Exploring Biased Probability Using Loaded Dice: An Active Learning

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Exploring Biased Probability Using Loaded Dice: An Active Learning Exercise with Analogy to Entropic and Energetic Determinants of Equilibria in Chemical Systems James A. Hebda* and Zachary Aamold Department of Chemistry, Austin College, Sherman, Texas 75092, United States

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S Supporting Information *

ABSTRACT: The equilibrium position of a chemical system is based on two distinct considerations, namely, the entropy and enthalpy differences between the states. Loaded dice provide a physical system for exploring the probability of observing different interconverting states on the basis of two distinct factors: the number of sides of the die (entropy) and the asymmetric distribution of mass in the die (gravitational potential energy). Although potential energy analogies such as pushing a rock up a hill have often been used for teaching differences in energy in chemistry, and dice have likewise often been used to explain entropy, such analogies can be unsatisfactory when entropic and enthalpic considerations favor different states. We present an activity that allows students to explore how potential energy and entropy together contribute to observed probability and then draw parallels to chemical systems. Students collect statistics for observed rolls of loaded dice, rationalize the different patterns as distinct states, and determine the difference in entropy between the states. These observations are then related to chemical reactions and to the definition of entropy. The effect of varying the loaded mass within the dice is related to the role enthalpy plays in determining equilibrium. This analogy and associated active learning exercises are applicable to a wide variety of college chemistry and physics courses; provide a hands-on approach to introducing concepts such as probability, entropy, and energy differences; and can be used to draw parallels to chemical reactions, enthalpy, and Gibbs free energy. KEYWORDS: First-Year Undergraduate/General, Second-Year Undergraduate, Upper-Division Undergraduate, Biochemistry, Physical Chemistry, Analogies/Transfer, Collaborative/Cooperative Learning, Hands-On Learning/Manipulatives, Biophysical Chemistry, Demonstrations, Equilibrium, Statistical Mechanics, Thermodynamics

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BACKGROUND AND RATIONALE Chances are if you teach chemistry, you introduce or expand upon concepts such as entropy and chemical energy. Although these topics are an essential part of introductory chemistry courses, challenges exist in properly introducing concepts that synthesize these topics, such as Gibbs free energy.1 The use of a loaded die as a model system has many potential applications in physics and chemistry, and is here developed to explore concepts of chemical potential energy and entropy as related to the observed distribution of states at equilibrium and Gibbs free energy. Although laboratories and activities have been designed to explore Gibbs free energy and equilibrium concepts,2,3 there are very few physical models that combine contributions from both enthalpy and entropy.4 In this exercise, students count outcomes to calculate probability. The results are extended to chemical systems as an analogy to explore equilibrium constants, Gibbs free energy, entropy, and enthalpy. This exercise combines common analogies for entropy (rolling a die5,6) and chemical energy (gravitational potential energy) to help students see that differences in chemical energy and entropy work together to determine equilibrium. In chemistry, entropy and enthalpy are usually introduced with a few limited analogies. Entropy is often introduced in terms of disorder (i.e., the “messy room” analogy). In physical © XXXX American Chemical Society and Division of Chemical Education, Inc.

chemistry, the concept of microstates helps relate the concept to molecules. Microstates can be demonstrated using a regular six-sided die. The six sides with six different numbers yields one microstate (side) per macrostate (number). Regular dice exhibit maximum entropy;6 they are equally likely to land on each side. Differences in observed probability for high and low entropy states could be simulated if a six-sided die were alternatively labeled with one “1” and five “2”s. For such a system you would expect to roll a 2 more often than a 1 because there are five microstates that all give the same value (i.e., “state 2” is higher in entropy but equal in potential energy to “state 1”). Although dice analogies have been often used in physics and chemistry as examples of radioactive decay7 and even quantum superposition,8 they are intrinsically limited when explaining equilibrium positions of chemical reactions, as few chemical reactions involve no differences in chemical energy between states. Enthalpy changes, due to altered bonds or electronic states, can be introduced using the analogy of rolling a boulder up a hill, thereby increasing gravitational potential energy. This analogy is helpful but fails to address how changes in entropy affect the extent of the reaction. A Received: January 24, 2019 Revised: June 19, 2019

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DOI: 10.1021/acs.jchemed.9b00074 J. Chem. Educ. XXXX, XXX, XXX−XXX

