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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

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.