Approach to Equilibrium: The Wasp and Beetle Model Statistical Insight into How Equilibrium Is Achieved and the Stability of an Equilibrium State
-
Non equilibrium
Equilibrium
Room 0
Alberl Changl and Russell D. Larsen2 Texas Tech University. Lubbock. TX 79409
I t is a central precept of chemistry certainly, and indeed all of science, that equilibrium is the state to which all processes eo under natural or svontaneous conditions. A nonequilib;ium state may be maintained only under wnditions that restrict spontaneity. The precept that there is a multiplicity of natuialor spontaneous processes that ..go toequilibrium" is widely appreciated, and even accepwd as intuitive and obvious. C o m i o n examples are that: rain falls, a ball rolls down a hill, ink mixes in a beaker of water, and iron rusts. The final "restine state" is the eouilibrium state. It a h is well known that the equilibrium phenomenon IS dynamic and not static and therefore not "restina". - With macroscopic systems dynamic equilibrium certainly is not intuitive: with microsco~icsvstems i t is an abstract concevt taken on faith. perhaps the central exhibit for illustratikg the existence of dynamic equilibrium (due to molecular interactions) is the phenomenon of Brownian motion. Numerous attempts have been made to portray features of the equilibrium state using various models of molecular hehavior. A recent article by Cullen (I) in this Journal portrays the equilihrium state through a simulation that the author calls "The Great Chemical Bead Game". One of the first models. called the "doe-flea model". was devised hv P. and T. ~ h r k n f e s tin 1907;~ithas been described and discussed bv M. Kac (2) and others (3.4). These models are able to be i l l h a t e d th*ough comp~t'e~simulation. One of the earliest uses for a com~uter.in fact, was the simulation of the Ehrenfest model by work& a t LOS Alamos using the Los Alamos MANIAC computer. An instructive illustrition of the problem of how equilibrium is approached and ultimately achieved is in a section of Arthur Engel's chapter, "Teaching probability in intermediate grades", which appears in the somewhat obscure, The Teaching of Probability and Statistics (5).The present article expands upon and extends Engel's model. I have been using this model in my introductory chemistry courses for some years. I t nicely illustrates several concepts, notably the increase in entropy in a spontaneous process, how a state of equilibrium is attained, and the stability of the equilibrium state once i t is reached. The Was0 Model A state of equilibrium is approached through a "mixing process". An equilibrium state is the "most mixed up state". In order to illustrate the astounding complexities associated with the mixinaprocess of large numbers of atoms and molecules and the-ultimate attainment of a "most mixed up -
-
' 1988 Robert A. Welch Foundation Summer Research Scholar.
Author to whom correspondence should be addressed. Present address: Department of Epidemiology. Graduate School of Public Health, Vniverslty of Pittsburgh, Pittsburgh, PA 15261.
Room 1
I Non equilibrium
Figure 1. Wasps in raarnri model. Randomly selected wasps mwt change rWmS every "minute" until all wasps move to m m 1.
state".. heeinners first need to be introduced to a verv " -simnle . three-particle system and a mechanism for the random mixing of those particles. Whereas deviations from equilibrium behavior are substantial when only three or four particles are considered, the equilihrium principle using this model is nicely illustrated for beginning students who quickly obtain a perception of the meaning of increase in entropy and the role played by the equilihrium state. Two rooms are separated by a door. See Figure 1.Initially, in the left-hand room, denoted 0, there are three labeled (numbered, or colored) wasps, flying around. The rule is that. once each "minute". one was0 must chanee rooms. which wasp changes rooms is deternkned a t random. I have students randomlv withdraw a colored ball from a coffee can that contains three differently colored balls. The ball is then replaced (sampling with replacement) and subsequent random drawings with replacement simulate a room change for other (or the same, wasps until thestopping configuration is reached. This class participation illustrates for the students what the "selection process" is, how the bookkeeping takes place (to be discussed below), and how it is carried o;t on a larger scale by computer. How is relative "mixedupedness" quantified? It is convenient to follow wasp movements from the initial, ordered arrangement (all in room 0) through the equilibrium states of mixed up configurations hack to a final, ordered arrangement of all wasns in room 1.The ouestion becomes: how lone does it take, on the average, for all three wasps in the lefthand room to change to the right-hand room. denoted I ? The connection with the moiecular world is'that the wasps are, of course, molecules. If the wasus are denoted hv the letters, a,b,c, then the state of the twdrooms can be denoted by {a,b,cl. As all three wasps are in room 0, initially, the initial state is {0,0,0).If wasp b flies to room 1, then the state of the wasps is (O,l,Ol. The stopping rule a t which time the game is over is when the state {1,1,1)is achieved-all three wasps are in room 1. Each "move" is a "change of state". Because a wasp is selected a t random each time, the state that follows (0,1,0), above, could be {0,0,0) again-wasp b might immediately return to room 0. Thus, how long does it take for the state (1,1,1)to be reached? Obviously, each time the game is played (for each trial) a different numher of moves will be involved. The minimum number is 3, corre-
