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Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 43

Molecules and Solids

Educators


Problem 1

Potassium chloride is an ionically bonded molecule that is sold as a salt substitute for use in a low-sodium diet. The electron affinity of chlorine is 3.6 eV. An energy input of 0.70 eV is required to form separate $\mathrm{K}^{+}$ and $\mathrm{Cl}^{-}$ ions from separate $\mathrm{K}$ and $\mathrm{Cl}$ atoms. What is the ionization energy of $\mathrm{K}$ ?

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Problem 2

$A K^{+}$ ion and a $C l^{-}$ ion are separated by a distance of $5.00 \times 10^{-10} \mathrm{m} .$ Assuming the two ions act like charged particles, determine (a) the force each ion exerts on the other and (b) the potential energy of the two-ion system in electron volts.

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Problem 3

A van der Waals dispersion force between helium atoms produces a very shallow potential well, with a depth on the order of 1 meV. At approximately what temperature would you expect helium to condense?

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Problem 4

In the potassium iodide (KI) molecule, assume the K and I atoms bond ionically by the transfer of one electron from K to I. (a) The ionization energy of K is 4.34 eV, and the electron affinity of I is 3.06 eV. What energy is needed to transfer an electron from $\mathrm{K}$ to I, to form $\mathrm{K}^{+}$ and $\mathrm{I}^{-}$ ions from neutral atoms? This quantity is sometimes called the activation energy $E_{a}$ (b) A model potential energy function for the KI molecule is the Lennard-Jones potential:
$$U(r)=4 \epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]+E_{a}$$
where $r$ is the inter nuclear separation distance and $\epsilon$ and $\sigma$ are adjustable parameters. The $E_{a}$ term is added to ensure the correct asymptotic behavior at large $r$ . At the equilibrium separation distance, $r=r_{0}=0.305 \mathrm{nm}, U(r)$ is a minimum, and $d U / d r=0 .$ In addition, $U\left(r_{0}\right)$ is the negative of the dissociation energy: $U\left(r_{0}\right)=-3.37$ eV. Find $\sigma$ and $\epsilon$ (c) Calculate the force needed to break up a KI molecule. (d) Calculate the force constant for small oscillations about $r=r_{0} .$ Suggestion: Set $r=r_{0}+s,$ where $s / r_{0}<<1$ , and expand $U(r)$ in powers of $s / r_{0}$ up to second-order terms.

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Problem 5

One description of the potential energy of a diatomic molecule is given by the Lennard–Jones potential,
$$U=\frac{A}{r^{12}}-\frac{B}{r^{6}}$$
where $A$ and $B$ are constants and $r$ is the separation distance between the atoms. For the $\mathrm{H}_{2}$ molecule, take $A=0.124 \times 10^{-120} \mathrm{eV} \cdot \mathrm{m}^{12}$ and $B=1.488 \times 10^{-60} \mathrm{eV} \cdot \mathrm{m}^{6} .$ Find (a) the separation distance $r_{0}$ at which the energy of the molecule is a minimum and (b) the energy $E$ required to break up the $\mathrm{H}_{2}$ molecule.

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Problem 6

One description of the potential energy of a diatomic molecule is given by the Lennard–Jones potential,
$$U=\frac{A}{r^{12}}-\frac{B}{r^{6}}$$
where $A$ and $B$ are constants and $r$ is the separation distance between the atoms. Find, in terms of $A$ and $B,$ (a) the value $r_{0}$ at which the energy is a minimum and (b) the energy $E$ required to break up a diatomic molecule.

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Problem 7

Assume the distance between the protons in the $\mathrm{H}_{2}$ molecule is $0.750 \times 10^{-10} \mathrm{m} .$ (a) Find the energy of the first excited rotational state, with $J=1 .$ (b) Find the wavelength of radiation emitted in the transition from $J=1$ to $J=0$ .

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Problem 8

The cesium iodide (CsI) molecule has an atomic separation of 0.127 $\mathrm{nm}$ (a) Determine the energy of the second excited rotational state, with $J=2 .$ (b) Find the frequency of the photon absorbed in the $J=1$ to $J=2$ transition.

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Problem 9

An HCl molecule is excited to its second rotational energy level, corresponding to $J=2 .$ If the distance between its nuclei is 0.1275 nm, what is the angular speed of the molecule about its center of mass?

