Edwin F. Meyer
and Edward Glass DePaul University Chicago, lllinois 60614
II
Demonstrating the Relation between Rate Constants and the Equilibrium Constant
A n apparatus which is able to illustrate quite vividly several fundamental principles covered in the freshman chemistry course is described by Weigang.' We present here in some detail the approach we have used with a similar apparatus to demonstrate quantitatively the relationship between rate constants and the equilibrium constant for simple reversible reactions. The apparatus is represented schematically in the figure. Two glass tubes (23-mm a d . , about a foot long) are mounted adjacent to sections of a meter stick on a vertical board. At the bottom of each tube is a one-hole rubber stopper into which has been inserted a constricted piece of glass tubing. The diameters of the constrictions are different, so that water flows through each at a different rate. Immediately after flowing out of one of the tubes, the water is pumped to the top of the other. There is a close analogy between the flow of water from tube A to tube B and the chemical reaction of compound A to from compound B. The height of the column of water in tube A is analogous to the concentration of compound A in a reaction mixture, while the cross-sectional area of the constriction is analogous to the rate constant of the chemical reaction forming compound B from compound A. Furthermore, when water is added and the pumps are turned on, the levels of water in the tubes quickly approach fixed l W ~ ~ ~JR., ~ 0. ~ E., ( i J., CAEM.EDUC., 391 146 (1962). 'This "rate" is actually the velocity of watex issuing from the tube in om/sec. To obtain the more relevant quantity, g/seo, v must be multiplied by the crossaectiond area of the constriction and the density of water role of "reaclim" in
he0
= "A = (2gh)'hA = kh1/2
where, as mentioned earlier, the raw romtanr i dirertly related to the rrossbertional area of the conarricrion.
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Schematic diegram of oppomtur.
values whose ratio is analogous - to an eauilibrium constant. We now consider the chemical and mechanical theory necessary to carry the analogy to its quantitative conclusion. If we write for the chemical reaction in question A S B
the equilibrium constant is given by
where the brackets represent concentrations. Now if we know that the forward and reverse reactions follow rate expressions of the type ?ate = k . (concentration of reactant)-
where m is the order of the reaction, we can obtain an
expression for the equilibrium constant in terms of the forward and reverse rate constants as follows rate of f m a r d reaction = k,[AIm
rate of veverse ~eaction= k,[BIm
At equilibrium these rates must be equal, and we can solve for [BI/ [A] [Bl/[Al = K, = (k,/k,)"'" (1) Thus, g!yen k,, k,, and m, we can calculate a value for the equihbrium constant which should agree with the value we measure directly from an equilibrium experiment. How do we evaluate these quantities for our apparatus? First we establish the order of the "reaction," i.e., the dependence of the rate of flow of water (rate of reaction) on the height of the water (concentration of reactant) in the tube. This provides an interesting example of the utility of the principle of conservation of energy: when a J-tube (23 mm o.d., about one and a half feet long) with a constriction to about 3 mm a t the tip of the J is filled with water, a vertical stream issues from it. It is readily observed that the apex of this stream always coincides with the height of the column of water in the long arm of the J-tube. Since the kinetic energy of a given mass of water as it issues from the constriction is converted totally to potential energy by the time the same mass of water reaches its apex, we can write
height of the column a t time t = 0. Comparing this b, the students to the straight line equation y = mz see that a plot of 2h"' versus t should be a straight line of slope -k . The necessary data for each column are collected as follows. (The pumps are not turned on for this part of the demonstration.) One student is given a stopwatch, and another is called upon to read the level of water in the tube upon a given signal. Tube A is filled with water while a finger at the constriction prevents any flow until everyone is ready. At t = 0 the finger is removed, and readings of h are taken every 5 sec. (A countdown of 3 sec prior to each reading helps the precision considerably.) The values of ho, h, h"', and t are tabulated on the chalkboard, and the procedure is repeated for tube B. Actually plotting 2h'/' versus t is too time-consuming t o complete in class, so the students are given this as an exercise to be handed in at the next class meeting. Instead, several values of k are calculated from eqn. (3) by rearranging
+
k =
2h1/2 - 2hJh -t
(5
and it is seen that k is actually quite constant. An average value for k is obtained for both tubes. Using the observation that m = eqn. (1) gives K, = (k,/k,)2 (6) The constant for tube A is k,; that for tube B is k,. ' / m v ' = mgh (2) We are now able to calculate the equilibrium constant for our apparatus. After doing so, the pumps are where the symbols have their usual meaning. But h turned on and water is added. The equilibrium conalso equals the height of the water in the tube, so we have the expression for the rate of our "rea~tion"~ stant is the ratio of the heights (concentrations) at equilibrium. Good agreement is obtained between the value calculated from the rate constants and that measured in an apparently independent experiment. This means the chemical reaction we are simulating Besides the demonstration in question, the dynamic must be of the one-half order, or m = '/2. nature of equilibrium is made quite apparent, since, Writing while the levels do not change a t equilibrium, it is clear that A is being converted to B and B to A a t an appreciable rate (emphasis on the singular). Furthermore, once equilibrium has been established, it is a simple we can rearrange and integrate just as for the first-and matter to illustrate Le Chatelier's Principle by pouring second-order reactions discussed previously in class. more water into one of the tubes. I n a matter of The result is moments the levels have shifted to new equilibrium 2h'/3 - 2A01/2= -kt (4) positions, and a measurement of the new levels produces the same value for K,, as before the addition of more where h is the height of the column of water (measured water. from the tip of the constriction) at time t, and ho is the
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