Chapter 42

If the energy of the \(\mathrm{H}_{2}\) covalent bond is \(-4.48 \mathrm{eV},\) what wavelength of light is needed to break that molecule apart? In what part of the electromagnetic spectrum does this light lie?

An Ionic Bond. (a) Calculate the electric potential energy for a \(\mathrm{K}^{+}\) ion and a Br \(^{-}\) ion separated by a distance of \(0.29 \mathrm{nm},\) the equilibrium separation in the KBr molecule. Treat the ions as point charges. (b) The ionization energy of the potassium atom is 4.3 \(\mathrm{eV}\) . Atomic bromine has an electron affinity of 3.5 eV. Use these data and the results of part (a) to estimate the binding energy of the KBr molecule. Do you expect the actual binding energy to be higher or lower than your estimate? Explain your reasoning.

Light of wavelength 3.10 \(\mathrm{mm}\) strikes and is absorbed by a molecule. Is this process most likely to alter the rotational, vibrational, or atomic energy levels of the molecule? Explain your reasoning. (b) If the light in part (a) had a wavelength of 207 \(\mathrm{nm}\) , which energy levels would it most likely affect? Explain.

(a) A molecule decreases its vibrational energy by 0.250 \(\mathrm{eV}\) by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie? (b) An atom decreases its energy by 8.50 eV by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie? (c) A molecule decreases its rotational energy by \(3.20 \times 10^{-3} \mathrm{eV}\) by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie?

A hypothetical \(\mathrm{NH}\) molecule makes a rotational-level transition from \(l=3\) to \(I=1\) and gives off a photon of wavelength 1.780 \(\mathrm{nm}\) in doing so. What is the separation between the two atoms in this molecule if we model them as point masses? The mass of hydrogen is \(1.67 \times 10^{-27} \mathrm{kg},\) and the mass of nitrogen is \(2.33 \times 10^{-26} \mathrm{kg} .\)

The water molecule has an \(I=1\) rotational level 1.0 \(\mathrm{I} \times\) \(10^{-5} \mathrm{eV}\) above the \(I=0\) ground level. Calculate the wavelength and frequency of the photon absorbed by water when it undergoes a rotational- level transition from \(I=0\) to \(I=1 .\) The magnetron oscillator in a microwave oven generates microwaves with a frequency of 2450 MHz. Does this make sense, in view of the frequency you calculated in this problem? Explain.

Two atoms of cesium (Cs) can form a Cs \(_{2}\) molecule. The equilibrium distance between the nuclei in a \(\mathrm{Cs}_{2}\) molecule is 0.447 \(\mathrm{nm} .\) Calculate the moment of inertia about an axis through the center of mass of the two nuclei and perpendicular to the line joining them. The mass of a cesium atom is 2.2 \(\mathrm{I} \times 10^{-25} \mathrm{kg}\) .

If a sodium chloride (NaCl) molecule could undergo an \(n \rightarrow n-1\) vibrational transition with no change in rotational quantum number, a photon with wavelength 20.0\(\mu \mathrm{m}\) would be emitted. The mass of a sodium atom is \(3.82 \times 10^{-26} \mathrm{kg},\) and the mass of a chlorine atom is \(5.81 \times 10^{-26} \mathrm{kg}\) . Calculate the force constant \(k^{\prime}\) for the interatomic force in NaCl.

The maximum wavelength of light that a certain silicon photocell can detect is 1.11\(\mu \mathrm{m}\) . (a) What is the energy gap (in electron volts) between the valence and conduction bands for this photocell? (b) Explain why pure silicon is opaque.

The gap between valence and conduction bands in diamond is 5.47 eV. (a) What is the maximum wavelength of a photon that can excite an electron from the top of the valence band into the conduction band? In what region of the electromagnetic spectrum does this photon lie? (b) Explain why pure diamond is transparent and colorless. (c) Most gem diamonds have a yellow color. Explain how impurities in the diamond can cause this color.

The gap between valence and conduction bands in silicon is 1.12 eV. A nickel nucleus in an excited state emits a gamma-ray photon with wavelength \(9.31 \times 10^{-4}\) nm. How many electrons can be excited from the top of the valence band to the bottom of the conduction band by the absorption of this gamma ray?

For a solid metal having a Fermi energy of 8.500 eV, what is the probability, at room temperature, that a state having an energy of 8.520 eV is occupied by an electron?