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Problem 10

The photon frequency that would be absorbed by the NO molecule in a transition from vibration state $v=0$ to $v=1$ with no change in rotation state, is 56.3 THz. The bond between the atoms has an effective spring constant of 1 530 N/m. (a) Use this information to calculate the reduced mass of the NO molecule. (b) Compute a value for $\mu$ using Equation 43.4. (c) Compare your results to parts (a) and (b) and explain their difference, if any.

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Problem 11

The CO molecule makes a transition from the $J= 1$ to the $J=2$ rotational state when it absorbs a photon of frequency $2.30 \times 10^{11} \mathrm{Hz}$ . (a) Find the moment of inertia of this molecule from these data. (b) Compare your answer with that obtained in Example 43.1 and comment on the significance of the two results.

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Problem 12

Why is the following situation impossible? The effective force constant of a vibrating HCl molecule is $k=480 \mathrm{N} / \mathrm{m} .$ A beam of infrared radiation of wavelength $6.20 \times 10^{3} \mathrm{nm}$ is directed through a gas of HCl molecules. As a result, the molecules are excited from the ground vibrational state to the first excited vibrational state.

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Problem 13

The effective spring constant describing the potential energy of the HI molecule is 320 N/m and that for the HF molecule is 970 N/m. Calculate the minimum amplitude of vibration for (a) the HI molecule and (b) the HF molecule.

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Problem 14

A diatomic molecule consists of two atoms having masses $m_{1}$ and $m_{2}$ separated by a distance $r$ . Show that the moment of inertia about an axis through the center of mass of the molecule is given by Equation $43.3, I=\mu r^{2} .$

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Problem 15

The atoms of an NaCl molecule are separated by a distance $r=0.280 \mathrm{nm} .$ Calculate (a) the reduced mass of an NaCl molecule, (b) the moment of inertia of an NaCl molecule, and (c) the wavelength of radiation emitted when an NaCl molecule undergoes a transition from the $J=2$ state to the $J=1$ state.

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Problem 16

The rotational spectrum of the HCl molecule contains lines with wavelengths of 0.060 4, 0.069 0, 0.080 4, 0.096 4, and 0.120 4 mm. What is the moment of inertia of the molecule?

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Problem 17

The nuclei of the $\mathrm{O}_{2}$ molecule are separated by a distance $1.20 \times 10^{-10} \mathrm{m} .$ The mass of each oxygen atom in the molecule is $2.66 \times 10^{-26} \mathrm{kg} .$ (a) Determine the rotational energies of an oxygen molecule in electron volts for the levels corresponding to $J=0,1,$ and $2 .$ (b) The effective force constant $k$ between the atoms in the oxygen molecule is $177 \mathrm{N} / \mathrm{m} .$ Determine the vibrational energies (in electron volts) corresponding to $v=0,1,$ and $2 .$

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Problem 18

Figure P 43.18 is a model of a benzene molecule. All atoms lie in a plane, and the carbon atoms $\left(m_{\mathrm{C}}=1.99 \times 10^{-26} \mathrm{kg}\right)$ form a regular hexagon, as do the hydrogen atoms $\left(m_{\mathrm{H}}=\right.1.67 \times 10^{-27} \mathrm{kg} ) .$ The carbon atoms are 0.110 nm apart center to center, and the adjacent carbon and hydrogen atoms are 0.100 nm apart center to center. (a) Calculate the moment of inertia of the molecule about an axis perpendicular to the plane of the paper through the center point $O.$ (b) Determine the allowed rotational energies about this axis.

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Problem 19

(a) In an HCl molecule, take the $\mathrm{Cl}$ atom to be the isotope $^{35} \mathrm{Cl}$ The equilibrium separation of the H and Cl atoms is 0.127 46 nm. The atomic mass of the H atom is 1.007 825 u and that of the $^{35} \mathrm{Cl}$ atom is 34.968 853 u. Calculate the longest wavelength in the rotational spectrum of this molecule. (b) What If? Repeat the calculation in part (a), but take the Cl atom to be the isotope $^{37} \mathrm{Cl}$ which has atomic mass 36.965 903 u. The equilibrium separation distance is the same as in part (a). (c) Naturally occurring chlorine contains approximately three parts of $^{55} \mathrm{Cl}$ to one part of $^{37} \mathrm{Cl}$ . Because of the two different Cl masses, each line in the microwave rotational spectrum of HCl is split into a doublet as shown in Figure P 43.19. Calculate the separation in wavelength between the doublet lines for the longest wavelength.

