Can a computer program calculate how fast atoms spread through a solid? Diffusion (the slow movement of particles from crowded areas to empty ones) is described by a second-order differential equation. Solving that equation by hand is slow and difficult.
You write programs in the Scheme language that solve first-order and then second-order differential equations. Both programs return their expected results. Changing the constants in the equation changes the rate of diffusion.
Because the diffusion equation is a type of second-order equation, a working solver for that type can be extended to model real diffusion between solids.
Diffusion — the slow movement of particles from crowded areas to empty ones — is described by a second-order differential equation. Solving that equation by hand is slow and difficult, but a computer program can solve it numerically. When you change the constants in the formula, the predicted rate of diffusion between solids changes too.
Atoms in a solid move from crowded areas to empty ones at a rate governed by a second-order differential equation. Solving that equation with a computer program lets you calculate how fast diffusion happens and predict how the spread changes. When you adjust the constants in the equation, the rate of diffusion changes too.
A math equation becomes a set of step-by-step rules a computer can follow thousands of times faster than any human. Writing programs in the Scheme language, you translate first-order and then second-order differential equations into those rules. That speed makes it practical to model how atoms spread through a solid — a calculation that would be impossible to do by hand.
Method & Materials
You will learn how to program using "How to Design Programs" and Dr. Scheme, design a program to compute first-order and second-order differential equations, and work on the diffusion equation.
You will need a computer, a programming language such as Scheme, and knowledge of calculus.
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The results of the two programs designed showed that the constants in the second-order differential equation caused the results to change. This suggests that a different diffusion constant would change the rate of diffusion, and this should be researched further.
Why do this project?
This science project is unique because it uses computer programming to solve a complex equation and determine the rate of diffusion between solids.
Also Consider
Experiment variations to consider include researching the Improved Euler method, which is a more accurate way of solving differential equations, and researching the effects of different diffusion constants on the rate of diffusion.
Full project details
Additional information and source material for this project are available below.
