Solved: Find the energies of the first four rotational states of the HCl molecule

If you push two atoms together to form a molecule, the exclusion principle results in a repulsive interaction between the atoms. How does this repulsion come about?

Why do ionically bonded materials have high melting points?

The electrostatic attraction between oppositely charged ions is what binds atoms in an ionic molecule. Is the electric force involved in covalent bonding? Explain.

Does it make sense to distinguish individual NaCl molecules in a salt crystal? What about individual H2O molecules in an ice crystal? Explain.

Is it useful to think of the highest-energy electrons as belonging to individual atoms in an ionically bonded molecule? In a covalently bonded molecule?

What are the approximate relative magnitudes of the energies associated with electronic excitation of a molecule, with molecular vibration, and with molecular rotation?

Radio astronomers have discovered many complex organic molecules in interstellar space. Why were these discoveries made with radio telescopes and not optical telescopes?

In Fig. 18.17, why are rotational states excited at lower temperatures than vibrational states?

Would you expect solid hydrogen to conduct electricity? Why or why not?

The Fermi energy in metals is much higher than the thermal energy at typical temperatures. Why does this make the mean speed of conduction electrons nearly independent of temperature?

Why does the size of the band gap determine whether a material is an insulator or a semiconductor?

How would you expect the conductivity of an undoped semiconductor to depend on temperature? Why?

Name some technological innovations that might result from a room-temperature superconductor

Suppose a room-temperature superconductor were discovered, but it had a very low critical field. In what way would this limit its practical applicability?

How do type I and type II superconductors differ?

Find the energies of the first four rotational states of the HCl molecule described in Example 37.1.

The rotational inertia of oxygen 1O22 is 1.95*10-46 kg# m2 . Find the wavelength of electromagnetic radiation needed to excite oxygen molecules to their first rotational excited state

Find the wavelength of a photon emitted in the l = 5 to l = 4 transition of a molecule whose rotational inertia is 1.75*10-47 kg# m2 .

Photons of wavelength 1.68 cm excite transitions from the rotational ground state to the first rotational excited state in a gas. Whats the rotational inertia of the gas molecules?

The classical vibration frequency for diatomic hydrogen 1H22 is 1.32*1014 Hz. Find the spacing between its vibrational energy levels.

The energy between adjacent vibrational levels in diatomic nitrogen is 0.293 eV. Whats the classical vibration frequency of N2?

Use the 2.16@g/cm3 density of NaCl to calculate the ionic spacing r0 in the NaCl crystal. (Hint: Consult Appendix D.)

Express the 7.84-eV ionic cohesive energy of NaCl in kilocalories per mole of ions

Lithium fluoride, LiF, has the same crystal structure as NaCl and therefore has essentially the same Madelung constant a. Its ionic cohesive energy is -10.5 eV and the value of n in Equation 37.4 is 6.25. Find the equilibrium ionic separation in LiF.

Find the wavelength of light emitted by a gallium phosphide (GaP) light-emitting diode. (Hint: See Table 37.1.)

Whats the shortest wavelength of light that could be produced by electrons jumping the band gap in a material from Table 37.1? What is that material?

Which material in Table 37.1 would provide the longest wavelength of light in a light-emitting diode? Whats that wavelength?

A common light-emitting diode is made with gallium arsenide phosphide (GaAsP) and emits red light at 650 nm. Whats its band gap?

A molecule drops from the l = 2 to the l = 1 rotational level, emitting a 2.68-meV photon. If the molecule then drops to the rotational ground state, what energy photon will it emit?

A molecule absorbs a photon and jumps to the next higher rotational state. If the photon energy is three times what would be needed for a transition from the rotational ground state to the first rotational excited state, between what two levels is the transition?

Find an expression for the energy of a photon required for a transition from the 1l - 12th level to the lth level in a molecule with rotational inertia I.

A molecule with rotational inertia I undergoes a transition from the lth rotational level to the 1l - 12th level. Show that the wavelength of the emitted photon is l = 4p2 Ic/hl

The rotational spectrum of diatomic oxygen shows spectral lines spaced 0.356 meV apart in energy. Find O2s atomic separation. (Hint: See Example 37.1, and remember that the oxygen atoms have equal mass.)

Use the answer in the back of the book for Problem 59 to find the bond length in carbon monoxide (CO), given that excitation of the first rotational state requires photons of wavelength 2.59 mm

For the HCl molecule of Example 37.2, determine (a) the energy of the vibrational ground state and (b) the energies of photons emitted in transitions among adjacent vibrational states, for the cases l = +1 and l = -1.

Diatomic deuterium has classical vibration frequency 9.35*1013 Hz and rotational inertia 9.17*10-48 kg# m2 . Find (a) the energy and (b) the wavelength of a photon emitted in a transition between the n = 1, l = 1 state and the n = 0, l = 2 state.

