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Chapter 3: Q. 3.35 (page 119)

In the text I showed that for an Einstein solid with three oscillators and three units of energy, the chemical potential is μ=-ϵ(where ϵis 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 -ϵ ? (Don't try to get an actual value for the chemical potential; just explain whether it is more or less than -ϵ.)

Short Answer

Expert verified

μ<-∈

Step by step solution

01

Explanation of Solution

Given:

For N=3and q=3the chemical potential is μ=-∈

02

Calculation

The chemical potential formula is

μ=ΔUΔNS

The entropy for N=3and q=3is

S=klnΩ

S=kln∣N+q-1|q-1|q∣3+3-1

S=kln|3-1|3∣5¯

S=kln|2|3|2||2|

S=kln(10)

The entropy must be constant throughout.

The entropy for N=3and q=4is,

S=kln⁡ΩS=kln⁡∣N+q−1|q−1|q∣3+3−1S=kln⁡|3−1|3∣5_S=kln⁡|2|3|2||2|S=kln⁡(10)

Thus, the entropy increases so to reduce the entropy to its original value.

03

Conclusion 

The chemical potential is lowered.

μ<-∈

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Most popular questions from this chapter

Suppose you have a mixture of gases (such as air, a mixture of nitrogen and oxygen). The mole fraction xiof any species iis defined as the fraction of all the molecules that belong to that species: xi=Ni/Ntotal. The partial pressure Piof species iis then defined as the corresponding fraction of the total pressure: Pi=xiP. Assuming that the mixture of gases is ideal, argue that the chemical potential μiof species i in this system is the same as if the other gases were not present, at a fixed partial pressure Pi.

Use a computer to study the entropy, temperature, and heat capacity of an Einstein solid, as follows. Let the solid contain 50 oscillators (initially), and from 0 to 100 units of energy. Make a table, analogous to Table 3.2, in which each row represents a different value for the energy. Use separate columns for the energy, multiplicity, entropy, temperature, and heat capacity. To calculate the temperature, evaluate ΔU/ΔSfor two nearby rows in the table. (Recall that U=qϵfor some constant ϵ.) The heat capacity (ΔU/ΔT)can be computed in a similar way. The first few rows of the table should look something like this:

(In this table I have computed derivatives using a "centered-difference" approximation. For example, the temperature .28is computed as 2/(7.15-0).) Make a graph of entropy vs. energy and a graph of heat capacity vs. temperature. Then change the number of oscillators to 5000 (to "dilute" the system and look at lower temperatures), and again make a graph of heat capacity vs. temperature. Discuss your prediction for the heat capacity, and compare it to the data for lead, aluminum, and diamond shown in Figure 1.14. Estimate the numerical value of εin electron-volts, for each of those real solids.

A liter of air, initially at room temperature and atmospheric pressure, is heated at constant pressure until it doubles in volume. Calculate the increase in its entropy during this process.

In the experiment of Purcell and Pound, the maximum magnetic field strength was 0.63Tand the initial temperature was 300K. Pretending that the lithium nuclei have only two possible spin states (in fact they have four), calculate the magnetization per particle, M/N, for this system. Take the constant μto be 5×10-8eV/T. To detect such a tiny magnetization, the experimenters used resonant absorption and emission of radio waves. Calculate the energy that a radio wave photon should have, in order to flip a single nucleus from one magnetic state to the other. What is the wavelength of such a photon?

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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