In the Classroom
W
Playing-Card Equilibrium Robert M. Hanson Department of Chemistry, St. Olaf College, Northfield, MN 55057-1098;
[email protected] Equilibration is a statistical phenomenon much like the shuffling of cards. It has been noted by Richardson (1) in this Journal that the game of “52-card pick up” can be used to demonstrate the relationship between entropy, irreversibility, and work. However, a simple deck of playing cards can be fun and effective in demonstrating much more, including the concepts of equilibrium fluctuation, the most probable distribution, the equilibrium constant, Le Châtelier’s principle, and the statistical effect of the narrowing range of fluctuations as the size of the equilibrating system increases. It may seem surprising that all of these concepts can be modeled in a system that takes no account of energy. But the fact that these concepts can be modeled without regard to energy clearly demonstrates that the phenomenon of equilibrium per se is not energy based. Indeed, the fact that energy is conserved should be enough to convince oneself that the position of chemical equilibrium cannot possibly be energy dependent. (The energy of a system plus its surroundings is constant throughout a chemical reaction.) Energy simply supplies another “shufflable” quantity to chemical equilibrium. Thus, in real chemical systems, atoms are not only shuffled among themselves (to make different molecules), but quanta of energy are shuffled among molecules in accord with Boltzmann’s law. In effect, Boltzmann’s law and the concepts of entropy, enthalpy, and free energy are all derivable from just the sort of statistical considerations discussed here, as described in detail by Davies (2), Nash (3), and Hanson and Green (4). Fluctuations and the Most Probable Distribution Consider the following isotope exchange equilibrium:
H2 + D 2
2HD
It is not hard to calculate what the most probable distribution of cards will be. Simply calculate the total number of ways, W, to distribute 24 red and 24 black cards (or 24 H and 24 D atoms) in each of the various possibilities, then compare these “thermodynamic” probabilities. The calculation is done using factorials and the following general equation:
W =
(2)
The results for nH = 24 and nD = 24 are summarized in Table 1, where we have also included the reaction quotient, Q, and the probability of the distribution, P. These quantities are defined as: Q =
2
nHD ; nH2 nD2
P =
W Wtotal
(3)
where Wtotal is the sum of all possible ways the atoms can be distributed. This number is simply the sum of all possible W, but may also be calculated using the following formula:
Wtotal =
(nH
+ nD )! nH! nD!
(4)
The value obtained for Q will vary. In fact, though, it has a relatively high probability of being 4 (31.7% in this case). This value is obtained when there are 6 pairs of red cards, 6 pairs of black cards, and 12 red兾black pairs.
(1)
The equilibrium constant, K, for this equilibrium is 3.3 at 25 ⬚C and rises to 3.8 at 500 ⬚C (3). While playing cards cannot be used to get a precise measurement of K, we can nonetheless demonstrate the origin of the phenomenon. To do this, form a deck containing 24 red cards and 24 black cards. Shuffle the cards thoroughly (at least seven times) and deal all of the cards out into hands of two cards each, separating the pairs into three piles: (a) red兾red (H 2), (b) black兾black (D2), and (c) either red兾black or black兾red (HD). If we repeat this experiment several times, the outcome will vary. Sometimes the distribution of cards will be such that there will be only a few red兾red and black兾black pairs; sometimes there will be more. That is, there will be fluctuations in the result. However, if you do this exercise yourself, you may be surprised at how small the fluctuations are. In fact, the odds are 3:1 that from 5 to 7 red兾red pairs will be obtained, and the odds are over 25:1 that from 4 to 8 red兾red pairs will be obtained.
ntotal molecules! nHD 2 nH2! nD2 ! nHD!
Table 1. Possible Distributions of 24 H and 24 D Atoms into Molecules n(H2) n(D2) n(HD) 12
12
0
%HD 0.0
Q
W
P
0.00
2.7 × 10
8.39 × 10᎑8
6
11
11
2
8.3
0.03
7.8 × 10
2.42 × 10᎑5
10
10
4
16.7
0.16
3.1 × 1010
9.74 × 10᎑4 0.0130
8
9
9
6
25.0
0.44
4.2 × 10
8
8
8
33.3
1.0
2.4 × 1012
0.0751
7
7
10
41.7
2.0
6.9 × 1012
0.214
6
6
12
50.0
4.0
1.0 × 1013
0.317
7.8
12
11
5
5
14
58.3
8.1 × 10
0.251
4
4
16
66.7
16.
3.4 × 1012
0.105
3
3
18
75.0
36.
7.1 × 1011
0.0219
2
2
20
83.3
100.