Carbon dioxide contributes to global warming because the triatomic CO2 molecule exhibits many vibrational and rotational excited states, and transitions among them occur in the infrared region where Earth emits most of its radiation. Among the strongest IR-absorbing transitions is one that takes CO2 from its ground state to the first excited state of a bending vibration and sets the molecule rotating in its first rotational excited state. The energy required for this transition is 82.96 meV. What IR wavelength does this transition absorb?

An oxygen molecule is in its vibrational and rotational ground states. It absorbs a photon of energy 0.19653 eV and jumps to the n = 1, l = 1 state. It then drops to n = 0, l = 2, emitting a 0.19546-eV photon. Find (a) the classical vibration frequency and (b) the rotational inertia of the molecule

The internuclear spacing in diatomic hydrogen (H2) is 74.14 pm. Find (a) the energy and (b) the wavelength of a photon emitted in a transition from the first rotational excited state to the ground state. (c) In what spectral region does this wavelength lie?

Biological macromolecules are complex structures that exhibit many more vibrational modes than the diatomic molecules considered in this chapter. DNA has a low-frequency breathing mode whose associated photon wavelength is 330 m. Find the corresponding (a) frequency and (b) photon energy in eV

What wavelength of infrared radiation is needed to excite a transition between the n = 0, l = 3 state and the n = 1, l = 2 state in KCl, for which the rotational inertia is 2.43*10-45 kg# m2 and the classical vibration frequency is 8.40 THz?

Find the wavelengths emitted in all allowed transitions between the first three rotational states in the n = 1 level to any states in the n = 0 level in H2, whose rotational inertia and classical vibration frequency are 4.60*10-48 kg# m2 and 3.69*1014 Hz, respectively.

Determine the constant n in Equation 37.4 for potassium chloride (KCl), which has the same crystal structure as NaCl and for which r0 = 0.315 nm and U0 = -7.21 eV.

A salt crystal contains 1021 sodiumchlorine pairs. How much energy would it take to compress the crystal to 90% of its normal size?

Lithium chloride, LiCl, has the same structure and therefore the same Madelung constant as NaCl. The equilibrium separation in LiCl is 0.257 nm, and n = 7 in Equation 37.4. Find the ionic cohesive energy of the LiCl crystal.

Youre researching the possibility of storing radioactive waste in underground salt formations. In support of this idea, youd like to demonstrate that salt is extremely resistant to compression. You differentiate Equation 37.4 to obtain an expression for the force on an ion in an ionic crystal, and then use your result to find the force on an ion in NaCl if the crystal were compressed to half its equilibrium spacing (see Example 37.3 for relevant parameters). You compare this with the electrostatic attraction at this compression. What do you find?

Integrating Equation 37.5 over all energies gives the total number of states per unit volume in a metal. Therefore, integrating from E = 0 to E = EFthat is, over the occupied states onlygives the number of conduction electrons per unit volume. Carry out this integration to show that the electron number density is given by n = a 29/2pm3/2 3h2 bE3/2

The Fermi energy in aluminum is 11.6 eV. Use the result of Problem 47 to find the density of conduction electrons in aluminum.

Use the result of Problem 47 to determine the Fermi energy for calcium, which has 4.6*1028 conduction electrons per cubic meter.

Find (a) the speed associated with an electron whose kinetic energy is equal to coppers 7.00-eV Fermi and (b) the thermal speed of an electron at room temperature (293 K). (c) What does the difference between your calculated speeds tell you about whether a quantum or a classical model best describes coppers electrical conductivity?

The Fermi temperature is defined by equating the thermal energy kT to the Fermi energy, where k is Boltzmanns constant. Calculate the Fermi temperature for silver, for which EF = 5.48 eV, and compare it with room temperature.

Photons with energy lower than a semiconductors band gap arent readily absorbed by the material, so a measurement of absorption versus wavelength gives the band gap. An absorption spectrum for silicon shows no absorption for wavelengths longer than 1090 nm. Use this information to calculate the band gap in silicon, and verify its value in Table 37.1.

Calculate the median wavelength lmedian for sunlight, treating the Sun as a 5800-K blackbody (see Equation 34.2b). Use your result to decide whether zinc selenide, with band gap 3.6 eV, would make a good photovoltaic cell.

Pure aluminum, which superconducts below 1.20 K, exhibits a critical field of 9.57 mT. Find the maximum current that can be carried in a 30-gauge (0.255-mm-diameter) aluminum superconducting wire without the field from that current exceeding the critical field. (Hint: Where is the field greatest? Consult Example 26.8.)

The critical field in a niobiumtitanium superconductor is 15 T. What current in a 5000-turn solenoid 75 cm long will produce a field of this strength?