Pure germanium has a band gap of 0.67 ev. The Fermienergy is in the middle of the gap. (a) For temperatures of 250 \(\mathrm{K}\) ,\(300 \mathrm{K},\) and 350 \(\mathrm{K}\) , calculate the probability \(f(E)\) that a state at the bottom of the conduction band is occupied. (b) For each temperature in part (a), calculate the probability that a state at the top of the valence band is empty.

Germanium has a band gap of 0.67 eV. Doping with arsenic adds donor levels in the gap 0.01 eV below the bottom of the conduction band. At a temperature of 300 \(\mathrm{K}\) , the probability is \(4.4 \times 10^{-4}\) that an electron state is occupied at the bottom of the conduction band. Where is the Fermi level relative to the conduction band in this case?

(a) Suppose a piece of very pure germanium is to be used as a light detector by observing, through the absorption of photons, the increase in conductivity resulting from generation of electron-hole pairs. If each pair requires 0.67 \(\mathrm{eV}\) of energy, what is the maximum wavelength that can be detected? In what portion of the spectrum does it lie? (b) What are the answers to part (a) if the material is silicon, with an energy requirement of 1.14 eV per pair, corresponding to the gap between valence and conduction bands in that element?

A hypothetical diatomic molecule of oxygen (mass =\( 2.656 \times 10^{-26} \mathrm{kg} ) \quad \text { and } \quad \text { hydrogen } \quad\left(\text { mass }=1.67 \times 10^{-27} \mathrm{kg}\right)\) emits a photon of wavelength 2.39\(\mu \mathrm{m}\) when it makes a transition from one vibrational state to the next lower state. If we model this molecule as two point masses at opposite ends of a massless spring, (a) what is the force constant of this spring, and (b) how many vibrations per second is the molecule making?

When a diatomic molecule undergoes a transition from the \(l=2\) to the \(I=1\) rotational state, a photon with wavelength 63.8\(\mu \mathrm{m}\) is emitted. What is the moment of inertia of the molecule for an axis through its center of mass and perpendicular to the line connecting the nuclei?

CP (a) The equilibrium separation of the two nuclei in an NaCl molecule is 0.24 \(\mathrm{nm}\) . If the molecule is modeled as charges \(+e\) and \(-e\) separated by 0.24 \(\mathrm{nm}\) , what is the electric dipole moment of the molecule (see Section 21.7\() ?\) (b) The measured electric dipole moment arises from point charges \(+q\) and \(-q\) separated by 0.24 \(\mathrm{nm}\) , what is \(q ?\) (c) A definition of the fractional ionic character of the bond is \(q / e\) . If the sodium atom has charge \(+e\) and the chlorine atom has charge \(-e\) the fractional ionic character would be equal to 1. What is the actual fractional ionic character for the bond in NaCl? (d) The equilibrium distance between nuclei in the hydrogen iodide (HI) molecule is \(0.16 \mathrm{nm},\) and the measured electric dipole moment of the molecule is \(1.5 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}\) . What is the fractional ionic character for the bond in \(\mathrm{HI}\) ? How does your answer compare to that for NaCl calculated in part (c)? Discuss reasons for the difference in these results.

The rotational spectrum of HCl contains the following wavelengths (among others): \(60.4 \mu \mathrm{m}, \quad 69.0 \mu \mathrm{m}, \quad 80.4 \mu \mathrm{m},\) \(96.4 \mu \mathrm{m},\) and 120.4\(\mu \mathrm{m} .\) Use this spectrum to find the moment of inertia of the HCl molecule about an axis through the center of mass and perpendicular to the line joining the two nuclei.

When an OH molecule undergoes a transition from the \(n=0\) to the \(n=1\) vibrational level, its internal vibrational energy increases by 0.463 eV. Calculate the frequency of vibration and the force constant for the interatomic force. (The mass of an oxygen atom is \(2.66 \times 10^{-26} \mathrm{kg},\) and the mass of a hydrogen atom is \(1.67 \times 10^{-27} \mathrm{kg} .\) )

Compute the Fermi energy of potassium by making the simple approximation that each atom contributes one free electron. The density of potassium is \(851 \mathrm{kg} / \mathrm{m}^{3},\) and the mass of a single potassium atom is \(6.49 \times 10^{-26} \mathrm{kg}\) .

