Chapter 9: Statistical Mechanics

Show that in the limit$\mathit{h}{\mathit{\omega }}_{\mathbf{0}}\mathbf{\ll }{\mathit{k}}_{\mathbf{\text{B}}}\mathit{T}$. Equation (9.15) becomes (9.28).

Given an arbitrary thermodynamic system, which is larger. the number of possible macro-states. or the numberof possible microstates, or is it impossible to say? Explain your answer. (For most systems, both are infinite, but il is still possible to answer the question,)

We claim that the famous exponential decrease of probability with energy is natural, the vastly most probable and disordered state given the constraints on total energy and number of particles. It should be a state of maximum entropy ! The proof involves mathematical techniques beyond the scope of the text, but finding support is good exercise and not difficult. Consider a system of$\mathbf{11}$oscillators sharing a total energy of just$\mathbf{5}{\mathbf{h\omega }}_{\mathbf{0}}$ . In the symbols of Section 9.3. $\mathbf{N}\mathbf{=}\mathbf{11}$and$\mathbf{M}\mathbf{=}\mathbf{5}$ .

1. Using equation$\mathbf{(}\mathbf{9}\mathbf{-}\mathbf{9}\mathbf{)}$ , calculate the probabilities of$\mathbf{n}$ , being$\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}$ and$\mathbf{3}$ .
2. How many particles${\mathbf{N}}_{\mathbf{n}}$ , would be expected in each level? Round each to the nearest integer. (Happily. the number is still $\mathbf{11}$. and the energy still$\mathbf{5}{\mathbf{h\omega }}_{\mathbf{0}}$ .) What you have is a distribution of the energy that is as close to expectations is possible. given that numbers at each level in a real case are integers.
3. Entropy is related to the number of microscopic ways the macro state can be obtained. and the number of ways of permuting particle labels with${\mathbf{N}}_{\mathbf{0}}$ ,${\mathbf{N}}_{\mathbf{1}}\mathbf{,}{\mathbf{N}}_{\mathbf{2}}$ , and ${\mathbf{N}}_3$fixed and totaling$\mathbf{11}$ is$\frac{\mathbf{11}\mathbf{!}}{\mathbf{(}{\mathbf{N}}_{\mathbf{0}}\mathbf{!}{\mathbf{N}}_{\mathbf{1}}\mathbf{!}{\mathbf{N}}_{\mathbf{2}}\mathbf{!}{\mathbf{N}}_{\mathbf{3}}\mathbf{!}\mathbf{)}}$ . (See Appendix J for the proof.) Calculate the number of ways for your distribution.
4. Calculate the number of ways if there were$\mathbf{6}$ particles in$\mathbf{n}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5}$ in$\mathbf{n}\mathbf{=}\mathbf{1}$ and none higher. Note that this also has the same total energy.
5. Find at least one other distribution in which the $\mathbf{11}$ oscillators share the same energy, and calculate the number of ways.

6. What do your finding suggests?

Figure 9.8 cannot do justice to values at the very highspeed end of the plot. This exercise investigates how small it really gets. However, although integrating the Maxwell speed distribution over the full range of speeds from 0 to infinity can be carried out (the so-called Gaussian integrals of Appendix K), over any restricted range, it is one of those integrals that. unfortunately. cannot be done in closed form. Using a computational aid of your choice. show that the fraction of molecules moving faster thanis; faster than$6v_{\text{rms}}$is$-{10}^{-23}$; and faster than$10{\text{v}}_{\text{ms}}$is$~{10}^{-64}$. where$v_{rms}$" from Exercise 41, is$\sqrt{3k_{\text{B}}T/m}$. (Exercise 48 uses these values in an interesting application.)

For a room$3.0m$tall, by roughly what percent does the probability of an air molecule being found at the ceiling differ from that of an equal speed molecule being found at the floor? Ignore any variation in temperature from floor to ceiling.

To obtain the Maxwell speed distribution, we assumed a uniform temperature. kinetic-only energy of $\mathbf{E}=\left\{m\left(v_x^2+v_y^2+v_z^2\right)\right.$, and we assumed that we wished to find the average of an arbitrary function of X. Along the way, we obtained probability per unit height speed,$P\left(v\right)$.

a) Assuming a uniform temperature and an energy of$\left.E=\frac{1}{2}m\left(v_x^2+v_y^2+v_z^3\right)+mgy\right)$and assuming we wish to find the average of an arbitrary function of Y, obtain a probability per unit height,$P\left(y\right)$ .

b) Assuming a temperature of$300\text{K}$. how much less the density of the atmosphere'sat an altitude of(about$3000$ft) than at sea level'?

(c) What of the${\text{O}}_2$in the atmosphere?

A particle subject to a planet's gravitational pull has a total mechanical energy given by $E_{\text{mechanical}}=\frac{1}{2}m{v}^2-\frac{GMm}{r}$, whereis the particle's mass.M the planet's mass, and Gthe gravitational constant$6.67\times {10}^{-11}$$\text{N}·{\text{m}}^3/{\text{kg}}^2$. It may escape if its energy is zero that is, if its positive KE is equal in magnitude to the negative PE holding if to the surface. Suppose the particle is a gas molecule in an atmosphere.

(a) Temperatures in Earth's atmosphere may reach $1000\text{K}$. Referring to the values obtained in Exercise 45 and given that $R_{\text{Earth}}=6.37\times$${10}^6\text{m}$and $M_{\text{Earth}}=5.98\times {10}^{24}\text{kg}$. should Earth be able to "hold on" to hydrogen $(1\text{g}/\text{mol})$? 10 nitrogens $(28\text{g}/\text{mol})$? (Note: An upper limit on the number of molecules in Earth's atmosphere is about ${10}^{{}^{-18}}$).

(b) The moon's mass is $0.0123$times Earth's. its radius 0.26 times Earth's, and its surface temperatures rise to $370K$. Should it be able to hold on to these gases?

Verify that the probabilities shown in Table 9.1 for four distinguishable oscillators sharing energy $\mathbf{2}\mathbf{\delta E}$agree with the exact probabilities given by equation (9-9).

A scientifically untrained but curious friend asks, "When I walk into a room, is there a chance that all the air will be on the other side?" How do you answer this question?

You have six shelves, one above the other and all above the floor, and six volumes of an encyclopedia, A, B, C, D, E and F.

(a) list all the ways you can arrange the volumes with five on the floor and one on the sixth/top shelf. One way might be$(ABCDE\_,\_,\_,\_,\_F)$.

(b) List all the ways you can arrange them with four on the floor and two on the third shelf.

(c) Show that there are many more ways, relative to pans (a) and (b), to arrange the six volumes with two on the floor and two each on the first and second shelves. (There are several ways to answer

this, but even listing them all won't take forever it's fewer than.)

(d) Suddenly, a fantastic change! All six volumes are volume X-it's impossible to tell them apart. For each of the three distributions described in parts (a), (b), and (c), how many different (distinguishable) ways are there now?

(e) If the energy you expend to lift a volume from the floor is proportional to a shelf's height, how do the total energies of distributions (a), (b), and (c) compare?

(I) Use these ideas to argue that the relative probabilities of occupying the lowest energy states should be higher for hosons than for classically distinguishable particles.

(g) Combine these ideas with a famous principle to argue that the relative probabilities of occupying the lowest states should he lower for fermions than for classically distinguishable particles.