The transition from the ground state to the first rotational excited state in diatomic oxygen 1O22 requires 356eV. At what temperature would the thermal energy kT be sufficient to set diatomic oxygen into rotation? Would you ever find diatomic oxygen exhibiting the specific heat of a monatomic gas at normal pressure?

Green fluorescent protein (GFP) is a substance that was first extracted from jellyfish; variants are used to tag biological molecules for study. The original wild GFP absorbs 395-nm light, undergoing an upward transition to an excited state. Movement of a proton within the protein then excites it to 2.44 eV above the ground state. Photons emitted in the subsequent downward transition to the ground state provide a visual indication of the GFPs location as seen in a microscope. Whats the wavelength of these photons?

The density of rubidium iodide (RbI) is 3.55 g/cm3 , and its ionic cohesive energy is -145 kcal/mol. Determine (a) the equilibrium separation and (b) the exponent n in Equation 37.4 for RbI.

Youre troubled that Example 37.1 neglects the mass of the hydrogen, and you wonder how much error this introduces. So you consider a diatomic molecule consisting of different atoms with masses m1 and m2, separated by a distance R, and derive an expression for the molecules rotational inertia about its center of mass. You then calculate a more accurate value for the HCl bond length in Example 37.1. Your results?

What fraction of conduction electrons in a metal at absolute zero have energies less than half the Fermi energy?

The Madelung constant (Section 37.3) is notoriously difficult to calculate because its the sum of an alternating series of nearly equal terms. But it can be calculated for a hypothetical one-dimensional crystal consisting of an evenly spaced line of

The lower-energy states in a covalently bound diatomic molecule can be found approximately from the so-called Morse potential U1r2 = U01e21r-r02/a - e-21r-r02/a 2, where r is the atomic separation and U0, r0, and a are constants determined from experimental data. Calculate dU/dr and d2 U/dr2 to show that U has a minimum, and find expressions for (a) Umin and (b) the separation rmin at the minimum energy.

(a) Count the number of electron states N1E2 with energy equal to or less than E in Equation 35.8 by finding the volume available to such states in the space with Cartesian coordinate axes nx, ny, nz. (Hint: Consider each set of positive integers, at the corner of a unit cube, and that lies inside a radius 2nx 2 + n 2 y + n 2 z , and remember that there are two spin values per state.) (b) Differentiate N1E2 with respect to E to obtain Equation 37.5.

Use Equation 37.5 to calculate the average energy of a conduction electron at T = 0 in terms of the Fermi energy.

Youre designing a new medical MRI imager. The design calls for a long solenoid wound with 75 turns per meter of niobium titanium superconductor. The upper critical field for your particular Nb-Ti alloy is 12 T. To avoid a disastrous loss of superconductivity (see Example 27.9), you want to limit the actual field to half the upper critical field. What maximum current do you specify for your device?

Squeezing a particular solid in all directions reduces the interatomic spacing to 76.6% of its equilibrium value; the result is that the solids total potential energy becomes zero. Find the value of the exponent n in Equation 37.4 for this solid.

The table below shows the wavelengths of photons emitted when identical molecules drop from the lth rotational level to the (l - 1)th level. Find quantities that, when plotted, should yield a straight line. Make your plot, determine a best-fit line, and use the result to find the rotational inertia of the molecules. Initial state, l 2 3 4 5 6 Wavelength, l (mm) 0.24 0.17 0.12 0.095 0.078

Problem 53 shows that the median wavelength in the solar spectrum is 710 nm, at the visible-IR boundary. What percentage of the incident solar energy can a silicon PV cell absorb? (Hint: See Exercise 36.31.) a. about 25% b. about 50% c. about 75%

How does the percentage of the number of incident solar photons that a PV cell absorbs compare with the energy percentage in the preceding problem? a. Its less than the energy percentage. b. Its the same as the energy percentage. c. Its more than the energy percentage

Making PV cells with a semiconductor whose band gap is lower than silicons will a. increase the fraction of solar energy absorbed while decreasing the amount of absorbed energy lost as heat. b. increase both the fraction of solar energy absorbed and the amount of absorbed energy lost as heat. c. decrease the fraction of solar energy absorbed while increasing the amount of absorbed energy lost as heat. d. decrease both the fraction of solar energy absorbed and the amount of absorbed energy lost as heat

One way to improve PV efficiency is to make multilayer cells with several PN junctions using semiconductors with different band gaps. For a multilayer PV cell to be effective, a. the junction with the largest band gap should be closest to the top of the PV cell. b. the junction with the largest band gap should be closest to the bottom of the PV cell. c. the largest band gap should correspond to infrared wavelengths. d. the smallest band gap should correspond to ultraviolet wavelengths.