Computer Simulation of Chemical Equilibrium. John F. Cullen, Jr. Ricks College, Rexburg, ID 83460. Of all the topics covered in an introductory chemis...

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Computer Simulation of Chemical Equilibrium John F. Cullen, Jr. Ricks College. Rexburg. ID 83460 Of all the topics covered in an introductory chemistry course, the areas of equilihrium and kinetics are some of the most difficult in which to design meaningful quantitative lahoratory experiments. Accurate quantitative data usually demand the use of instruments such as pH meters or spectronhotometers. The costs of these t w e s of instruments in theAnumbersneeded for an entire class are beyond the budeets of most hieh schools and small collenes. Further, the condepts that are t o he demonstrated are often clouded by the theory and use of the instrumentation. Computer simulation of chemical reactions is an alternative to an actual lahoratory experiment in these areas. I have had a great deal of success with a simulation I call The Great Chemical Bead Game. This game can he used a t a range of levels and can be used to generate data that is treated qualitatively or quantitatively depending on the desire of the instructor. Since the game requires no instruments, the concepts of equilihrium and kinetics are more clearly presented than through an experiment. Rules ol the Game My experience has shown that most students do not understand whv eauilihrium occurs in a chemical system. As a result manist"dents benefit from physically playing the game prior to running the computer simulation. Students are supplied with a game board that has 10 columns and five rows. The game is played with three different colored heads: red. hlue. and ereen. I t is necessarv that these beads be facdted so that they do not roll easily. Faceted plastic beads are available a t most craft stores. Half of the students are given 50 green beads and told t o place one head in each square of the game hoard. They are also given a handful of red and hlue heads to he used once the- game starts. The other half are eiven 25 red and 25 hlue heads. Thev are instructed to pike the heads randomly on the game bbard, one bead to a box. Thev are also -eiven a handful of green beads. A computer is programmed to generate pairs of random numhers. One numher has oossihle values from 1t o 10. and the other from 1to 5. This ;air of numbers is used to locate a hox on the game hoard. A second pair of random numhers is generated to locate a second box on the game hoard. The color of the head in each box so selected is examined and the following rules applied: ~~

~~

~

~

1. If one head is red and one is hlue, bath heads are removed and

replaced with two green heads.

2. If both beads are meen, - the two heads are replaced with one red

and one blue.

3. If the colors are any other combination, no changes are made.

These rules correspond to the head "reaction": red + hlue = 2 green

Another two pairs of random numbers are selected and the rules applied to these two beads. The game continues in this manner. Students are told that the first nroup . . to convert their hoard over to the other starting type is the winner. The fact that virtually all students will enthusiastically play the game clearly indicates a lack of understanding of prohability. A t intervals of about 25 turns. the eame is stonoed. and the heads of each type are counted. ~ f G100 r turni,students are convinced that there is no hone of winninn and the physical game is stopped. At the p o k the gamc L i stopped, there is no difference between the came boards that started with all green and those that started with a mixture of red and hlue. The same equilihrium state is clearly approached from either direction. The reason equilihrium has occurred is also clearly seen. Students notice that a t first every change they make is in the "right"direction to win the game. After a while, however, there are about as many "right" conversions as there are "wrone" conversions. At this ooint it would appear to an observ& who could not actual6 see the game heine nlaved - that the eame had been stooned. .. The concent of dynamic equilibrium producing static concentrations is thus demonstrated. Once the game is ended, a computer is programmed to play the same game. On the IBM PC the program takes about 45 s for every 1000 turns. The program is run with 50 heads and printed every 25 turns. Students can see that the results of the computer game are the same as their results and are ready to progress to the computer simulation phase of the game.

--

The Rate Law One of the strengths of the beadgame is that i t can be used auantitativelv to nredict the values of rates. orders.. snecific rate constants, equilibrium constants, and other important properties of kinetics and equilihrium. The head game is basically a single-step, himolecular reaction in both the forward and reverse directions. I t is, therefore, first order in red beads, first order in blue heads, and second order in green heads. These orders can he derived in one of two ways.