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chemistry class. Depending on the class, students used 3Dprinted loaded dice (see the Supporting Information) with two pennies providing the load or commercial trick dice.13,14 Dice were distributed, and students were asked to determine what was special about their die. Students were led to consider the way that the loaded die felt pulled to one stable position by the weight within it. Each group rolled their die 50 times and used those results to obtain rough percentages for each side (Table 1). A class average for each side was determined. Students then

more holistic understanding would be aided by an analogy that considers how differences in both potential energy and entropy work together to determine probability. This exercise uses loaded dice. Loaded dice analogies have previously been used to explore concepts as diverse as activation energy9 and statistical reasoning.10−12 They are natural models of energetically biased chance, as the probability of each roll can be determined by differences in both gravitational potential energy and the number of possible states because of a small amount of mass added to one side, such that the opposing side is more likely to be observed. Loaded dice retain six possible microstates while introducing differences in potential energy. For a standard six-sided loaded die, a roll will result in one of three distinct potential energy macrostates: the load at the top of the die, the side, or on the bottom (Figure 1). This creates a system with macrostates that

Table 1. Tally of Rolls for Loaded Diea Side

Average Observations

Standard Deviation

Total Observations

Probability (%)

1 2 3 4 5 6

28.1 5.4 4.7 5.8 4.2 1.8

4.0 2.4 2.0 2.9 2.2 1.6

178 28 23 31 25 15

59 9 8 10 8 5

a

Result totals from student groups using 3D-printed, two penny loaded dice. Dice were loaded to bias side 1. Average rolls and standard deviations are given to reflect expected variation among groups.

grouped the six different sides into three distinct states, combining the probabilities of similar states. For this activity, each roll was considered a random observation from a system at equilibrium. Students proposed a physical basis for their observations and compared each state’s potential energy to the observation percentage of each state (Table 2). Comparing the Figure 1. Analogy scheme summary. Each potential energy state of the loaded die can be assigned as a macrostate (the shading represents the orientation of the added weight). Each macrostate consists of 1 or 4 microstates and is defined by relative differences in potential energy and entropy. An example model chemical reaction is given to aid in extending the loaded die analogy to a chemical reaction. Obtaining loaded dice for this activity can be accomplished through 3D printing of loadable dice (see Supplementary Figure S1), using commercial dice,13,14 or simulating expected rolls with a biased number generator (see Supplementary Figure S2). All data presented here use the author’s 3D-printed, loadable dice.

Table 2. Roll Totals Grouped by Potential Energy Based Macrostatesa State: Sides

Energy (ΔE)b

Entropy (S)c

Total Rolls per State

Probability (%)

3:1 2:2−5 1:6

Low Medium High

Low High Low

178 107 15

59.3 35.7 5.0

a Results of rolls are shown in Table 1. bHigh, med, or low. cHigh or low.

differ in entropy because of an unequal distribution of microstates. For this activity, State 1 was defined as side 6 face up, with the weight furthest from the table. State 2 consisted of sides 2−5 up, with the weight at an intermediate distance from the table. State 3 showed side 1 up, with the weight closest to the table (Figure 1). These three macrostates will be interpreted as three interconverting chemical states and represented by a model chemical reaction as follows: State 1 as a high energy, low entropy state referred to as D; State 2 as a lower energy, high entropy state referred to as A + B; and State 3 as the lowest energy, low entropy state, C.

individual microstates (six possible sides) to the different macrostates (three potential energy positions) illustrates the influence of entropy in biasing equilibrium distributions in a hypothetical chemical reaction at constant pressure (allowing the energy term to be analogous to enthalpy). Students then mapped their three states onto a model chemical reaction (Figure 1). A bimolecular reaction for State 2 was used to indicate a difference in entropy for the model reaction moving to either State 1 or State 3. To further illustrate the role of energy differences in observed probability, dice can be rolled with only one penny to alter potential energy differences and therefore the observed result (see Supplementary Tables S1 and S2). Finally, students were asked to predict how changing the load or the number of sides would affect the observed probability of the lowest energy state and connect these analogies to chemical concepts such as Gibbs free energy, entropy, and enthalpy.

