Modern Physics

Raymond A. Serway, Clement J. Moses, Curt A. Moye

Chapter 10

Statistical Physics - all with Video Answers

Educators

Chapter Questions

Problem 1

The Maxwell-Boltzmann Distribution
Verify that for a system of six distinguishable particles with total energy $8 E_{y}$ the probabilities of finding a particle with energies $1 E$ through $8 E$ are: 0.256,0.167 0.0978,0.0543,0.0272,0.0117,0.00388,0.000777.

Mihajlo Grcic

Mihajlo Grcic

Numerade Educator

Problem 2

Show that the most probable speed of a gas molecule is
$$
v_{\mathrm{mp}}=\sqrt{\frac{2 k_{\mathrm{B}} T}{m}}
$$
Note that the most probable speed corresponds to the point where the Maxwellian speed distribution curve, $n(v),$ has a maximum.

Mihajlo Grcic

Numerade Educator

Problem 3

Figure $\mathrm{P} 10.3$ shows an apparatus similar to that used by Otto Stern in 1920 to verify the Maxwell speed distribution. A collimated beam of gas molecules from an oven, $O,$ is allowed to enter a rapidly rotating cylinder when slit $S$ is coincident with the beam. The pulse of molecules created by the rapid rotation of $S$ then strikes and adheres to a glass plate detector, $D$. The velocity of a molecule may be determined from its position on the glass plate (fastest molecules to the right). The number of molecules arriving with a given velocity may be determined by measuring the density of molecules deposited on $D$ at a given position. Suppose that the oven contains a gas of bismuth molecules $\left(\mathrm{Bi}_{2}\right)$ at $850 \mathrm{K},$ and that the cylinder is $10 \mathrm{cm}$ in diameter and rotates at 6250 rpm. (a) Find the distance from $A$ of the impact points of molecules traveling at $\bar{v}, v_{\mathrm{rms}},$ and $v_{\mathrm{mp}} \cdot(\mathrm{b})$ Why do you suppose that measurements were originally made with $\mathrm{Bi}_{2}$ instead of $\mathrm{O}_{2}$ or $\mathrm{N}_{2} ?$

Mihajlo Grcic

Numerade Educator

Problem 4

Energy levels known as tunneling levels have been observed from $\mathrm{CN}^{-}$ ions incorporated into $\mathrm{KCl}$ crystals. These levels arise from the rotational motion of $\mathrm{CN}^{-}$ ions as they tunnel, at low temperatures, through barriers separating crystalline potential minima. According to one model, the tunneling levels should consist of four equally spaced levels in the far IR with spacing of $12.41 \times 10^{-5} \mathrm{eV}\left(1 \mathrm{cm}^{-1}\right) .$ Assuming that the CN $^{-}$ ions obey Maxwell-Boltzmann statistics and using Figure $\mathrm{P} 10.4,$ calculate and sketch the expected appearance of the absorption spectra (strength of absorption vs. energy) at $4 \mathrm{K}$ and $1 \mathrm{K}$. Assume equal transition probabilities for all allowed transitions, and make the strength of absorption proportional to the peak height. Use $k_{\mathrm{B}}=8.62 \times 10^{-5} \mathrm{eV} / \mathrm{K}$
$$\Delta=1 \mathrm{cm}^{-1}=12.41 \times 10^{-5} \mathrm{eV}$$
Tunneling energy levels. Allowed transitions are indicated by vertical arrows. The degeneracy of each level is indicated in parentheses.

Mihajlo Grcic

Numerade Educator

Problem 5

Fit an exponential curve $P(E)=A e^{-B E}$ to Figure 10.2 to see how closely the system of six distinguishable particles comes to an exponential distribution. Use the values at energies of 0 and $1 E$ to determine $A$ and $B.$

Mihajlo Grcic

Numerade Educator

Problem 6

The energy difference between the first excited state of mercury and the ground state is $4.86 \mathrm{eV}$. (a) If a sample of mercury vaporized in a flame contains $10^{20}$ atoms in thermal equilibrium at $1600 \mathrm{K},$ calculate the number of atoms in the $n=1$ (ground) and $n=2$ (first excited) states. (Assume the Maxwell-Boltzmann distribution applies and that the $n=1$ and $n=2$ states have equal statistical weights.) (b) If the mean lifetime of the $n=2$ state is $\tau$ seconds, the transition probability is $1 / \tau$ and the number of photons emitted per second by the $n=2$ state is $n_{2} / \tau,$ where $n_{2}$ is the number of atoms in state 2.
If the lifetime of the $n=2$ state is 100 ns, calculate the power (in watts) emitted by the hot atoms.

