Question: The rotational energy levels of CO are
The rotational energy levels of CO are calculated in Example 42.2. If the energy of the rotating molecule is described by the classical expression K = 1/2 Iω2, for the l = 1 level what are a. the angular speed of the rotating molecule; b. the linear speed of each atom; c. the rotational period (the time for one rotation)? From Example 42.2 
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EXAMPLE 42.2 ROTATIONAL SPECTRUM OF CARBON MONOXIDE The two nuclei in the carbon monoxide (CO) molecule are 0.1128 nm The rotational levels are given by Eq. (42.3): apart. The mass of the most common carbon atom is 1.993 × 10-26 kg; that of the most common oxygen atom is 2.656 × 10-26 kg. (a) Find the energies of the lowest three rotational energy levels of CO. Express your results in meV (1 meV = 10-³eV). (b) Find the wavelength of the photon emitted in the transition (1 meV = 10-³ eV.) Substituting 1 = 0, 1, 2, we find (1.0546 × 10-34 J ·s)? 2(1.449 × 1046 kg • m²) = |(! + 1)(3.838 × 10-23 J) = 1(I + 1)0.2395 meV E = {(I + 1) - 1(1 + 1); from the I = 2 to the l = 1 level. E, = 0 E, = 0.479 meV E2 = 1.437 meV SOLUTION (b) The photon energy and wavelength are IDENTIFY and SET UP: This problem uses the ideas developed in this section about the rotational energy levels of molecules. We are given the distance rɔ between the atoms and their masses m1 and m2. We find the reduced mass m, from Eq. (42.4), the moment of inertia I from Eq. (42.6), and the energies E¡ from Eq. (42.3). The energy E of the emitted photon is equal to the difference in energy between the l = 2 and I = 1 levels. (This transition obeys EVALUATE: The differences between the first few rotational en- the Al = +1 selection rule, since Al = 1 – 2 = -1.) We deter- mine the photon wavelength by using E = hc/A. E = E2 – E, = 0.958 meV hc (4.136 x 10-15 eV · s)(3.00 × 10 m/s) 0.958 × 10-³ eV E = 1.29 × 10³ m = 1.29 mm ergy levels of CO are very small (about 1 meV = 10-3 eV) com- pared to the differences between atomic energy levels (typically a EXECUTE: (a) From Eqs. (42.4) and (42.6), the reduced mass and few eV). Hence a photon emitted by a CO molecule in a transition moment of inertia of the CO molecule are: from the I = 2 to the I = 1 level has very low energy and a very long wavelength compared to the visible light emitted by excited atoms. photon wavelengths for rotational transitions in molecules are typically in the microwave and far infrared regions of the spectrum. In this example we were given the equilibrium separation between the atoms, also called the bond length, and we used it to calculate one of the wavelengths emitted by excited CO mol- ecules. In experiments, scientists work this problem backward: By measuring the long-wavelength emissions of a sample of diatomic molecules, they determine the moment of inertia of the molecule and hence the bond length. m, mi + m2 (1.993 × 10-26 kg)(2.656 × 10-26 kg) (1.993 × 10 26 kg) + (2.656 × 10 26 kg) 1.139 x 10 26 kg = m,rở (1.139 × 10-20 kg)(0.1128 × 10º m)² - 1.449 × 10 46 kg•m² UTION
> A particle of mass m in a one-dimensional box has the following wave function in the region x = 0 to x = L: Here ψ1(x) and ψ3(x) are the normalized stationary-state wave functions for the n = 1 and n = 3 levels, and E1 and E3 are
> Consider the wave packet defined by Let B(k)= e-a2k2. a. The function B(k) has its maximum value at k = 0. Let kh be the value of k at which B(k) has fallen to half its maximum value, and define the width of B(k) as wk = kh. In terms of α
> On the basis of the Pauli exclusion principle, the structure of the periodic table of the elements shows that there must be a fourth quantum number in addition to n, l, and ml. Explain.