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STUDENT EXERCISE This active learning exercise was performed in groups of two to four students and took approximately 25 min for a qualitative explanation and an additional 20 min for the complete quantitative analysis (see the Supporting Information for student worksheets and the instructor guide). This exercise has been conducted in six different classes, including three general chemistry classes, two biochemistry classes, and one physical B

DOI: 10.1021/acs.jchemed.9b00074 J. Chem. Educ. XXXX, XXX, XXX−XXX

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Figure 2. Calculation and interpretation summary. Each possible transition has been calculated for the two penny dice rolls in Table 1 following the example of the transition between State 2 and State 3. Interpretations of ΔPd, ΔPd*, and ΔS are given.

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QUANTITATIVE TREATMENT OF STUDENT EXERCISE DATA

59.3 = −0.507 35.7 % State 3 %C K= = % State 2 % (A + B)

ΔP d = −ln K = −ln

As an analogy of biased probability, this system can be used to explore many chemical concepts. Consider Gibbs free energy, a thermodynamic state function that can be derived from the ratio of observed populations of two states at equilibrium. Data from Table 1 can be used to calculate values analogous to Gibbs free energy, enthalpy, and entropy to allow further application of the loaded die exercise. This analysis can be done by comparing any two states (Figure 2). The following is an analysis of the conversion between States 2 and 3. Students work through the calculations and then compare and contrast their results to Gibbs free energy, enthalpy, and entropy. Rolls using dice loaded with two pennies were treated as observations of a chemical system at equilibrium. The equilibrium ratio considering only observations of States 2 and 3 is given as K. As this model is used as an analogy for Gibbs free energy, the negative natural logarithm of K is taken to give a unitless Gibbs-like probability value, ΔPd (eq 1). The value ΔPd does not include an R value, as found in true ΔG equations, because there is no appropriate R constant for this analogous macroscopic system. In this analogy, temperature is proportional to the ratio of the surface area of microstates between each macrostate. Because the dice are cubes, the ratio is unity and constant. Dice that deviate from cubic form but have that same number of sides could be interpreted as alternative temperature conditions (see Supplementary Figure S3). As temperature and entropy are intrinsically interconnected, so are the differences in surface area and the number of sides between states. Probability distributions for any given die are considered constant because different students can roll the same die and arrive at similar probabilities, provided students do not just drop the die (resulting in kinetically trapped rolls that do not tumble). In this analogy, ΔPd values are interpreted relative to each other on the basis of sign and magnitude.

where

(1)

Using the representative data from Table 2, ΔPd was determined to be −0.507, considering State 2 as the “reactant” and State 3 as the “product”. The value determined in this step is not Gibbs free energy but rather an analogous probabilitybased value useful for predicting the likelihood of rolling “reactant” and “product” macrostates. The negative value agrees with the observation that the “product” (State 3) was observed more frequently at equilibrium than the “reactant” (State 2), and therefore “free energy” decreases when moving to the lower-energy “product” state. Under constant pressure, the free energy difference between two macrostates depends on changes in enthalpy and entropy. Given our hypothetical chemical reaction and the physical nature of a loaded die, the gravitational potential energy caused by two pennies, ΔE2p, correlates to enthalpy, whereas the number of sides per macrostate determines entropy (ΔS), resulting in an equation analogous to the Gibbs free energy equation (eq 2). Temperature is dropped from eq 2 as this is a macroscopic physical analogy. ΔP d = ΔE2p − ΔS

(2)

The difference in potential energy can be isolated by comparing two microstates within different macrostates. The probability difference between two microstates, ΔPd*, is dependent only on the potential energy driving force caused by the loading of two pennies, ΔE2p. Because of the relatively low sample number, picking any of the four equivalent sides that define State 2 may give slightly different values for ΔE2p. The per-side average observation for State 2 limited the variability of low sample numbers on the basis of the assumption that the four sides for State 2 were essentially degenerate. ΔPd* and, therefore, ΔE2p were determined to be −1.897 (eq 3). C

DOI: 10.1021/acs.jchemed.9b00074 J. Chem. Educ. XXXX, XXX, XXX−XXX

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behave as state functions and help demonstrate the path independent property of state functions. The results of each transition can be interpreted using chemical language. For example, the transition from State 1 to State 2 is analogous to a chemical reaction that is spontaneous (negative ΔPd) and exothermic (negative ΔE2p) and has a gain of entropy (positive ΔS due to a gain in the number of microstates). This analysis was also followed through with the one penny die and a hypothetical 20-sided loaded die (see Supplementary Figures S4 and S5).