Mihajlo Grcic

Numerade Educator

Problem 7

Consider a molecule with a permanent electric dipole moment $\mathbf{p}$ placed in an electric field $\mathcal{E},$ with $\mathbf{p}$ aligned either parallel or antiparallel to $\mathcal{E}$. (a) Recall that the energy of a dipole in an electric field is given by $E=-\mathbf{p} \cdot \mathcal{E},$ and show that this system has two allowed energy states separated by $2 p \mathcal{E}$. (b) Assume a groundstate energy of 0 and an excited-state energy of $2 p \mathcal{E}$ and degeneracies in the ratio $g(2 p \varepsilon) / g(0)=2/1.$
For a collection of $N$ molecules obeying MaxwellBoltzmann statistics, calculate the ratio of the number of molecules in the excited state to the number in the ground state at temperature $T$. (c) For high $T$ such that $k_{\mathrm{B}} T \gg 2 p \varepsilon,$ the ratio of the number of molecules in the upper state to the number in the lower is 2 to 1 . Taking reasonable estimates of $p=1.0 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}$ and $\mathcal{E}=1.0 \times 10^{6} \mathrm{V} / \mathrm{m},$ find the temperature at which the ratio has fallen by a measurable $10 \%$ to
1.9 to $1 .$ (d) Calculate the average energy $\bar{E}$ at $T$ and show that $\bar{E} \rightarrow 0$ as $T \rightarrow 0$ and $\bar{E} \rightarrow 4 p \mathcal{E} / 3$ as $T \rightarrow \infty$
(e) Find $E_{\text {total from }} \bar{E},$ and show that the heat capacity for this two-level system is
$$
C=\left(\frac{N k_{\mathrm{B}}}{2}\right)\left(\frac{2 p \mathcal{E}}{k_{\mathrm{B}} T}\right)^{2}\left(\frac{e^{2 p \varepsilon / k_{\mathrm{B}} T}}{\left(1+\frac{1}{2} e^{2 p E / k_{\mathrm{B}} T}\right)^{2}}\right)
$$
(f) Sketch $C$ as a function of $2 p \varepsilon / k_{\mathrm{B}} T$. Find the value of $2 p \varepsilon / k_{\mathrm{B}} T$ at which $C$ is a maximum, and explain, in physical terms, the dependence of Con $T.$

Raj Bala

Numerade Educator

Problem 8

Use the distribution function given in Exercise 10.1
$$
n(E) d E=\frac{2 \pi(N / V)}{\left(\pi k_{\mathrm{B}} T\right)^{3 / 2}} E^{1 / 2} e^{-E / k_{\mathrm{p}} T} d E
$$
to find (a) the most probable kinetic energy of gas molecules at temperature $T,$ (b) the mean kinetic energy at $T,$ and $(c)$ the root mean square kinetic energy at $T.$

Mihajlo Grcic

Numerade Educator

Problem 9

The light from a heated atomic gas is shifted in frequency because of the random thermal motion of light emitting atoms toward or away from an observer. Estimate the fractional Doppler shift $\left(\Delta f / f_{0}\right),$ assuming that light of frequency $f_{0}$ is emitted in the rest frame of each atom, that the light-emitting atoms are iron atoms in a star at temperature $6000 \mathrm{K},$ and that the atoms are moving relative to an observer with the mean speed
$$
\bar{v}=\pm \sqrt{\frac{8 k_{\mathrm{B}} T}{\pi m}}
$$
Must we use the relativistic Doppler shift formulas
$$
f=f_{0} \frac{\sqrt{1 \pm v / c}}{\sqrt{1 \mp v / c}}
$$
for this calculation? Such thermal Doppler shifts are measurable and are used to determine stellar surface temperatures.

Mihajlo Grcic

Numerade Educator

Problem 10

Helium atoms have spin 0 and are therefore bosons.
(a) Must we use the Bose-Einstein distribution at standard temperature and pressure to describe helium gas, or will the Maxwell-Boltzmann distribution suffice?
(b) Helium becomes a liquid with a density of $0.145 \mathrm{g} / \mathrm{cm}^{3}$ at $4.2 \mathrm{K}$ and atmospheric pressure. Must the Bose-Einstein distribution be used in this case? Explain.