6.7 × 1010
0.00207
1
1
22
91.7
484.
2.3 × 109
7.18 × 10᎑5
—
1.7 × 10
5.20 × 10᎑7
0
0
24
100.0
7
NOTE: The most probable distribution is bolded.
JChemEd.chem.wisc.edu • Vol. 80 No. 11 November 2003 • Journal of Chemical Education
1271
In the Classroom
Le Châtelier’s Principle and the Equilibrium Constant The most probable value for Q turns out to be what we call the equilibrium constant, K, for the reaction. From Table 1, we see K = 4. That this value is, indeed, constant, is illustrated by considering the effect of a perturbation on the system. If we now start with a system in the most probable distribution (in this case, 6 H2, 6 D2, and 12 HD) and “stack the deck” by adding another 24 black (D atom) cards, we are effecting a “perturbation” of the system. The distribution is now 6 H2, 18 D2 and 12 HD molecules, and Q, calculated from eq 3, is 1.3, which is less than the value of K. The system, according to Le Châtelier’s principle, should react in a way that counteracts that perturbation, and, since Q < K, the system should shift to the right (see eq 1). This is perfectly reasonable, for it is highly improbable that if we shuffle the deck starting with so many pairs of black cards (D2) that we will be left with as many as we started with.1 More HD should be produced at the expense of both H2 and D2 and that should increase the value of Q. Indeed, if this experiment is carried out a few times it will become clear that fewer H2, fewer D2, and more HD are present at equilibrium than in the initial distribution. Interestingly, the most probable distribution still has Q = 4, as shown in Table 2. The net reaction leading to the new most probable distribution that occurs involves two units of forward reaction:
6 H2
Initial Dist: Reaction: Final Dist:
18D2
12HD
2 H2 + 2 D2
4HD
4 H2
16HD
16D2
Q = 1.33
Q=4
This is the law of mass action in operation: there is one and only one most probable reaction quotient for any given chemical reaction at a given temperature (disregarding effects of nonideality). It is this number we call the equilibrium constant.
We can do the same for any isotope exchange reaction (ignoring energy factors), although it is advisable to carefully plan the number of cards used so that there actually is a distribution corresponding to the “equilibrium” distribution. For example, for an ammonia H兾D isotope exchange, there are multiple active equilibria:
N H3 + N D3
n(H2) n(D2) n(HD) %HD 12
24
0
1.57 × 10᎑10
11
0.016 7.2 × 10
9.07 × 10᎑8
0.073 6.1 × 1013
7.65 × 10᎑6
2
5.6
10
22
4
11.1
6
P
0.000 1.3 × 109
23 21
W
0.0
11 9
Q
16.7
0.19
1.8 × 10
2.24 × 10᎑4
16
15
8
20
8
22.2
0.40
2.4 × 10
0.0030
7
19
10
27.8
0.75
1.7 × 1017
0.0215
6
18
12
33.3
1.3
6.9 × 1017
0.0868
5
17
14
38.9
2.3
1.6 × 1018
0.206
4
16
16
44.4
4.0
2.3 × 1018
0.292
3
15
18
50.0
7.2
1.9 × 1018
0.244
2
14
20
55.6
14.
9.2 × 1017
0.116
1
13
22
61.1
37.
2.2 × 10
0.028
0
12
24
66.7
—
2.1 × 1016
0.0026
17
NOTE: The most probable distribution is bolded.