-
Volume 68
Number 4
Am11 1991
297
- - -
sponding to three successive draws of different wasps. One scenario (trial) would be: (0,O.O) (0,1,0) (O,1,1) (1,1,1). Actually, this problem is fairly complicated. From the viewpoint of elementary probability theory one wants to find the ex~ectationvalue, E(X), of X, which is a random variahle. he expectation is the weighted average of the X values. each weight being the probability of X. In thisexample, is a "waiting time;', or length of time before all wasps move to the right-hand room, for a given realization of wasp movements. k ( ~in )the average length of time required for a very larne number of realizations (or trials) of this random pro>ess.-~nthe average, how many moves (or minutes-at 1 min per move--or students) does this take? The expectation va~uk(for three wasps in two rooms) is 10 (10 moves are required, on the average). That this is the result, even for this simple case, is not a t all obvious. We will show below, using a decision tree, how this result arises. Computer Slmulatlon of Wasp Movements
If r e ~ e a t e dtrials of the above game of wasp movements are made, because each trial will risult in a different number of "moves" until the I1,1,1l state is reached, i t is instructive to carry out the above class demonstration twice, a t least, and then to generalize the drawing-colored-balls-from-acoffee-can game through a computer simulation. Binary random digits can be produced by flipping a "Laplace coin" having faces 0 and 1, or by spinning a "Laplace wheel of fortune" with congruent sectors of 0 and 1. Of course, in a comouter simulation the random numbers are internally generated through the programmed random number tor. The results are shown in Tables 1 and 2 for a large number of random trials. ~
~
Table 1. Average Number of Moves for Flrrt Six of 50 Sets of 100 trlab. Total Number o( Trials Is 5000. m e Average tor There 5000 Trlals is 9.91.
Table 2.
Moves
3 5
7 9 11 35 69
Set Number
Average Number of Moves
1 2 3 4 5 6
10.02 10.28 10.60 9.02 10.32 8.82
Frequency of Occurrence ot Varlous Moves for Data In Table 1 Number of Occurrences
Percent of Tml Trials
1087 886 656 537 442 11 1
21.74% 17.72% 13.12% 10.74% 8.84% 0.22% 0.02%
The Beetle Model, An Equlvalenl Mlxlng Model
Engelalso showed that random walks can be performed by "beetles", having no memory, that walk along the edges of regular 2-dimensional polygons, such as triangles and squares, or along the edges of regular 3-dimensional polytopes, such as tetrahedra or cubes. In our case of three wasps we require a cube with eight vertices. The vertices are labeled with binary numbers. Starting from vertex (0,0,0), how 298
Journal of Chemical Education
long does it take, on the average, to reach vertex (1,1,1) if there is an equal probability of going to an adjacent vertex or returning to a previous vertex? In other words, if the beetle "dies" a t vertex (1,1,1), what is the lifetime of the beetle, assuming that the beetle must move to another adjacent vertex each minute. selected randomlv? The problem of wasps moving between rooms is equivamodel. We show this lent to the beetle-walkina-on-an-edre equivalent representation as another example of how random movement leads from one ordered (noneauilibrium) state, throughasuccession of mixed upstates (vieked differentlv), to another ordered, nonequilihrium state. There are Z3 pbs&ilities with three wasps in two rooms corresponding t o eight vertices on a 3-cube. These eight vertices can be laheled with eight binary numhers, 000 to 111.See Figure 2. The model is more physical in the "beetle form" than in the "wasps form" because the concept of microstates and macrostates is easily introduced and a decision tree can be constructed from which the expectation value of the lifetime, E(X), can be found. The eight binary triplets (abc)may be considered in terms of their "digit sum", a b c, which is either 0, corresponding to the initial state, (0001, 1corresponding to either 10011, (0101, or (1001, 2 corresponding to either (0111, (1011, or (1101, or 3, corresponding to the final state, (111).The states (or vertices) of equal digit sum are "macrostates". They correspond to a certain number of wasps in each room: for example, (OOl), 10101, or (1001, all correspond to a single wasp in room 1. Each of these three states is a "microstate" and specifies which wasp is in room 1.