- -

Initial Rate Method In the initial rate method the forward and reverse reactions are studied separately. A certain numher of red and blue heads are selected and the program run for about 1000 turns. On the IBM PC this takes a little less than a minute. There must be sufficient beads so that the numbers of red and blue l~cadsdo not significantly change during the 1000 turns. The initial rate of the forward reaction iscalculated in one of three ways: -A[R]/# turns or -A[B]/# turns or '/zA[G]/# turns. The concentration is taken as the number of heads of a given color divided by the total numher of Volume 66

Number 12 December 1989

1023

beads. A second trial is run with the number of red beads half the initial number. To keep the remaining concentrations unchanged, sufficient white heads are added to keep the total beads constant. White beads represent inert or solvent molecules and do not ~ a r t i c i ~ aint ethe eame. T w i cal data clearly show a first-eider process. If theconce~~i;ation of the blue beads is changed, kcepina the concentration of red constant, the data ind;a& a filst-order reaction with respect to blue. In a similar manner the reverse reaction can be shown to be second order in green. Typical data are shown in Table 1.

Table 1. Typlcal Experimental Reactlon Orders Calculated from lnltlal Rate Data for Bead Game wlth 1 Billion Beads alter 1000 Turnsa

Graphical Analysls

At a more advanced level the data can be treated graphically. The forward reaction is first order in both red and blue beads. The rate law is

-- d[R1

d(turns)

a

= kf(R][B]

initial

rate

WI

[Blue]

[c+eenl

0.500 0.250 0.125 0.500 0.500 0 0 0

0.500 0.500 0.500 0.250 0.125 0 0 0

0 0 0 0 0 1.000 0.500 0.250

White be*

order

(X 5.03 2.60 1.22 2.50 1.27 10.00 2.42 0.63

0.97 1.01 1.03 0.99 2.07 1.98

are Bdded in some vials to k s e me ~ total bedo constant.

Table 2. Typlcal Velues ot Speclllc Rate Constants and Equlllbrlum Consants Derlved from Rate Constantsa

Since both red and blue beads enter into the rate law, a pseudo-first-order experiment must be used where the [B] is sufficiently large that it does not change during the course of the reaction. Under these conditions the rate law becomes

Source of Data initial rate --

graph -

hitorward)

hireverse)

2.01 X 1 0 P 2.12 X 1 0 P

1.00 X 9.95 X

lo-e lo-'o

K 2.01 2.13

.For m m p ~ r l s m .me vslws 01 the graphically derived rete constants have been mnverted to l billi~nbeads fatal.

Integration of this equation gives In [R], = -k;(tums)

+ In [R],,

(1) A plot of In [R] vs. turns gives a straight line graph with a typical correlation coefficient of 0.9999. A similar plot of a pseudo-first-order reaction where [R] does not change will give a linear plot for in [B] vs. turns. The reverse reaction is second order in green and so obeys the rate law:

Table 3. Comparison ot Rates Predicted from Rate Law and Observed Rate of a Typlcal Trial for 1 Bllllon Total Beads

Rate [RdI

[Blue]

(X

lo-'?

Rate

(X loPo)

Integration of this function gives the expression:

A plot of l/[G] vs. turns should be linear. Typical results give a straight line with correlation coefficient of 0.9999. Specific Rate Constants

Once the rate law has been determined, the specific rate constants for the forward and reverse reactions can be calculated from the data used to determine the orders of the reaction. Twical values are shown in Table 2. Since the degree of reaction is based on chance, the values of the rate constants will varv sliahtlv from trial to trial. Unlike a chemical reaction, incr;asgg the total number of beads does not cause the reaction to proceed any faster. Beads "collide" only two at a time. As result, one will calculate a different value for k as the total number of beads is changed. For all kinetic data, the total number of beads must be kept constant. If a graphical method has been used to determine the orders, the rate constants are determined from the slopes of the corresponding graphs. From eq 1it is evident that the slope of In [R] vs. turns is equal to -k(forward). As eq 2 shows. the slope of the 1IIG1vs. turns is eoual to 2kheverse). ~ y p i c values h are s h o k in Table 2. Once the rate law has been determined completely, it can be used to predict the rates of forward and reverse reactions for other concentrations. As Table 3 shows, the nredicted and observed rates are in close agreement.

a

Equllibriurn Constant

The usual graphs of concentration of reactants and products vs. time can be plotted in order to show the variation of 1024

Journal of Chemical Education

the rate of reaction (slope) and the approach to equilibrium as time proceeds. The effect of dilution on the time needed to achieve equilibrium can be shown by diluting the beads with white "solvent" beads. The relationship between equilibrium constant and the specific rate constants is easily established. When the game has come to equilibrium, the forward qnd reverse rates are equal: ratef = rate,