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Problem 20

Estimate the moment of inertia of an HCl molecule from its infrared absorption spectrum shown in Figure P 43.19.

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Problem 21

An $\mathrm{H}_{2}$ molecule is in its vibrational and rotational ground states. It absorbs a photon of wavelength 2.2112$\mu \mathrm{m}$ and makes a transition to the $v=1, J=1$ energy level. It then drops to the $v=0, J=2$ energy level while emitting a photon of wavelength $2.4054 \mu \mathrm{m} .$ Calculate (a) the moment of inertia of the $\mathrm{H}_{2}$ molecule about an axis through its center of mass and perpendicular to the $\mathrm{H}-\mathrm{H}$ bond, (b) the vibrational frequency of the $\mathrm{H}_{2}$ molecule, and (c) the equilibrium separation distance for this molecule.

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Problem 22

Photons of what frequencies can be spontaneously emitted by CO molecules in the state with $v=1$ and $J=0 ?$

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Problem 23

Most of the mass of an atom is in its nucleus. Model the mass distribution in a diatomic molecule as two spheres of uniform density, each of radius $2.00 \times 10^{-15} \mathrm{m}$ and mass $1.00 \times 10^{-26} \mathrm{kg},$ located at points along the $y$ axis as in Active Figure $43.5 \mathrm{a},$ and separated by $2.00 \times 10^{-10} \mathrm{m}$ . Rotation about the axis joining the nuclei in the diatomic molecule is ordinarily ignored because the first excited state would have an energy that is too high to access. To see why, calculate the ratio of the energy of the first excited state for rotation about the $y$ axis to the energy of the first excited state for rotation about the $x$ axis.

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Problem 24

Use a magnifying glass to look at the grains of table salt that come out of a salt shaker. Compare what you see with Figure 43.10a. The distance between a sodium ion and a nearest-neighbor chlorine ion is 0.261 nm. (a) Make an order-of-magnitude estimate of the number N of atoms in a typical grain of salt. (b) What If? Suppose you had a number of grains of salt equal to this number N. What would be the volume of this quantity of salt?

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Problem 25

Use Equation 43.18 to calculate the ionic cohesive energy for $\mathrm{NaCl}$ . Take $\alpha=1.7476, r_{0}=0.281 \mathrm{nm},$ and $m=8$.

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Problem 26

Consider a one-dimensional chain of alternating singly-ionized positive and negative ions. Show that the potential energy associated with one of the ions and its interactions with the rest of this hypothetical crystal is
$$U(r)=-k_{e} \alpha \frac{e^{2}}{r}$$
where the Madelung constant is $\alpha=2 \ln 2$ and $r$ is the distance between ions. Suggestion: Use the series expansion for $\ln (1+x) .$

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Problem 27

Sodium is a mono valent 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 and (b) the Fermi energy of sodium.

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Problem 28

(a) State what the Fermi energy depends on according to the free-electron theory of metals and how the Fermi energy depends on that quantity. (b) Show that Equation 43.25 can be expressed as $E_{\mathrm{F}}=\left(3.65 \times 10^{-19}\right) n_{e}^{2 / 3}$ where $E_{\mathrm{F}}$ is in electron volts when $n_{e}$ is in electrons per cubic meter. (c) According to Table 43.2, by what factor does the free-electron concentration in copper exceed that in potassium? (d) Which of these metals has the larger Fermi energy? (e) By what factor is the Fermi energy larger? (f) Explain whether this behavior is predicted by Equation 43.25.

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Problem 29

When solid silver starts to melt, what is the approximate fraction of the conduction electrons that are thermally excited above the Fermi level?

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Problem 30

(a) Find the typical speed of a conduction electron in copper, taking its kinetic energy as equal to the Fermi energy, 7.05 eV. (b) Suppose the copper is a current-carrying wire. How does the speed found in part (a) compare with a typical drift speed (see Section 27.1) of electrons in the wire of 0.1 mm/s?

