In below Problem 1 you showed that the multiplicity of an Einstein solid containing N oscillators and q energy units is approximately
(a) Starting with this formula, find an expression for the entropy of an Einstein solid as a function of N and q. Explain why the factors omitted from the formula have no effect on the entropy, when N and q are large.
(b) Use the result of part (a) to calculate the temperature of an Einstein solid as a function of its energy. (The energy is U = qϵ, where ϵ is a constant.) Be sure to simplify your result as much as possible.
(c) Invert the relation you found in part (b) to find the energy as a function of temperature, then differentiate to find a formula for the heat capacity.
(d) Show that, in the limit T →∞ the heat capacity is C = Nk. (Hint: When x is very small, ex ≈ 1 + x.) Is this the result you would expect? Explain.
(e) Make a graph (possibly using a computer) of the result of part (c). To avoid awkward numerical factors, plot C/Nk vs. the dimensionlcss variable t = kT/ϵ, for t in the range from 0 to about 2. Discuss your prediction for the heat capacity at low temperature, comparing to the data for lead, aluminum, and diamond shown in Figure. Estimate the value of ϵ, in electron-volts, for each of those real solids.
(f) Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through x 3 in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than (ϵ/kT)2 in the final answer. When the smoke clears, you should find .
Use Stirling’s approximation to show that the multiplicity of an Einstein solid, for any large values of N and q, is approximately
The square root in the denominator is merely large, and can often be neglected. However, it is needed in below Problem 2 (Hint: First show that Do not neglect the in Stirling’s approximation.)
This problem gives an alternative approach to estimating the width of the peak of the multiplicity function for a system of two large Einstein solids.
a) Consider two identical Einstein solids, each with N oscillators, in thermal contact with each other. Suppose that the total number of energy units in the combined system is exactly 2N. How many different macrostates (that is, possible values for the total energy in the first, solid) are there for this combined system?
b) Use the result of above Problem to find an approximate expression, for the total number of microstates for the combined system. (Hint: Traet the combined system as a single Einstein solid. Do not throw away factors of “large” numbers, since you will eventually be dividing two “very large” numbers that are nearly equal.
c) The most likely macrostate for this system is (of course) the one in which the energy is shared equally between the two solids. Use the result of above Problem to find an approximate expression for the multiplicity of this macrostate.
d)You can get a rough idea of the “sharpness” of the multiplicity function by comparing your answers to parts (b) and (c). Part (c) tells you the Height of the peak, while part (b) tells yon the total area under the entire graph. As a very crude approximation, pretend that the peak’s shape is rectangular. In this case, how wide would it be? Out of all the macrostates, what fraction have reasonably large probabilities? Evaluate this fraction numerically for the case N = 1023.
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).
Chapter 12 Lecture Notes Things to remember o Remember that properties of gases can be related by PV=nRT (ideal gas law) o Remember that molecules of gases are far apart, while molecules in liquid and solid are closely packed o Remember that molecules can have polar bonds, due to differences in electronegativity of bonded atoms o Intramolecular bonds are strong Covalent: 100-400 kJ/mol Ionic: 700 -1100 kJ/mol o Intramolecular forces: forces holding atoms together to form molecules (covalent bonds, ionic bonds) o Intermolecular forces: electrostatic interactions between molecules that are weaker than forces between oppos