+ +
More Comnuter
Simulations
Consider, now, the macrostates corresponding to various confieurations of three wasm distributed between the two room;. Table 3 shows representative results for the comput-
Figwe 2.Cube with binav labeled vertlces for the random walk of a beetle without me-. The beetle must stad at vertex (0.0.0) and may walk only along an edge to a randomly chosen, adlacent vertex. The beetle "dies" at vertex (1.1.1l.
Table 3. Frequency of Dig# Sums tor Selections trom 500 Trials of Three-WaspITwo-Room Model (Elght-Vertex Beetle Model) Showing Predomlnance of Tlme Spent In Equllibrlum Stater for Which the Dlglt Sum Is 1 or 2. Nonequlllbrlum Stater Have Dlglt Sums of 0 or 3. The Average Number of Moves for the 500 Trlals Was 10.42. Trial number
Moves
Sum = 0
Sum = 1
Sum = 2
Sum = 3
1 2 3 4 10 17 27 44
3 13 19 3 29 3 63 5
0 2 1 0 5 0 9 0
1 6 9 1 14 1 31 2
1 4 8 1 9 1 22 2
1 1 1
1 1 1 1 1
er simulation of 500 trials of random wasp movements between the two rooms. Theinitial stateand the final state are earb "nonequilibrium statesw-not normally obtained by a spontaneous or natural process. The equilibrium Jtates (the mixed-up states) are se& to he much-more frequently obtained. Figure 4 shows a histogram that summarizes the occurrence of the equilibrium states compared to the nonequilibrium states (for the simolest case of the three-wasn/twor o o k model). In fact, the'two equilibrium states (dig6 sums of 1and 2) should be added together in order to see the far greater likelihood of being in an equilibrium state than in the initial or final nonequilibrium states. Table 4 extends the results of Table 3 to four wasps instead of three wasps. The equilibrium states are now even more numerous and frequentk visited than for three wasps. Only selections of 100 trials are shown in this table. For Y."
0c.un.nc..
,or D"l.r.n,
Table 4. Frequency . . of DigH Sums tor Selections trom 100 Trlals 01 ~ o ~ r - ~ a s p I T w o - ~Model o o m (Eight-Vertex Beetle Model) Showlng Predominance ot Time Spent In Equlllbrlum States tor Which the Dlglt Sum Is 1, 2, or 3. Nonequlllbrlum States Have Dlglt Sums of 0 or 4. The Average Number of Moves tor the 100 Trlals was 22.37. A Run of 500 Gals Ylelded an Average of 21.26. The Expectallon is 21% Moves. Trial number Moves Sum = 0 Sum = 1 Sum = 2 Sum = 3 Sum = 4 1
6
2 3 4
40 10 20 76 6 66 114
5
6 7 11
0 2 0 4 10 0
6 2
2 12 1 9 32 1 23 34
2 17 4 5
1 4 1
27 2
6 2
8
26
10
54
23
1 1 1 1 1 1 1 1
st.,.
Figure 3. Truncated declslon bee for randam walk of beetle on edges of cube. [0.1,2.3] are dlgit sums of binary labeled cube vertices to which beetle has access an successive moves. Fractions are probabllltles of reaching a given Vertex from a previous vertex. another 500-trial run, an average of 21.26 moves was obtained. The expected number from the decision tree is 21%. Dlscusslon
The auestion arises: Wbv not choose a stopvine rule that corresponds to an equilibrLm state rather than & t h e noneauilihrium state,. 11,1,1I? no from one noneauilibrium . . ~ Why . state (an ordered (0,0,0)statejtiirough a successi& of equilibrium states until another nonequilibrium state (the ordered (1,1,1)state) is reached? This question lies a t the very heart of what an equilibrium state is! For a large number of wasps (vertices or atoms) there is going to be an astoundingly large number of mixed-up (equilibrium) states and very few ordered (noneouilibriuml states. The ~robahilitvof spontaneously going from an ordered state to a mixed-up state is very high, whereas the probability of a fluctuation from a mixed-up state to an ordered state is very small. The stability of equilibrium is not only due to the enormous number of mixed-up states that are possible but also due to the tinv number of possible ordered states. We have seen that the misleadingiy simple three-wasp/eight-vertex scenario requires 10 moves on the average to reach another nonequilibrium state. We also simulated the four-waspIl6vertex scenario that required 21% moves. The 100-wasp model involves 2100 (or 1030) vertices (5) and an even larger number of moues t o encounter a nonequilibrium state by chance alone! The simulations illustrated in this article are intended show the stability of the equilibrium state as well as portray the complexity of the mixing process (shown
