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Chapter 3: Q. 3.18 (page 103)

Use a computer to reproduce Table 3.2 and the associated graphs of entropy, temperature, heat capacity, and magnetization. (The graphs in this section are actually drawn from the analytic formulas derived below, so your numerical graphs won't be quite as smooth.)

Short Answer

Expert verified

Table 3.2 can be reproduced by using a computer as follows:

The graphs can also be made as:

Step by step solution

01

Given Information

Table 3.2 showing the thermodynamic properties of a two-state paramagnet consisting of 100 elementary dipoles is given as follows:

N=100dipoles

02

Calculation

The table values are calculated with the help of the following formulae:

N=N↓+N↑

U=μBN↓-N↑

M=μN↓-N↑=-UB

Ω=N!N↓!(N−N↓)!=N!N!!N↓!

T=ΔUΔS

C=ΔUΔT

The table can be computed as follows:

Also,

The graphs are computed as follows:

Entropy as a function of energy for a two-state paramagnetic material can be made as:

The graph of Temperature as a function of energy can be made as:

Graph of Heat capacity can be made as:

Graph of magnetization can be made as:

03

Final answer

The table is reconstructed as:

The graphs are made based on the table values as:

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Most popular questions from this chapter

Can a "miserly" system, with a concave-up entropy-energy graph, ever be in stable thermal equilibrium with another system? Explain.

Polymers, like rubber, are made of very long molecules, usually tangled up in a configuration that has lots of entropy. As a very crude model of a rubber band, consider a chain of N links, each of length ℓ(see Figure 3.17). Imagine that each link has only two possible states, pointing either left or right. The total length L of the rubber band is the net displacement from the beginning of the first link to the end of the last link.

(a) Find an expression for the entropy of this system in terms of N and NR, the number of links pointing to the right.
(b) Write down a formula for L in terms of N and NR.
(c) For a one-dimensional system such as this, the length L is analogous to the volume V of a three-dimensional system. Similarly, the pressure P is replaced by the tension force F. Taking F to be positive when the rubber band is pulling inward, write down and explain the appropriate thermodynamic identity for this system.
(d) Using the thermodynamic identity, you can now express the tension force F in terms of a partial derivative of the entropy. From this expression, compute the tension in terms of L, T, N, and ℓ.
(e) Show that when L << Nℓ, the tension force is directly proportional to L (Hooke's law).
(f) Discuss the dependence of the tension force on temperature. If you increase the temperature of a rubber band, does it tend to expand or contract? Does this behavior make sense?
(g) Suppose that you hold a relaxed rubber band in both hands and suddenly stretch it. Would you expect its temperature to increase or decrease? Explain. Test your prediction with a real rubber band (preferably a fairly heavy one with lots of stretch), using your lips or forehead as a thermometer. (Hint: The entropy you computed in part (a) is not the total entropy of the rubber band. There is additional entropy associated with the vibrational energy of the molecules; this entropy depends on U but is approximately independent of L.)

In Problem 2.18 you showed that the multiplicity of an Einstein solid containing N oscillators and q energy units is approximately

Ω(N,q)≈q+Nqqq+NNN

(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 ϵis 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/Nkvs. the dimensionless 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 1.14. 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 x3 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 C=Nk1-112(ϵ/kT)2.

Consider a monatomic ideal gas that lives at a height z above sea level, so each molecule has potential energy mgzin addition to its kinetic energy.

(a) Show that the chemical potential is the same as if the gas were at sea level, plus an additional term mgz:

μ(z)=-kTlnVN2πmkTh23/2+mgz.

(You can derive this result from either the definition μ=-T(∂S/∂N)U,Vor the formula μ=(∂U/∂N)S,V.

(b) Suppose you have two chunks of helium gas, one at sea level and one at height z, each having the same temperature and volume. Assuming that they are in diffusive equilibrium, show that the number of molecules in the higher chunk is

N(z)=N(0)e-mgz/kT

in agreement with the result of Problem 1.16.

A bit of computer memory is some physical object that can be in two different states, often interpreted as 0 and 1. A byte is eight bits, a kilobyte is 1024=210bytes, a megabyte is 1024 kilobytes, and a gigabyte is 1024 megabytes.

(a) Suppose that your computer erases or overwrites one gigabyte of memory, keeping no record of the information that was stored. Explain why this process must create a certain minimum amount of entropy, and calculate how much.

(b) If this entropy is dumped into an environment at room temperature, how much heat must come along with it? Is this amount of heat significant?
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