Hydrogen is found in two naturally occurring isotopes; normal hydrogen (containing a single proton in its nucleus) and deuterium (having a proton and a neurron). Assuming that both molecules are the same size and that the proton and neutron have the same mass (which is almost the case), find the ratio of (a) the energy of any given rotational state in a diatomic hydrogen molecule to the energy of the same state in a diatomic deuterium molecule and (b) the energy of any given vibrational state in hydrogen to the same state in deuterium (assuming that the force constant is the same for both molecules) Why is it physically reasonable that the force constant would be the same for hydrogen and deuterium molecules?

Metallic lithium has a bce crystal structure. Each unit cell is a cube of side length \(a=0.35 \mathrm{nm}\) . (a) For a bce lattice, what is the number of atoms per unit volume? Give your answer in terms of \(a\) . (Hint: How many are there per unit cell?) (b) Use the result of part (a) to calculate the zero- temperature Fermi energy \(E_{\mathrm{PO}}\) for metallic lithium. Assume there is one free electron per atom.

CALC The one-dimensional calculation of Example 42.4 (Section 42.3\()\) can be extended to three dimensions. For the three-dimensional fce NaCl lattice, the result for the potential energy of a pair of \(\mathrm{Na}^{+}\) and \(C^{-}\) ions due to the electrostatic interaction with all of the ions in the crystal is \(U=-\alpha e^{2} / 4 \pi \epsilon_{0} r,\) where \(\alpha=1.75\) is the Madelung constant. Another contribution to the potential energy is a repulsive interaction at small ionic separation \(r\) due to overlap of the electron clouds. This contribution can be represented by \(A / r^{8},\) where \(A\) is a positive constant, so the expression for the total potential energy is $$U_{\mathrm{tot}}=-\frac{\alpha e^{2}}{4 \pi \epsilon_{0} r}+\frac{A}{r^{8}}$$ (a) Let \(r_{0}\) be the value of the ionic separation \(r\) for which \(U_{\text { ot }}\) is a minimum. Use this definition to find an equation that relates \(r_{0}\) and \(A,\) and use this to write \(U_{\text { ot }}\) in terms of \(r_{0 .}\) For \(\mathrm{NaCl}\) , \(r_{0}=0.281 \mathrm{nm} .\) Obtain a numerical value (in electron volts) of \(U_{\mathrm{tot}}\) for NaCl. (b) The quantity \(-U_{\text { tot }}\) is the energy required to remove a \(\mathrm{Na}^{+}\) ion and a \(\mathrm{Cl}^{-}\) ion from the crystal. Forming a pair of neutral atoms from this pair of ions involves the release of 5.14 eV (the ionization energy of \(\mathrm{Na}\) ) and the expenditure of 3.61 \(\mathrm{eV}\) (the electron affinity of Cl). Use the result of part (a) to calculate the energy required to remove a pair of neutral Na and Cl atoms from the crystal. The experimental value for this quantity is \(6.39 \mathrm{eV} ;\) how well does your calculation agree?

CP A variable DC battery is connected in series with a \(125-\Omega\) resistor and a \(p-n\) junction diode that hat hat a saturation current of 0.625 \(\mathrm{mA}\) at room temperature \(\left(20^{\circ} \mathrm{C}\right) .\) When a voltmeter across the \(125-\Omega\) resistor reads 35.0 \(\mathrm{V}\) , what are (a) the voltage across the diode and (b) the resistance of the diode?

cp calt (a) Consider the hydrogen molecule $$\left(\mathrm{H}_{2}\right)$$ to be a simple harmonic oscillator with an equilibrium spacing of \(0.074 \mathrm{nm},\) and estimate the vibrational energy-level spacing for H. The mass of a hydrogen atom is \(1.67 \times 10^{-27} \mathrm{kg}\) . Hint: Estimate the force constant by equating the change in Coulomb repulsion of the protons, when the atoms move slightly closer together than \(r_{0}\) to the "spring" force. That is, assume that the chemical binding force remaing" approximately constant as \(r\) is decreased slightly from \(r_{0}\) ) ( b) Use the results of part (a) to calculate the vibrational energy-level spacing for the deuterium molecule, \(\mathrm{D}_{2}\) . Assume that the spring constant is the same for \(D_{2}\) as for \(\mathrm{H}_{2}\) . The mass of a deuterium atom is \(3.34 \times 10^{-27} \mathrm{kg}\)