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Problem 31

The Fermi energy of copper at 300 K is 7.05 eV. (a) What is the average energy of a conduction electron in copper at 300 K? (b) At what temperature would the average translational energy of a molecule in an ideal gas be equal to the energy calculated in part (a)?

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Problem 32

Consider a cube of gold 1.00 mm on an edge. Calculate the approximate number of conduction electrons in this cube whose energies lie in the range 4.000 to 4.025 eV.

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Problem 33

Calculate the energy of a conduction electron in silver at 800 K, assuming the probability of finding an electron in that state is 0.950. The Fermi energy of silver is 5.48 eV at this temperature.

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Problem 34

(a) Consider a system of electrons confined to a three-dimensional box. Calculate the ratio of the number of allowed energy levels at 8.50 eV to the number at 7.05 eV. (b) What If? Copper has a Fermi energy of 7.05 eV at 300 K. Calculate the ratio of the number of occupied levels in copper at an energy of 8.50 eV to the number at the Fermi energy. (c) How does your answer to part (b) compare with that obtained in part (a)?

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Problem 35

For copper at 300 K, calculate the probability that a state with an energy equal to 99.0% of the Fermi energy is occupied.

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Problem 36

For a metal at temperature $T$ , calculate the probability that a state with an energy equal to $\beta E_{\mathrm{F}}$ is occupied where $\beta$ is a fraction between 0 and 1.

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Problem 37

An electron moves in a three-dimensional box of edge length $L$ and volume $L^{3}$ . The wave function of the particle is $\psi=A \sin \left(k_{x} x\right) \sin \left(k_{y} y\right) \sin \left(k_{z} z\right) .$ Show that its energy is given by Equation 43.20 ,
$$E=\frac{\hbar^{2} \pi^{2}}{2 m_{e} L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)$$
where the quantum numbers $\left(n_{x}, n_{y}, n_{z}\right)$ are integers $\geq 1 .$ Suggestion: The Schrodinger equation in three dimensions may be written
$$\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+\frac{\partial^{2} \psi}{\partial z^{2}}\right)=(U-E) \psi$$

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Problem 38

Why is the following situation impossible? A hypothetical metal has the following properties: its Fermi energy is 5.48 eV, its density is $4.90 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ , its molar mass is 100 $\mathrm{g} / \mathrm{mol}$ , and it has one free electron per atom.

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Problem 39

Show that the average kinetic energy of a conduction electron in a metal at 0 $\mathrm{K}$ is $E_{\mathrm{arg}}=\frac{3}{5} E_{\mathrm{F}} .$ Suggestion: In general, the average kinetic energy is
$$E_{\mathrm{avg}}=\frac{1}{n_{e}} \int_{0}^{\infty} E N(E) d E$$
where $n_{e}$ is the density of particles, $N(E) d E$ is given by Equation 43.22, and the integral is over all possible values of the energy.

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Problem 40

The longest wavelength of radiation absorbed by a certain semiconductor is $0.512 \mu \mathrm{m} .$ Calculate the energy gap for this semiconductor.

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Problem 41

The energy gap for silicon at 300 K is 1.14 eV. (a) Find the lowest-frequency photon that can promote an electron from the valence band to the conduction band. (b) What is the wavelength of this photon?

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Problem 42

Light from a hydrogen discharge tube is incident on a CdS crystal. (a) Which spectral lines from the Balmer series are absorbed and (b) which are transmitted?

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Problem 43

A light-emitting diode (LED) made of the semiconductor GaAsP emits red light $(\lambda=650 \mathrm{nm}) .$ Determine the energy-band gap $E_{g}$ for this semiconductor.

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Problem 44

Most solar radiation has a wavelength of 1$\mu \mathrm{m}$ or less. (a) What energy gap should the material in a solar cell have if it is to absorb this radiation? (b) Is silicon an appropriate solar cell material (see Table 43.3)? Explain your answer.

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Problem 45

You are asked to build a scientific instrument that is thermally isolated from its surroundings. The isolation container may be a calorimeter, but these design criteria could apply to other containers as well. You wish to use a laser external to the container to raise the temperature of a target inside the instrument. You decide to use a diamond window in the container. Diamond has an energy gap of 5.47 eV. What is the shortest laser wavelength you can use to warm the sample inside the instrument?