The theoretical value of the equilibrium constant for the bead game is 2.00. The calculated values of the equilibrium constant from the specific rate constants are quite close to this value. Typical results are shown in Table 2. The bead game can be used to show clearly that the value of K is not absolutelv constant. but. rather. fluctuatessliehtly around an averagd value. he bead game is run with 2fred and 25 blue, with 2,500 red and 2,500 blue, and 250,000 red and 250,000 blue. To come to equilibrium, the game requires a number of turns about four times the total number of beads. Since trials containing 500,000 beads will require several million turns, this data will need to be supplied to students rather than having them generate their own data. Typical data is shown in Table 4. From the data it is clear that the larger the number of beads, the larger the variation

Table 4.

Varlatim In Equlllbrlum State and Equlllbrlum Constant as a Functlon of Number of Total Beads

totalbeads 50

Table 5.

#turns

red-blue

100 150 200 250

15 10 12 17

green

variation ingreen

20 30 26 16

K 1.78 9.00 4.69 0.89

14

Changes In Equllibrlum Concentrations as Le ChatellerType Stresses are Applled

mess

red

innla1 blue

green

white

red

fins1 blue

green

K

none

1450 1450 1450 1450 1450 800

1450 1450 1450 2450 450 1450

2100 3100 1500 2100 2100 2100

1000 0 1600 0 2000 1650

1472 1741 1278 1253 1708 965

1472 1741 1276 2253 706 1615

2056 2518 1600 2494 1584 1770

2.0 2.1 2.1 2.1 2.0 2.0

>G

>B

Table 6. Calculated and Observed Equillbrlum Constants for Bead Games Where Forward and Reverse Chances Are Not Equally Probable

of red. blue. and meen beads. However. when the data are graphid, it'is apparent that the fluct"ations are far less noticeable as the number of heads is increased. Extra~olation to mole-sized samples suggests that the equilibrium concentrations may vary by millions of molecules every second. Such variation, however, would not be noticeable from normal measurements of concentration. As a result, the equilibrium constant appears constant when the number of objects is extremely great.

Le Chatelier's Principle The shifting of the equilibrium concentrations and the constancy of the equilibrium constant can be demonstrated either qualitatively or quantitatively using the computer program. If one starts with 25 red and 25 blue beads, the eauilihrium constant of 2.00 reouires about 14.6 red. 14.6 biue and 20.8 green as the equili&ium condition. ~ a k &15 red, 15 blue, and 15 green as an approximate equilibrium mixture, "stresses" can be placed on the system by changing the number of beads of any one color. The total number of beads is kept constant by addition or subtraction of white beads. The observed results are in ameement with Le Chatelier's principle. Students can read& understand the reason for the shifts in terms of the probabilities of picking the various beads as the number is chaneed. Quantitatively, the shifts in the e:uilibrium state can be shown to have no effect on the value of & For this demonstration, a large number of beads must be used so that the variability of the observed K is small. Typical results are shown in Table 5. AMering the Rules The rules of the name can he easily altered. For exam~le, if a red and blue beid have been selected, a random number from I to 25 can be generated. If the random numher is 1, then the conversion to two green beads is made. If the random number is 2 through 25, then no change is made. This has the effect of hinderine the forward reaction and corresponds to a chemical reacGon for which the activation enerev of the forward reaction is much lareer than the reverse. a ... typical endothermic reaction. The resulting theoretical value of K is %S of the orieinal K. or a value of 0.08. If instead of the forward reaction the reverse reaction is hindered in a similar manner, the resulting K should have a value of 25 times the original K, or a value of 50. This corresponds to a chemical reaction for which the activation enerm of the reverse reaction is larger than the forward, or a typical exothermic reaction. Data shown in Tahle 6 show that the new equilihrium constants calculated from specific rate constants and those actually observed are close to the theoretical values, The rules can be easily altered to include other types of changes such as multi-step mechanisms or different orders.

The Great Chemical Bead Game has enormous potential for demonstrating kinetic and equilibrium concepts in either a qualitative or quantitative way. The game can be used on a variety levels. Since the game is conceptually simple, students are better able to grasp the abstractions presented because they are not distracted by the theory of the measuring instruments. A copy of the program c& be obtained by contacting the author.

Volume 66

Number 12 December 1989

1025