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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鈫�+N鈫�

U=渭BN鈫�-N鈫�

M=渭N鈫�-N鈫�=-UB

惟=N!N鈫�!(N鈭�N鈫�)!=N!N!!N鈫�!

T=螖U螖S

C=螖U螖T

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

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

惟(N,q)鈮�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=q系, where 系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鈫�鈭�, the heat capacity is C=Nk. (Hint: When x is very small, ex鈮�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/系, 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 系, 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(系/kT)2 in the final answer. When the smoke clears, you should find C=Nk1-112(系/kT)2.

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?

When the sun is high in the sky, it delivers approximately 1000 watts of power to each square meter of earth's surface. The temperature of the surface of the sun is about 6000K, while that of the earth is about 300K.

(a) Estimate the entropy created in one year by the flow of solar heat onto a square meter of the earth.

(b) Suppose you plant grass on this square meter of earth. Some people might argue that the growth of the grass (or of any other living thing) violates the second law of thermodynamics, because disorderly nutrients are converted into an orderly life form. How would you respond?

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:

渭(z)=-kTlnVN2蟺mkTh23/2+mgz.

(You can derive this result from either the definition 渭=-T(鈭�S/鈭�N)U,Vor the formula 渭=(鈭�U/鈭�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.

In Problem 2.32you computed the entropy of an ideal monatomic gas that lives in a two-dimensional universe. Take partial derivatives with respect to U,A, and N to determine the temperature, pressure, and chemical potential of this gas. (In two dimensions, pressure is defined as force per unit length.) Simplify your results as much as possible, and explain whether they make sense.
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