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NH3 + ND2H
2NH2D
ND3 + NH2D
2ND2H
All three of these equilibria should have characteristic equilibrium constants. To explore these equilibria, we need only equilibrate a deck containing 27 H cards and 27 D cards, this time dealing out hands of three cards instead of two. For a demonstration of Le Châtelier’s principle, add another 27 H cards and see what happens. Predicting the Equilibrium Constant Using Probability It is not hard to rationalize why the equilibrium constant in the H2兾D2兾2HD equilibrium system is not 1. Every time a H2 molecule and a D2 molecule react, a reaction occurs. However, only half of the time that two HD molecules react are H2 and D2 produced. The other half of the time the two HD molecules simply exchange atoms to become two different HD molecules. But why should the most probable distribution always have Q = 4? The fact that Q = 4 for the most probable distribution of H2, D2, and HD molecules arises from laws of probability. The number of molecules of H2, D2, and HD in the most probable state are related to the probabilities of those molecules being found, which are themselves related to the probabilities of finding the specific atoms that are exchanging, H and D. Thus, we can write:
Q = Table 2. Possible Distributions of 24 H and 48 D Atoms into Molecules
NH2D + ND2H
2
2
nHD PHD = = nH2 nD2 PH2 PD2
(PHPD + PDPH )2 (PHPH ) (PDPD )
= 4 (5)
Notice that in calculating the probability that a molecule is HD, we must consider two independent possibilities, that the first atom is H and the second atom is D, and the reverse. A similar analysis for the ammonia equilibrium involving NH3, ND3, NH2D, and ND2H suggests that in that case Q = 9. Increasing the Size of the System It is important to realize that the “most probable distribution” is not really the “equilibrium distribution”. Equilibrium is a dynamic state, not a static distribution. Equilibrium arises from constant shuffling, which results in a wide variety of possible distributions, only one of which is most probable. The power of probability in relation to equilibrium, however, arises when the number of interacting particles in the system is increased. Thus, it is interesting to predict what would happen if we were to equilibrate a larger deck of cards. For example, with 240 H and 240 D cards, we can calculate
Journal of Chemical Education • Vol. 80 No. 11 November 2003 • JChemEd.chem.wisc.edu
In the Classroom
1 1 ln (n !) ≈ n ln (n) − n + ln 1 + n + ln (2π n) 12 2
(6)
Notice that 98% of all possible distributions have a percent HD between 43% and 58%, and a reaction quotient, Q, between 2.3 and 7.4. Clearly the ranges of expected values for both %HD and Q have decreased upon increasing the number of cards involved. At the same time, the probability of the single most probable distribution is decreased relative to slightly different distributions. This effect has been discussed in depth (2–4). While it may not be difficult to detect fluctuations on the order of a few percent, by the time we get to systems involving macroscopic numbers of particles—even a mere million particles—the probability of actually observing any distribution significantly different from the most probable one is extremely small. For a mole of particles, while it is very unlikely to find exactly 50% HD in the system at any given time, it can be shown2 that the chances of finding less than 49.999999% or greater than 50.000001% HD can be estimated to be less than one part in 1050,000,000. So, it isn’t that the most probable distribution is the equilibrium distribution; rather, the most probable distribution is representative of all distributions with any significant expectation of being observed at equilibrium when large numbers of interacting particles are involved. Although it is impractical to actually use decks of hundreds or thousands of cards, we have found that shuffling and distributing a deck of 48 cards several hundred times is a reasonable approximation to a deck of thousands of cards, and even that number is large enough for a trend to be seen. Admittedly, many possible distributions are ignored in this manner, but the effect is nonetheless quite dramatic. Thus, as part their first laboratory experiment in Chemistry 126, Energies and Rates of Chemical Reactions, 180 students working in groups of two or three shuffled decks consisting of 24 H and 24 D cards.3 Each group shuffled and distributed their cards eight times. Results were compiled for each of seven laboratory sections (A–G). Over the course of the week, 960 distributions were made, roughly simulating a deck of 46,080 cards. The results are summarized in Table 4. Comparison of these results to individual groups’ data sets of only eight distributions (which varied considerably) clearly demonstrated to all involved the effect of system size on the range of expected values.4 Inclusion of Energy Of course, in this discussion of equilibrium we are considering only half of the picture. The actual “shuffling” of molecules includes the exchange of energy as well as position. Our analogy with playing cards only deals with the latter. The differences between playing-card equilibrium and reality are summarized in Figure 1. Real molecules do not have ground states of the same energy, nor do they have only
Table 3. The 18 Most Probable Distributions of 240 H and 240 D Atoms into Molecules n(H2) n(D2) n(HD)
%HD
Q
W
P
68
68
104
43.3
2.3
1.3 × 10
0.012
67
67
106
44.2
2.5
2.2 × 10141
0.019
66
66
108
45.0
2.7
3.4 × 10141
0.030
65
65
110
45.8
2.9
4.9 × 10141
0.043
141
141
64
64
112
46.7
3.1
6.6 × 10
0.059
63
63
114
47.5
3.3
8.5 × 10141
0.074
62
62
116
48.3
3.5
1.0 × 10142
0.089
61
61
118
49.2
3.7
1.1 × 10142
0.099 0.103
60
60
120
50.0
4.0
1.2 × 10142
59
59
122
50.8
4.3
1.1 × 10142
0.100
58
58
124
51.7
4.6
1.0 × 10142
0.091
57
57
126
52.5
4.9
8.9 × 10141
0.078
141
56
56
128
53.3
5.2
7.1 × 10
0.062
55
55
130
54.2
5.6
5.3 × 10141
0.047
54
54
132
55.0
6.0
3.7 × 10141
0.033
53
53
134
55.8
6.4
2.4 × 10141
0.021
52
52
136
56.7
6.8
1.5 × 10141
0.013
7.3
8.5 × 10
0.007
51
51
138
57.5
140
NOTE: These possibilities account for 98% of the ways the atoms can be distributed. The most probable distribution is bolded.