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Problem 46

When a phosphorus atom is substituted for a silicon atom in a crystal, four of the phosphorus valence electrons form bonds with neighboring atoms and the remaining electron is much more loosely bound. You can model the electron as free to move through the crystal lattice. The phosphorus nucleus has one more positive charge than does the silicon nucleus, however, so the extra electron provided by the phosphorus atom is attracted to this single nuclear charge $+e .$ The energy levels of the extra electron are similar to those of the electron in the Bohr hydrogen atom with two important exceptions. First, the Coulomb attraction between the electron and the positive charge on the phosphorus nucleus is reduced by a factor of 1$/ \mathrm{K}$ from what it would be in free space (see Eq. 26.21 ), where $\kappa$ is the dielectric constant of the crystal. As a result, the orbit radii are greatly increased over those of the hydrogen atom. Second, the influence of the periodic electric potential of the lattice causes the electron to move as if it had an effective mass $m^{*},$ which is quite different from the mass $m_{e}$ of a free electron. You can use the Bohr as if it had an effective mass $m^{*},$ which is quite different from the mass $m_{e}$ of a free electron. You can use the Bohr important role in semiconductor devices. Assume $\kappa=11.7$ for silicon and $m^{*}=0.220 m_{c}$ (a) Find a symbolic expression for the smallest radius of the electron orbit in terms of $a_{0},$ the Bohr radius. (b) Substitute numerical values to find the numerical value of the smallest radius. (c) Find a symbolic expression for the energy levels $E_{n}^{\prime}$ of the electron in the Bohr orbits around the donor atom in terms of $m_{e}, m^{*},$ $\kappa,$ and $E_{n},$ the energy of the hydrogen atom in the Bohr model. (d) Find the numerical value of the energy for the ground state of the electron.

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Problem 47

Assuming $T=300 \mathrm{K},$ (a) for what value of the bias voltage $\Delta V$ in Equation 43.27 does $I=9.00 I_{0} ?$ (b) What If? What if $I=-0.900 I_{0} ?$

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Problem 48

A diode is at room temperature so that $k_{\mathrm{B}} T=0.0250 \mathrm{eV}$. Taking the applied voltages across the diode to be $+1.00 \mathrm{V}$ (under forward bias) and $-1.00 \mathrm{V}$ (under reverse bias), calculate the ratio of the forward current to the reverse current if the diode is described by Equation 43.27 .

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Problem 49

You put a diode in a microelectronic circuit to protect the system in case an untrained person installs the battery backward. In the correct forward-bias situation, the current is 200 mA with a potential difference of 100 mV across the diode at room temperature (300 K). If the battery were reversed, so that the potential difference across the diode is still 100 mV but with the opposite sign, what would be the magnitude of the current in the diode?

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Problem 50

A diode, a resistor, and a battery are connected in a series circuit. The diode is at a temperature for which $k_{\mathrm{B}} T=25.0 \mathrm{meV},$ and the saturation value of the current is $I_{0}=1.00 \mu \mathrm{A} .$ The resistance of the resistor is $R=745 \Omega$ and the battery maintains a constant potential difference of $\varepsilon=2.42 \mathrm{V}$ between its terminals. (a) Use Kirchhoff's loop rule to show that
$$\boldsymbol{\varepsilon}-\Delta V=I_{0} R\left(e^{e \Delta V / k_{\mathrm{B}} T}-1\right)$$
where $\Delta V$ is the voltage across the diode. (b) To solve this transcendental equation for the voltage $\Delta V,$ graph the left-hand side of the above equation and the right-hand side as functions of $\Delta V$ and find the value of $\Delta V$ at which the curves cross. (c) Find the current $I$ in the circuit. (d) Find the ohmic resistance of the diode, defined as the ratio $\Delta V / I,$ at the voltage in part (b). (e) Find the dynamic resistance of the diode, which is defined as the derivative $d(\Delta V) / d I$ , at the voltage in part (b).

