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'The Diffusion Game' - Using Symbolic Mathematics Software to Play the Game on a Large Scale

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Abstract

The primary goal of the activities in this document is to observe how the statistical 'microscopic' behavior of individual particles ultimately determines a certain 'macroscopic' behavior in a large system. Diffusion provides a simple example; a particle on one side of a container, through random kinetic collision events, is either kicked to the opposite side of the container or remains on its current side (assuming the container is split into two halves). This behavior is simulated here for a large number of particles by using a Mathcad random number generator. An interactive process is carried out whereby a random integer is generated during each step for each particle, and the particle is moved to the other side if a certain pre-selected outcome is obtained. By carrying out the simulation in Mathcad, one can easily alter the ultimate number of steps in the simulation, the number of particles in the simulation, and the 'probability criteria' that is used to determine whether a particle switches sides. The outcome of these simulations are analyzed within the context of equilibrium and non-equilibrium states, entropy maximization, and Fick's law of diffusion, reinforcing one's understanding of the microscopic origin of these phenomena.

This Mathcad document is designed for student use in a Junior-Senior level physical chemistry course. Students should have had at least one year of calculus and physics, as well as an introductory knowledge of thermodynamics. While not necessary, it will also be helpful if students are familiar with concepts such as equilibrium and non-equilibrium states, the statistical origin of entropy, and Fick's law of diffusion. Some basic Mathcad skills are assumed (how to enter and solve equations, both numerically and symbolically, as well as the ability to graph functions). This document requires Mathcad 11 or higher.

Figure 1. Variation in the left (dotted line) and right populations (solid line) as a function of steps for a two chamber box containing 50 particles. At each step particles move from one side of the box to the other based on the probability criterion determined using the upper ceiling whole number obtained from a random number generated using the probability criterion F=50. The probability fluctuates around 25 in a random fashion after equilibrium is established.

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Version	Download
Mathcad 11	diffusion.mcd

Version	Download
Mathcad 11	diffusion.pdf

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Keywords

Audience: Upper-Division Undergraduate;

Domain: Physical Chemistry;

Pedagogy: Computer-Based Learning;

Topics: Mathematics / Symbolic Mathematics; Statistical Mechanics; Transport Properties;

JCE Citation