Table 4. Actual Results of Seven Laboratory Sections Shuffling Multiple Decks Containing 24 H and 24 D Cards Section
n(H2)
n(D2)
n(HD)
%HD
Q
A
322
322
700
52.1
4.73
B
400
400
928
53.7
5.38
C
241
241
478
49.8
3.93
D
366
366
804
52.3
4.83
E
560
560
1184
51.4
4.47
F
406
406
724
47.1
3.18
G
506
506
1100
52.1
4.73
TOTAL
2801
2801
5918
51.3
4.46
Energy
the probabilities of each of the possible outcomes. The eighteen most probable distributions (comprising 98% of all possible distributions) are given in Table 3. Values of P, W, and Wtotal were calculated employing Stirling’s precise approximation (5) to assess n!, which is accurate to within 0.25% for all numbers and within 0.01% for all numbers greater than 5:
H2 D2 HD DH (a)
H2 D2 HD DH
surroundings
(b)
Figure 1. Comparison of the treatment of the H2/D2/HD system using (a) playing cards and (b) with the inclusion of energy.
JChemEd.chem.wisc.edu • Vol. 80 No. 11 November 2003 • Journal of Chemical Education
1273
In the Classroom
one possible energy state. In addition, energy in real equilibrium systems is exchanged between the system and its surroundings (with its own set of energy levels). Conclusions The Boltzmann law, which is the basis for modern interpretations of entropy, enthalpy, and free energy, relates specifically only to the most probable state of a system precisely because that state is representative of all states likely to be found at equilibrium for systems involving large numbers of particles. What is often overlooked in discussions of equilibrium is the fact that it is a dynamic, fluctuating state based on probability. Using a simple deck of standard playing cards, we can demonstrate effectively the basic principles underlying equilibration—the establishment of a dynamic state that is constantly fluctuating around a most probable distribution. Acknowledgments The playing cards used by our students were kindly donated by Gregory Nelson from the Department of Advanced Computing at Hoechst Celanese Corporation. W
Supplemental Material
Supplemental material for this article is available at http:// www.stolaf.edu/people/hansonr/jce/cards (accessed Sep 2003). Contents of this directory include copies of the description of the experiment performed by the students in Chemistry 126 (exp1.pdf ), the spreadsheet used for the preparation of the tables presented in this article (ways.xls), a derivation of Wf 兾Wmax (proof.pdf ), a Visual Basic program that can be used to explore various combinations of H and D atoms in the H2兾D2兾HD equilibrium system (wineq.exe), and a JavaScript version of the same (h2d2.htm).
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Notes 1. The probability of that outcome upon shuffling is just 8.7%. 2. The following formula, derived in the supplemental material, is the basis for this calculation. Here Wf is the number of ways of finding a system of n H atoms and n D atoms with fraction f of the molecules as HD, and Wmax is the number of ways of finding 50% HD in the system.
Wf Wmax
≈ e
−2n( f − 0.5)
2
3. Approximately 300 brown-backed cards and 300 silverbacked cards were used by each laboratory section. Such matched sets may be purchased as “bridge decks” or obtained from casinos for free. 4. In an actual deck of 46,080 cards, the probability of finding 51.3% HD is only about 0.04% of the probability of finding 50.0% HD.
Literature Cited 1. Richardson, W. S. J. Chem. Educ. 1982, 59, 649. 2. Davies, W. G. Introduction to Chemical Thermodynamics: A Non-Calculus Approach; Saunders: Philadelphia, 1972; Chapter 1. 3. Nash, L. K. Elements of Statistical Thermodynamics, 2nd ed.; Addison-Wesley: Reading, MA, 1974; Chapter 1. 4. Hanson, R. M.; Green, S. M. E. Introduction to Molecular Thermodynamics; Integrated Graphics: Northfield, MN, 2001; Chapter 1. See also http://www.stolaf.edu/depts/chemistry/imt (accessed Sep 2003). 5. Stirling, J. Methodus Differentialis; London, 1730; p 137. See also http://www.nist.gov/dads/HTML/stirlingappx.html (accessed Sep 2003) and references therein.
Journal of Chemical Education • Vol. 80 No. 11 November 2003 • JChemEd.chem.wisc.edu