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Problem 51

A superconducting ring of niobium metal 2.00 cm in diameter is immersed in a uniform 0.020 0-T magnetic field directed perpendicular to the ring and carries no current. Determine the current generated in the ring when the magnetic field is suddenly decreased to zero. The inductance of the ring is $3.10 \times 10^{-8} \mathrm{H} .$

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Problem 52

A direct and relatively simple demonstration of zero DC resistance can be carried out using the four-point probe method. The probe shown in Figure P 43.52 consists of a disk of YBa_ $\mathrm{Cu}_{3} \mathrm{O}_{7}$ (a high- $T_{c}$ superconductor) to which four wires are attached. Current is maintained through the sample by applying a DC voltage between points $a$ and $b,$ and it is measured with a DC ammeter. The current can be varied with the variable resistance $R$ . The potential difference $\Delta V_{c b}$ between $c$ and $d$ is measured with a digital volt-meter. When the probe is immersed in liquid nitrogen, the sample quickly cools to $77 \mathrm{K},$ below the critical temperature of the material, 92 $\mathrm{K}$ . The current remains approximately constant, but $\Delta V_{c d}$ drops abruptly to zero. (a) Explain this observation on the basis of what you know about super-conductors. (b) The data in the accompanying table represent actual values of $\Delta V_{c d}$ for different values of $I$ taken on the sample at room temperature in the senior author's laboratory. A $6-\mathrm{V}$ battery in series with a variable resistor $R$ supplied the current. The values of $R$ ranged from 10$\Omega$ to $100 \Omega .$ Make an $I-\Delta V$ plot of the data and determine whether the sample behaves in a linear manner. (c) From the data, obtain a value for the DC resistance of the sample at room temperature. (d) At room temperature, it was found that $\Delta V_{c d}=2.234 \mathrm{mV}$ for $I=100.3 \mathrm{mA},$ but after the sample was cooled to $77 \mathrm{K}, \Delta V_{c d}=0$ and $I=98.1 \mathrm{mA}$ . What do you think might have caused the slight decrease in current?

Current Versus Potential Difference $\Delta V_{c d}$ Measured in a Bulk Ceramic Sample of $Y B a_{2} C u_{3} O_{7-\delta}$ at Room Temperature

(Table Cant Copy)

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Problem 53

A thin rod of superconducting material 2.50 cm long is placed into a 0.540-T magnetic field with its cylindrical axis along the magnetic field lines. (a) Sketch the directions of the applied field and the induced surface current. (b) Find the magnitude of the surface current on the curved surface of the rod.

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Problem 54

The effective spring constant associated with bonding in the $\mathrm{N}_{2}$ molecule is $2297 \mathrm{N} / \mathrm{m} .$ The nitrogen atoms each have a mass of $2.32 \times 10^{-26} \mathrm{kg},$ and their nuclei are 0.120 nm apart. Assume the molecule is rigid. The first excited vibrational state of the molecule is above the vibrational ground state by an energy difference $\Delta E .$ Calculate the $J$ value of the rotational state that is above the rotational ground state by the same energy difference $\Delta E$ .

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Problem 55

The hydrogen molecule comes apart (dissociates) when it is excited internally by 4.48 eV. Assuming this molecule behaves like a harmonic oscillator having classical angular frequency $\omega=8.28 \times 10^{14} \mathrm{rad} / \mathrm{s}$ , find the highest vibrational quantum number for a state below the 4.48 -eV dissociation energy.

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Problem 56

The Fermi–Dirac distribution function can be written as
$$f(E)=\frac{1}{e^{\left(E-E_{F}\right) / k_{\mathrm{B}} T}+1}=\frac{1}{e^{\left(E / F_{\mathrm{F}}-1\right) T_{\mathrm{F}} / T}+1}$$
where $T_{\mathrm{F}}$ is the Fermi temperature, defined according to
$$k_{\mathrm{B}} T_{\mathrm{F}} \equiv E_{\mathrm{F}}$$
(a) Write a spreadsheet to calculate and plot $f(E)$ versus $E / E_{\mathrm{F}}$ at a fixed temperature $T .$ (b) Describe the curves obtained for $T=0.1 T_{\mathrm{F}}, 0.2 T_{\mathrm{F}},$ and 0.5$T_{\mathrm{F}}$.

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Problem 57

Under pressure, liquid helium can solidify as each atom bonds with four others, and each bond has an average energy of $1.74 \times 10^{-23} \mathrm{J}$ . Find the latent heat of fusion for helium in joules per gram. (The molar mass of He is 4.00 g/mol.)

