this simple physical analog to complex reaction kinetics would appear to be limited only by the patience of the Literature Cied
(6) Fmst, A A , and Pasrron. R.G.."Kinetics and Mechanism,' 2nd Ed, John Wilcyand Sons. NewYork. 1961.p. 186.
(9) Higgins,&,in "InveJtlgationofRsteaand MechanirmsofReaefions." (Editon:Fleias. S. L.. Lewis. E. S., and Weissbe~er.AI, lnte~ciencePublishers,New York, 1961. pp. 302311.
Volume 54, Number 9, September 1977 1 559
