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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

A cylinder contains one liter of air at room temperature ( 300K) and atmospheric pressure 105N/m2. At one end of the cylinder is a massless piston, whose surface area is 0.01m2. Suppose that you push the piston in very suddenly, exerting a force of 2000N. The piston moves only one millimeter, before it is stopped by an immovable barrier of some sort.

(a) How much work have you done on this system?

(b) How much heat has been added to the gas?

(c) Assuming that all the energy added goes into the gas (not the piston or cylinder walls), by how much does the internal energy of the gas increase?

(d) Use the thermodynamic identity to calculate the change in the entropy of the gas (once it has again reached equilibrium).

In the text I showed that for an Einstein solid with three oscillators and three units of energy, the chemical potential is 渭=-系(where 系is the size of an energy unit and we treat each oscillator as a "particle"). Suppose instead that the solid has three oscillators and four units of energy. How does the chemical potential then compare to -系 ? (Don't try to get an actual value for the chemical potential; just explain whether it is more or less than -系.)

The results of either of the two preceding problems can also be applied to the vibrational motions of gas molecules. Looking only at the vibrational contribution to the heat capacity graph for H2shown in Figure 1.13, estimate the value of 蔚for the vibrational motion of anH2 molecule.

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.

Use the result of Problem 2.42 to calculate the temperature of a black hole, in terms of its mass M. (The energy is Mc2. ) Evaluate the resulting expression for a one-solar-mass black hole. Also sketch the entropy as a function of energy, and discuss the implications of the shape of the graph.
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