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Problem 58

The dissociation energy of ground-state molecular hydrogen is 4.48 eV, but it only takes 3.96 eV to dissociate it when it starts in the first excited vibrational state with $J=0 .$ Using this information, determine the depth of the $\mathrm{H}_{2}$ molecular potential-energy function.

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Problem 59

Starting with Equation 43.17, show that the ionic cohesive energy of an ionically bonded solid is given by Equation 43.18.

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Problem 60

(a) Starting with Equation 43.17, show that the force exerted on an ion in an ionic solid can be written as
$$F=-k_{e} \alpha \frac{e^{2}}{r^{2}}\left[1-\left(\frac{r_{0}}{r}\right)^{m-1}\right]$$
where $\alpha$ is the Madelung constant and $r_{0}$ is the equilibrium separation. (b) Imagine that an ion in the solid is displaced a small distance $s$ from $r_{0} .$ Show that the ion experiences a restoring force $F=-K s,$ where
$$K=\frac{k_{e} \alpha e^{2}}{r_{0}^{3}}(m-1)$$
(c) Use the result of part $(\mathrm{b})$ to find the frequency of vibration of a $\mathrm{Na}^{+}$ ion in $\mathrm{NaCl}$ . Take $m=8$ and use the value $\alpha=1.7476 .$

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Problem 61

A particle moves in one- dimensional motion through a field for which the potential energy of the particle–field system is
$$U(x)=\frac{A}{x^{3}}-\frac{B}{x}$$
where $A=0.150 \mathrm{eV} \cdot \mathrm{nm}^{3}$ and $B=3.68 \mathrm{eV} \cdot \mathrm{nm} .$ The shape of this function is shown in Figure P 43.61 . (a) Find the equilibrium position $x_{0}$ of the particle. (b) Determine the depth $U_{0}$ of this potential well. (c) In moving along the $x$ axis, what maximum force toward the negative $x$ direction does the particle experience?

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Problem 62

A particle of mass $m$ moves in one-dimensional motion through a field for which the potential energy of the particle–field system is
$$U(x)=\frac{A}{x^{3}}-\frac{B}{x}$$
where $A$ and $B$ are constants. The general shape of this function is shown in Figure $\mathrm{P} 43.61$ . ( a) Find the equilibrium position $x_{0}$ of the particle in terms of $m, A,$ and $B$ . (b) Determine the depth $U_{0}$ of this potential well. (c) In moving along the $x$ axis, what maximum force toward the negative $x$ direction does the particle experience?

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Problem 63

As you will learn in Chapter $44,$ carbon- 14$\left(^{14} \mathrm{C}\right)$ is an unstable isotope of carbon. It has the same chemical properties and electronic structure as the much more abundant isotope carbon- $12\left(^{12} \mathrm{C}\right),$ but it has different nuclear properties. Its mass is $14 \mathrm{u},$ greater than that of carbon-12 because of the two extra neutrons in the carbon- 14 nucleus. Assume the $\mathrm{CO}$ molecular potential energy is the same for both isotopes of carbon and the examples in Section 43.2 contain accurate data and results for carbon monoxide with carbon-12 atoms. (a) What is the vibrational frequency of $^{14} \mathrm{CO}?$ (b) What is the moment of inertia of $^{14} \mathrm{CO} ?$ (c) What wavelengths of light can be absorbed by $^{14} \mathrm{CO}$ in the $(v=0,$ $J=10 )$ state that cause it to end up in the $v=1$ state?

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Problem 64

As an alternative to Equation 43.1, another useful model for the potential energy of a diatomic molecule is the Morse potential
$$U(r)=B\left[e^{-a\left(r-r_{0}\right)}-1\right]^{2}$$
where $B, a,$ and $r_{0}$ are parameters used to adjust the shape of the potential and its depth. (a) What is the equilibrium separation of the nuclei? (b) What is the depth of the potential well, defined as the difference in energy between the potential's minimum value and its asymptote as $r$ approaches infinity? (c) If $\mu$ is the reduced mass of the system of two nuclei and assuming the potential is nearly parabolic about the well minimum, what is the vibrational frequency of the diatomic molecule in its ground state? (d) What amount of energy needs to be supplied to the ground-state molecule to separate the two nuclei to infinity?

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