throueh the decision tree) for even the most simple cases. A beginning chemistry student should be able &appreciate the com~lexitiesof the mixing-. Drocess, increase in entropy, ... and stability of equilibrium. T o summarize, the mixed-up states-wasps in each room-are the most frequently visited states f o r a spontaneous process. In fact, they arevery highly more frequent when thereare many wasps (10 or more wnsps, or molecules, shows this point convincingly~;witha mole or so of molecules-the tvnical situation of the realm of chemistrv-the eauilibrium state is overwhelmingly more probahle).~hemost frequent state is one that has aneaual distribution of total wasas: . . e.e.. -. with six wasps in two rooms, the state in which three wasps are in each room. Extension of the model to lareer numbers of wasps (e.g., 10 wasps in room 0) shows that a uniform distribution ("eaui1ibrium"-one of many ~ o s s i b l e5 5 states) is reachedquickly. In this case, a be& would walk on a 10-cube with 2" = 1021 vertices, so that visiration of a nonequilibrium state (ordered state) is very unlikely. For 100 wasps, the beetle "random walks" on a 100-cube with >lo3"vertices (6).Similarly, starting with a pristine Rubik's cube one gets lost in a mixed-up regime very quickly3 and virtually cannot return toan ord&edstate with& a s k i f i c algorithm. However. Poincare's recurrence theorem permits ali wasos to return to room 0 if the eame is nlaved " . - lone enough'(3). These simple wasp-in-room and random-walk-of-a-beetle-on-the-edges-of-:-cube models confirm two very important conclusions from the field of molecular thermodynamics (statistical thermodynamics). These conclusions are (6, 8): 1. If a gas is not ins thermal equilibrium state (one in which it has a e and velociMaxwell-Boltzmann distribution of ~ a r t i d enereies ea in& this state.' ties). then it will almost alwavs , " 2. OIWthe gas is (n theequilthrium atate, it will almost always stay there, although fluctuations away from equilibrium wrll and must occur. ~~
~
~
These two conclusions constitute the so-called Boltzmann picture of approach to equilibrium by a macroscopic system of mechanical particles. This irreversible approach to equilibrium by reversible mechanical particles is so fundamental to thermodynamics that i t is called the "zeroth law of thermodynamics''-the law of attainment of (thermal) equilihrium. The most mixed-up states that are attained in a spontaneous process are those that are of highest entropy and that edtomize the truth of the tenet (the second law of thermod-ynamics actually) that entropy increases in a natural process. There are 5.2 X 1OZ0 arrangements or palterns and 4.3 X microstates, only one of which corresponds to complete order (7). Volume 66 Number 4
April 1991
299
Acknowledgment The Rohert A. Welch Foundation is gratefully acknowledged for their exemplary support of the Welch Summer Research Program and for their support of this research. Literature Clted 1. Cu1len.J. F. J. Chem.Edue. 1989,66.1043-1045. 2. Kae. M. Probobdiry ond Related Topics in Physieol Science; Interscience: New York. 1959: p 72ff. 3. Kac, M.: Rots. 0.-C.: Schwsrtz. J. T. In Dbcrata Thoughla. Essoys on Malhsmalirs. Science, and Philosophy: Newman, H., Ed.: Birkhauaer: Boston, 1985:p 42. 4. Kac, M. Enigmos o, Chonrs: Univ. of California: Berkeley, 1985: p 117; Kse. M. Am. Moth. Monlhiy 1947.51.263-291;Kse, M. Sci. Am. 1964,211.97A08. 5. Engel. A. In The Teaching o/ Pmbabiity and Statistics, Ride, L., Ed.; Wiley-lnterscimce: New YorklAlmqvist & Wiksell: Stockholm. 1970: p 145. C -. Ilhl~nh~rkGr'.:Fnd.C.W.LacfurrsinStali~ticolM~chhhiiiiA~~~iiiiM~thhmhtical Society: Providence, 1963:p 5. 7. Marx,G.;Gajzago,E.: Gnadig, P.EW. J . P h W 1982.3.39-43. 8. ~ h ~ e n f e sPt , and T. ~ h conceptual r ~oundotionao/ the Stotiaricai ~ p p m o c hin Mechoniea: Moraucsik. M. J.,Tran~l.;Cornell Univ.: Ithaca, 1959;p 30.
