Entropy and temperature


This program calculates the entropy of two Einstein solids that can exchange energy. The plot shows the entropy of the two systems, SA (blue curve) and SB (green curve), and the entropy S (red curve) of the composite system. NA and NB are the number of particles in each system. The total energy is given by E = EA + EB.

    The entropy is computed by using the binomial coefficients to determine the number of states as a function of energy. In each case the entropy is plotted versus EA, the energy of system A. The entropy of system A is given by

S(EA) = k ln Ω(EA),

where ΩA(EA) is the number of microstates of system A with energy EA and k is Boltzmann's constant. We will choose units such that k = 1. The entropy of system B is given by a similar expression. The entropy of the composite system is given by

S(EA) = k ln [ΩA(EA) ΩB(EB)] = SA + SB.

The total entropy of the system is given by


We will see that as the total number of particles increases, the total entropy can be approximated by


where is the most probable value of EA; that is, the value of EA for which S(EA) is a maximum.


  1. Choose various values of the parameters and describe the qualitative behavior of SA, SB, and S as a function of EA
  2. The message box gives the relative error in the total entropy when it is approximated by its maximum value as a function of EA. Describe the behavior of this error as you increase the size of the system and keep the energy per particle a constant.
  3. Discuss why the total entropy has a maximum. What can you say about the slopes of SA and SB at the value of EA where S(EA) is a maximum?
  4. The values of the inverse slopes dSA/dEA and dSB/dEB can be obtained by clicking your mouse on the corresponding curves. The values of the (inverse) slopes are given in the lower right corner. We know that the two systems are in equilibrium when their temperatures are equal. Discuss the relation of the slopes to the temperature.

Java Classes

Updated 28 December 2009.