Use a computer to study the entropy, temperature, and heat capacity of an Einstein solid, as follows. Let the solid contain 50 oscillators (initially), and from 0 to 100 units of energy. Make a table, analogous to Table 3.2, in which each row represents a different value for the energy. Use separate columns for the energy, multiplicity, entropy, temperature, and heat capacity. To calculate the temperature, evaluate ΔU/ΔS for two nearby rows in the table. (Recall that U = qϵ for some constant ϵ.) The heat capacity (ΔU/ΔT) can be computed in a similar way. The first few rows of the table should look something like this:

q |
Ω |
S/k |
kT/ϵ |
C/Nk |

0 |
1 |
0 |
0 |
— |

1 |
50 |
3.91 |
.28 |
.12 |

2 |
1275 |
7.15 |
.33 |
.45 |

(In this table I have computed derivatives using a “centered-difference” approximation. For example, the temperature .28 is computed as 2/(7.15 – 0).) Make a graph of entropy vs. energy and a graph of heat capacity vs. temperature. Then change the number of oscillators to 5000 (to “dilute” the system and look at lower temperatures), and again make a graph of heat capacity vs. temperature. Discuss your prediction for the heat capacity, and compare it to the data for lead, aluminium, and diamond shown in below Figure. Estimate the numerical value of ϵ, in electron-volts, for each of those real solids.

Figure: Measured heat capacities at constant pressure (data points) forone mole each of three different elemental solids. The solid curves show the heatcapacity at constant volume predicted by the model used in Section 7.5, with thehorizontal scale chosen to best fit the data for each substance. At sufficiently hightemperatures, CV for each material approaches the value 3R predicted by theequipartition theorem. The discrepancies between the data and the solid curvesat high T are mostly due to the differences between CP and CV. At T = 0 alldegrees of freedom are frozen out, so both CP and CV go to zero. Data from Y. S.Touloukian, ed., Thermophysical Properties of Matter (Plenum, New York, 1970).