ΔP d* = ΔE2p − 0 = −ln K = −ln

% State 3 59.3 = −ln % State 2 (average per side) 8.9

= −1.897 = ΔE2p

(3)

The difference in the probability difference considering all or only single microstates, ΔPd or ΔPd*, respectively, results in the isolation of a numerical value for the difference in entropy between these two macrostates (eq 4).

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ΔP d* − ΔP d = (ΔE2p − 0) − (ΔE2p − ΔS) = −1.897 − ( −0.507) = − 1.390 = ΔS

SUMMARY This exercise provides a physical model that helps students visualize and conceptualize a core principle in thermodynamics: differences in entropy and potential energy between two states determine the equilibrium distribution of those states. Student handouts (see the Supporting Information) have been developed that highlight purely conceptual, primarily numerical, and combined approaches, which broadens the applicability of this exercise from introductory chemistry to upper-level undergraduate courses with specific application to Gibbs free energy.

(4)

The value for ΔS was determined to be −1.390, which agrees with the loss of entropy moving from State 2 to State 3. The difference in entropy between two states is defined as the natural logarithm of the ratio of the number of microstates (w) between two macrostates, thus allowing the numerical deriving of this ratio using the previously determined ΔS value (eq 5). A ratio of 0.25 or 1/4 was found using these data. This derived value correctly reflects the number of microstates of each macrostate considered. ij w yz w 1 ΔS = lnjjj 3 zzz ∴ 3 = e−1.390 = 0.2491 = jw z w 4 2 k 2{

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S Supporting Information *

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The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.9b00074.

Loading the die with only one penny resulted in fewer observations of the lowest energy state because of the entropic driving force being greater than the potential energy drive (see Supplementary Table S2). The potential energy driving force for a die loaded with a single penny, ΔE1p, was determined to be −1.025 (eq 6). There is still a significant bias toward the low-energy State 3 using only a single penny, but when entropy is taken into account, the high-entropy State 2 is more likely to be observed (see Table S1). ΔE1p = −ln = −1.025

ASSOCIATED CONTENT

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% State 3 39 = −ln % State 2 (average per side) 14

Additional data tables and summary figures for differential loading of 3D-printed dice, figures of the 3D CAD file for printing the dice and the Java script for simulated dice rolls, illustration of altered die geometry as a mimic of temperature, instructor’s sheet, and student handouts (PDF, DOCX) Custom object file for 3D printing loadable dice and Java script for simulating loaded dice rolls (ZIP)

AUTHOR INFORMATION

Corresponding Author

(6)

*E-mail: [email protected].

The one penny example reflects a smaller potential energy drive, ΔE1p, toward the favored state compared with that of the two penny example. An analogy between this difference in potential energy between one and two pennies to two chemical transformations, where one involves a single favorable bond formation and the other involves two favorable bond formations, can then be made. The chemical transformation that forms two bonds would have a greater energetic drive, just as the die with two pennies has a greater potential energy drive. The difference in potential energy between the two systems, ΔΔEd, is approximately proportional to the difference in the mass of the loads (eq 7). The increase in the potential difference reflects the doubling of the loaded mass from one to two pennies.

ORCID

James A. Hebda: 0000-0001-7671-1480 Zachary Aamold: 0000-0001-9897-0969 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Z.A. wrote the code for the available Java script to simulate loaded dice rolls. Many thanks to Joanna Mathew, Danielle Rogan, and Clara Zee for rolling many dice and reading through the paper. Additional thanks to Karla McCain, Lindsay Zack, and Jefferson Knight for critical reading and constructive feedback.

ΔΔE d = ΔE2p − ΔE1p = −1.897 − ( − 1.025) = − 0.872

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(7)

Calculations for all possible state transitions were performed for the two penny example (Figure 2). The total changes in both ΔPd and ΔE2p from State 1 to State 3 are the same as the sums of the changes from States 1 to 2 and 2 to 3. These data confirm that the empirical values for ΔPd and ΔPd* (ΔE2p)

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DOI: 10.1021/acs.jchemed.9b00074 J. Chem. Educ. XXXX, XXX, XXX−XXX