In Problem 2.18 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 , where 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 , the heat capacity is . (Hint: When x is very small, .) 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 vs. the dimensionless variable , 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 1.14. Estimate the value of , 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 in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than in the final answer. When the smoke clears, you should find .
(d) It is shown that in the limit , the heat capacity is .
(e) The graph can be made as:
The multiplicity of an Einstein solid is given as:
The entropy is given as:
By substituting the given value in the above equation, we get,
The multiplicity for Einstein solid can be given as:
Stirling approximation of large n is given as:
For large and , when compared to the power terms, the square root terms are small. As they have a negligible effect, they can be neglected. Hence, the approximate value can be written as:
Hence, the entropy of an Einstein solid containing oscillators and energy units is given as:
For large values of and , the square root terms are small when compared to the power terms. Hence they are neglected as they have a very negligible effect on the entropy. Therefore the approximate value of multiplicity is given as:
The total energy of the system is given as:
The entropy of the system is given as:
The temperature can be given as:
The temperature as a function of its energy can be given as:
The temperature as a function of energy is given as:
On simplifying, we get,
Now, differentiating the energy with respect to temperature to get the heat capacity as:
The energy as a function of temperature can be given as:
Expression of Heat capacity is:
Heat capacity in terms of temperature is given as:
For very small values of .
By expanding the exponential for higher temperatures, we get
The heat capacity for very large values of can be given as:
The derived expression is analogous to the expression of heat capacity at high temperatures.
So, is approximately the expected result.
The heat capacity is given as:
The heat capacity for low temperature can be given as:
Based on the derived equation, the graph of versus can be made as follows:
For lead, heat capacity is maximum at
For aluminum, heat capacity is maximum at
For diamond, heat capacity is maximum at
The heat capacity for low temperatures can be given as:
The graph can be made as:
The values of are as follows:
The graph of heat capacity versus temperature for lead, aluminum, and diamond is approximately anomalous as in figure 1.14.
The heat capacity is given as:
The heat capacity can be modified as:
The Taylor expansion is given as:
Hence, the above equation becomes:
Using the Taylor series, the above equation becomes:
Solving further and neglecting the higher powers than 2, we get,
Hence, the required expression is:
In Problem 1.55 you used the virial theorem to estimate the heat capacity of a star. Starting with that result, calculate the entropy of a star, first in terms of its average temperature and then in terms of its total energy. Sketch the entropy as a function of energy, and comment on the shape of the graph.
As shown in Figure 1.14, the heat capacity of diamond near room temperature is approximately linear in T. Extrapolate this function up to , and estimate the change in entropy of a mole of diamond as its temperature is raised from to . Add on the tabulated value at (from the back of this book) to obtain .
In Section 2.5 I quoted a theorem on the multiplicity of any system with only quadratic degrees of freedom: In the high-temperature limit where the number of units of energy is much larger than the number of degrees of freedom, the multiplicity of any such system is proportional to , where is the total number of degrees of freedom. Find an expression for the energy of such a system in terms of its temperature, and comment on the result. How can you tell that this formula for cannot be valid when the total energy is very small?
Suppose you have a mixture of gases (such as air, a mixture of nitrogen and oxygen). The mole fraction of any species is defined as the fraction of all the molecules that belong to that species: . The partial pressure of species is then defined as the corresponding fraction of the total pressure: . Assuming that the mixture of gases is ideal, argue that the chemical potential of species in this system is the same as if the other gases were not present, at a fixed partial pressure .
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