Determinationof Concentrations of Species Whose Absorption Bands Overlap Extensively An Instrumental Analysis Laboratory Experiment C. Cappas, N. Holtman, J. Jones, and University of South Alabama. Mobile. AL 36688 Figure 4. Histogram ofmean occbnenceof lhe four states tor the l h r e w a s p l twwoom m0481. Thedlgll-sum 1 an0 2 states are squil'bro~mstates. 0 git-sum 0 and 3 states are nanequilibrium states. Histogram confirms preponderance of eq~ilibriumstates
Appendlx The surprisinge~~mplexity ofall hut thequalitative featuresofthe "simplr"thrre-wasp two-roomweight-vertex beetle model is much roo advanced for the average beginning college student hut shuuld serve 10 con\,inre interestrd r~ndemthat mixing procertse+,and approach toequdibrium, arc another of Nature's wonders. Engel (5) showed that thp average rim? fur repeated trials of n heetle walking the route. 10001 1111 .can he ohtained from a truncated derision tree ia prohahilily tree). This decisiun tree is shown in Figure 3. 11 the expectntwn fur thr lifetime i a e = EtX), then e can be obtained by considering thc pruhnbility of going r u a specific vertex mncroatate or branch of the tree, so that:
-
If this equation is solved for e, then e = 10. Thus, the expectation value for three wasps starting in the configuration (000)and ending in the configuration 1111)is e = 10. The derivation of this equation from the decision tree is quite subtle. The first term in the equation is for the event correspondingtogoing from the initial state {O,O,Olof digit sum 0 to the final state Il,l,l)ofdigit sum 3 involving the three necessary steps,O-1-2-thef first step,O-1 hesaprobability of 111,asa movement from 0-1 must occur. Whenin a digit sum state of 1, three movements to adjacent verticesare possible, one ofwhich can go hack to 0, but two others go to cube vertices with digit sums of 2. Thus, the probability is 213 for the second step, 1-2. Once in a digit sum state of 2 only one movement to the three eligible vertices goes from digit sum 2-3, the final state of (1,1,1].The total probabilitv for these three steps is their product, (111 X 213 X 1131.3. he second term involves the steps, 0-1-2-1, with respective probabilities of 1/1,2/3,213 giving a total probability of 419. However. is onlv - the - ,heetle ~ ~ ~ at ~a vertex with a dieit sum of 1 for which the to the final state isnow, e - 1. It took three probability of steps to get there, so the overall probability is 419(3 + e - 1). The last term involves the steps, 0-1-0, with respective probabilities of 111and 113, giving a total probability of 113. However, the beetle now is only at a vertex with a digit sum of 0 for which the probability of getting to the final state is e itself. It took two steps to get there, so the overall probability is 1/30 + el. ~~~
300
~
Journal of Chemical Education
S. Young
In mixtures of structurally similar compounds, determination of concentrations of components by absorption apectrophotometry often fails to produce accurate results due to considerable soectral overlao of the comoonents. I n such a case, the traditional techniqie of choosing N wavelengths to determine concentrations of Ncomponents results in an illconditioned matrix of molar ahsor&vities. In the theory of linear eauarions, use of an ill-conditioned matrix, defined as a matrix which has a large condition number (vide infra), leads to solutions which are extremely sensitive to errors in the input data ( I ) . Toovercome this problem, measurements are taken a t M wavelengths, where M > N; the technique of multivariate linear least squares is then used to determine both the concentrations and the estimated uncertainties of the comoonents. We have used this aooroach for auantita.. tive analysis of mixtures of rhodium(1) carhonyl complexes whose carhonvl stretchine bands over la^ sianificantlv ( 2 . 3 ) . Due to the iarge numb& of absorhan&vavelengih pairs required for this method, it is impractical to take the data either from a chart recorder or from an analogue or digital readout; instead, computer interfacing is required. Various methods are available, e.g., direct interfacing to a P C from an IEEE-488 or RS-232 interface bus or indirect interfacing by connecting output normally sent to the chart recorder to an AID interface board. By whatever method, the raw data can be sent to data files on the P C for analysis by the multivariate least-squares method. This experiment, appropriate for an instrumental analysis laboratory experiment, involves determination of the cons a mixture of ketones utilizcentrations of the c o m ~ o n e n tof ing 1R spectrophotometry. The data are transferred to a PC throueh an RS-232 interface. and the resulting data set is analyzed using the BASIC algorithm describedielow. Experlmental Mixtures of 2-hexanone, 3-hexanone, 3,3-dimethyl-2-hutanone, and 3-methyl-2-pentanone of various concentrations in carbon tetrachloride were prepared volumetrically using literature values of density. Special care was taken in handling the reagents due to their volatilitv. Table 1 summarizes the concentrations of all mixtures used fo; the six runs. Carbon tetrachloride was chosen as solvent because it does not absorb in the spectral wavelength region of interest and is of only moderate volatility. [Caution: Carbon tetrachloride is a suspected carcinogen, so sample preparation should be carried out in an efficient fume hood.] Samples of the individual components were prepared as 10.0%solutions in carbon tetrachlo-
