Problem 15.2: Compare the temperature of distributions
EJS Simulation currently not available.
This animation allows you to change the total number of particles and the total energy of a particular system of 11 boxes, each with an energy of 0 to 10. Restart. You can also change the number of states in a box currently set to 40. To make a change, you must push return (so the input box is no longer yellow) and you must push the "Set Value and Run" button. The animation calculates the values of α and β in the following expressions so that the total energy is constant and the total number of particles is constant (β = 1/kBT for all distributions):
|Bose-Einstein:||ni = g/(exp(α + βεi) - 1)|
|Fermi-Dirac:||ni = g/(exp(α + βεi) + 1)|
|Maxwell-Boltzmann (Classical):||ni = g/exp(α + βεi)|
where ni is the number of particles in region in phase space, εi is the energy of that region (or box) and g is the number of states per box, also called a density of states.
- For situations where the average energy per particle is the same, how does the temperature of Fermi-Dirac distribution compare with a classical system? Specifically, for an average energy per particle of 5, how do the temperatures compare? What about for an average energy per particle of 1?
- The average energy per particle of electrons in a metal (Fermi gas) are close to the Fermi level at very low temperatures (T near zero). If the electrons in a metal with a Fermi energy of 5 eV behaved classically (like ideal gas particles), what temperature would be necessary for them to have the same average energy per particle as they actually have at T near zero?
Problem by Anne J. Cox
Script by Anne J. Cox and William F. Junkin III
Applet built using EJS (Francisco Esquembre), Open Source Libraries