Mihajlo Grcic

Numerade Educator

Problem 11

To obtain a more clearly defined picture of the Fermi-Dirac distribution, consider a system of 20 Fermi-Dirac particles sharing 94 units of energy. By drawing diagrams like Figure $\mathrm{P} 10.11,$ show that there are nine different microstates. Using Equation 10.2 calculate and plot the average number of particles in each energy level from 0 to $14 E$. Locate the Fermi energy at $0 \mathrm{K}$ on your plot from the fact that electrons at $0 \mathrm{K}$ fill all the levels consecutively up to the Fermi energy. (At $0 \mathrm{K}$ the system no longer has 94 units of energy, but has the minimum amount of 90E.)
One of the nine equally probable microstates for 20 FD particles with a total energy of $94 E.$

Mihajlo Grcic

Numerade Educator

Problem 12

(a) Find the average energy per photon for photons in thermal equilibrium with a cavity at temperature $T.$
(b) Calculate the average photon energy in electron volts at $T=6000 \mathrm{K}$. Hint: Two useful integrals are $\int_{0}^{\infty} \frac{z^{2} d z}{e^{z}-1} \approx 2.41 \quad$ and $\quad \int_{0}^{\infty} \frac{z^{3} d z}{e^{z}-1}=\frac{\pi^{4}}{15}$

Mihajlo Grcic

Numerade Educator

Problem 13

(a) Show that the specific heat of any substance in the Einstein model equals 5.48 cal/mol\cdot $\mathrm{K}$ at $T=T_{\mathrm{E}}.$
(b) Using this result, estimate the Einstein temperatures of lead, aluminum, and silicon from Figure 10.9.
(c) Using the temperatures found in part (b), calculate the Einstein specific heats of each element at $50-\mathrm{K}$ intervals and compare with the experimental results shown in Figure $10.9 .$ You may wish to photocopy Figure 10.9 and plot your calculated values on this figure for easy comparison.

Shahab Ullah

Numerade Educator

Problem 14

The Fermi energy of copper at $0 \mathrm{K}$ is 7.05 eV. (a) What is the average energy of a conduction electron in copper at $0 \mathrm{K}$ ? (b) At what temperature would the average energy of a molecule of an ideal gas equal the energy obtained in (a)? (See Problem 16).

Mihajlo Grcic

Numerade Educator

Problem 15

The Fermi energy of aluminum is $11.63 \mathrm{eV}$. (a) Assuming that the free electron model applies to aluminum, calculate the number of free electrons per unit volume at low temperatures. (b) Determine the valence of aluminum by dividing the answer found in part (a) by the number of aluminum atoms per unit volume as calculated from the density and the atomic weight. Note that aluminum has a density of $2.70 \mathrm{g} / \mathrm{cm}^{3}.$

Mihajlo Grcic

Numerade Educator

Problem 16

Show that the average kinetic energy of a conduction electron in a metal at $0 \mathrm{K}$ is given by $\bar{E}=3 E_{\mathrm{F}} / 5 .$ By way of contrast, note that all of the molecules in an ideal gas at 0 K have zero energy. Hint: Use the standard definition of an average given by
$$
\bar{E}=\frac{\int_{0}^{\infty} E g(E) / \mathrm{FD}(E) d E}{N / V}
$$
where $E_{\mathrm{F}}$ is in electron volts when $n$ is in electrons per cubic meter.

Mihajlo Grcic

Numerade Educator

Problem 17

Although we usually apply Fermi-Dirac statistics to free electrons in a conductor, Fermi-Dirac statistics apply to any system of spin $\frac{1}{2}$ particles, including protons and neutrons in a nucleus. since protons are distinguishable from neutrons, assume that each set of nucleons independently obeys the Fermi-Dirac distribution and that the number of protons equals the number of neutrons. Using these ideas, estimate $E_{\mathrm{F}}$ and $E$ for the nucleons in Zn. (Zn has 30 protons, 34 neutrons, and a radius of $\left.4.8 \times 10^{-15} \mathrm{m} .\right)$ Are your answers reasonable? Explain.

