This is a model of two solids in thermal contact. When the solids are allowed to interact their internal energy is redistributed until thermal equilibrium is established. The model is designed to illustrate how equilibrium temperature is related to the initial temperatures and quantities of the interacting solids. It also illustrates how the entropy of each solid changes as a result of the interaction.
created with NetLogo
view/download model file: Einstein_Solid.nlogo
The solids are made up of patches, representing quantum oscillators, which have been randomly assigned discrete units of energy in such a way that the average energy per cell in each solid equates to the initial temperature specified by sliders. When the model is running, for each time step each patch takes its current amount of energy and distributes it randomly to its neighbours or to itself. Note the boundary cells do not gain or lose energy. In this way no energy is lost from the system. That is the solids are insulated from the environment and the system is closed. Over time the energy is redistributed throughout both solids until the a dynamic equilibrium is established.
Chose a divider location on the worldview using the divider-location slider. This specifies the boundary between the two solids and sets up their relative size. Set initial temperatures for the two solids using the T-left and T-right sliders. Then run the model using the go button.
This model can be used to illustrate a number of properties in thermodynamics.
First, the temperature of the solids at equilibrium lies between the initial temperatures of the solids. In this model the solids have the same specific heats, so the equilibrium temperature should follow the law m1(T-T1)=m2(T2-T) where m1 and m2 represent the "mass" of the left and right solid respectively. m1 can be represented by the percentage on the divider-location slider, and m2 is then the rest.
Second, the model can be used to illustrate and investigate entropy changes. As the system moves to equilibrium, the entropy of the hotter solid decreases but the entropy of the cooler solid increases by more so that the total entropy of the system increases. Entropy is calculated from the total energy and number of states (patches) of each solid.
Try varying the relative size of each solid. Make the hotter solid small or larger and observe how this affects final temperature and the entropy changes. Consider exploring the properties of the cooling/heating curve using the Behaviourspace feature of Netlogo to see if the curve fits the predictions based on Newton's law of cooling.
This model can be extended to included a slower rate of diffusion of energy, perhaps by only allowing some fraction of available energy in each patch to be shared with neighbours. You might consider allowing the different solids to diffuse their energy at different rates, for example. It might also be an idea to give the different solids different specific heat capacities, perhaps by giving the solid on the right patches which share the energy between two different modes of oscillation.
See the other Entropy Models in this series
Copyright 2006 David McAvity
This model was created at the Evergreen State College, in Olympia Washington
and is part of a series of applets designed to illustrate principles in physics and biology.
Funding was provided by the Plato Royalty Grant.
The model may be freely used, modified and redistributed provided this copyright is included and it not used for profit.
Contact David McAvity at mcavityd@evergreen.edu if you have questions about its use.