Mihajlo Grcic

Numerade Educator

Problem 18

Show that Equation 10.44 can be expressed as
$$
E_{\mathrm{F}}=\left(3.65 \times 10^{-19}\right) n^{2 / 3} \mathrm{eV}
$$
where $E_{\mathrm{F}}$ is in electron volts when $n$ is in electrons per cubic meter.

Mihajlo Grcic

Numerade Educator

Problem 19

Calculate the probability that a conduction electron in copper at $300 \mathrm{K}$ has an energy equal to $99 \%$ of the Fermi energy.

Mihajlo Grcic

Numerade Educator

Problem 20

Find the probability that a conduction electron in a metal has an energy equal to the Fermi energy at the temperature $300 \mathrm{K}.$

Mihajlo Grcic

Numerade Educator

Problem 21

Sodium is a monovalent metal having a density of $0.971 \mathrm{g} / \mathrm{cm}^{3}$ and a molar mass of $23.0 \mathrm{g} / \mathrm{mol} .$ Use this information to calculate (a) the density of charge carriers, (b) the Fermi energy, and (c) the Fermi speed for sodium.

Mihajlo Grcic

Numerade Educator

Problem 22

Calculate the energy of a conduction electron in silver at $800 \mathrm{K}$ if the probability of finding the electron in that state is $0.95 .$ Assume that the Fermi energy for silver is 5.48 eV at this temperature.

Mihajlo Grcic

Numerade Educator

Problem 23

Consider a cube of gold $1 \mathrm{mm}$ on an edge. Calculate the approximate number of conduction electrons in this cube whose energies lie in the range from 4.000 to $4.025 \mathrm{eV}$ at $300 \mathrm{K} .$ Assume $E_{\mathrm{F}}(300 \mathrm{K})=E_{\mathrm{F}}(0).$

Mihajlo Grcic

Numerade Educator

Problem 24

(a) Consider a system of electrons confined to a three-dimensional box. Calculate the ratio of the number of allowed energy levels at 8.5 eV to the number of allowed energy levels at $7.0 \mathrm{eV}$. (b) Cop per has a Fermi energy of 7.0 eV at 300 K. Calculate the ratio of the number of occupied levels at an energy of 8.5 eV to the number of occupied levels at the Fermi energy. Compare your answer with that obtained in part (a).

Mihajlo Grcic

Numerade Educator

Problem 25

Consider a system of $10^{4}$ oxygen molecules per cubic centimeter at a temperature $T$. Calculate and plot a graph of the Maxwell distribution, $n(v),$ as a function of $v$ and $T$. Use your program to evaluate $n(v)$ for speeds ranging from $v=0$ to $v=2000 \mathrm{m} / \mathrm{s}$ (in intervals of $100 \mathrm{m} / \mathrm{s}$ ) at temperatures of (a) $300 \mathrm{K}$ and
(b) $1000 \mathrm{K}$. (c) Make graphs of $n(v)$ versus $v$ and use the graph at $T=1000 \mathrm{K}$ to estimate the number of molecules per cubic centimeter having speeds between $800 \mathrm{m} / \mathrm{s}$ and $1000 \mathrm{m} / \mathrm{s}$ at $T=1000 \mathrm{K} .$ (d) Calculate
and indicate on each graph the root mean square speed, the average speed, and the most probable speed (see Problem 2 ).

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

Graph the Fermi-Dirac distribution function, Equation $10.25,$ versus energy. Plot $f(E)$ versus $E$ for (a) $T=0.2 T_{\mathrm{F}}$ and $(\mathrm{b}) T=0.5 T_{\mathrm{F}},$ where $T_{\mathrm{F}}$ is the Fermi temperature, defined by Equation 10.45 .

Dominador Tan

Numerade Educator

Problem 27

Copper has a Fermi energy of $7.05 \mathrm{eV}$ at $300 \mathrm{~K}$ and a conduction electron concentration of $8.49 \times 10^{28} \mathrm{~m}^{-3} .$ Calculate and plot (a) the density of states, $g(E)$; versus $E ;$ (b) the particle distribution function, $n(E),$ as a function of energy at $T=0 \mathrm{~K} ;$ and
(c) the particle distribution function versus $E$ at $T=1000 \mathrm{~K}$. Your energy scales should range from $E=0$ to $E=10 \mathrm{eV}$.

Chai Santi

Numerade Educator