> A particle is in the three-dimensional cubical box of Section 41.2. a. Consider the cubical volume defined by 0 ≤ x ≤ L/4, 0 ≤ y ≤ L/4, and 0 ≤ z ≤ L/4. What fraction of the total volume of the box is this cubical volume? b. If the particle is in the gr
> An electron is in a three-dimensional box with side lengths LX = 0.600 nm and LY = LZ = 2LX. What are the quantum numbers nX, nY, and nZ and the energies, in eV, for the four lowest energy levels? What is the degeneracy of each (including the degeneracy
> In terms of the ground-state energy E1,1,1, what is the energy of the highest level occupied by an electron when 10 electrons are placed into a cubical box?
> When a NaF molecule makes a transition from the l = 3 to the l = 2 rotational level with no change in vibrational quantum number or electronic state, a photon with wavelength 3.83 mm is emitted. A sodium atom has mass 3.82 * 10-26 kg, and a fluorine atom
> a. For the sodium chloride molecule (NaCl) discussed at the beginning of Section 42.1, what is the maximum separation of the ions for stability if they may be regarded as point charges? That is, what is the largest separation for which the energy of an N
> Which statement best explains the temperature dependence of the current–voltage characteristics that the graph shows? At higher temperatures: a. The band gap is larger, so the electron– hole pairs have more energy, which causes the current at a given vo
> The sensitivity of a diode thermometer depends on how much the voltage changes for a given temperature change, with the current remaining constant. What is the sensitivity for this diode thermometer, operated at 100 mA, for a temperature change from 25°C
> The current–voltage characteristics of a forward­biased p-n junction diode depend strongly on temperature, as shown in the figure. As a result, diodes can be used as temperature sensors. In actual operation, the voltage is ad
> What type of radioactive decay produces 131I from 131Te? a. Alpha decay; b. β- decay; c. β+ decay; d. gamma decay.
> Which reaction produces 131Te in the nuclear reactor? a. 130Te + n → 131Te; b. 130I + n → 131Te; c. 132Te + n → 131Te; d. 132I + n → 131Te.
> Use Table 41.3 to help determine the ground-state electron configuration of the neutral gallium atom (Ga) as well as the ions Ga+ and Ga-. Gallium has an atomic number of 31. From table 41.3: TABLE 41.3 Ground-State Electron Configurations Atomic E
> Why might 123I be preferred for imaging over 131I? a. The atomic mass of 123I is smaller, so the 123I particles travel farther through tissue. b. Because 123I emits only gamma-ray photons, the radiation dose to the body is lower with that isotope. c. T
> In the reaction that produces 123I, is there a minimum kinetic energy the protons need to make the reaction go? a. No, because the proton has a smaller mass than the neutron. b. No, because the total initial mass is smaller than the total final mass.
> How many 131I atoms are administered in a typical thyroid cancer treatment? a. 4.2 * 1010; b. 1.0 * 1012; c. 2.5 * 1014; d. 3.7 * 1015.
> In the Bohr model, what is the principal quantum number n at which the excited electron is at a radius of 1 µm? a. 140; b. 400; c. 20; d. 81.
> How many different possible electron states are there in the n = 100, l = 2 subshell? a. 2; b. 100; c. 10,000; d. 10.
> Assume that the researchers place an atom in a state with n = 100, l = 2. What is the magnitude of the orbital angular momentum L associated with this state? a. 2 ħ; b. 6 ħ; c. 200 ħ; d. 10100 ħ.
> Take the size of a Rydberg atom to be the diameter of the orbit of the excited electron. If the researchers want to perform this experiment with the rubidium atoms in a gas, with atoms separated by a distance 10 times their size, the density of atoms per
> One advantage of the quantum dot is that, compared to many other fluorescent materials, excited states have relatively long lifetimes (10 ns). What does this mean for the spread in the energy of the photons emitted by quantum dots? a. Quantum dots emit
> Dots that are the same size but made from different materials are compared. In the same transition, a dot of material 1 emits a photon of longer wavelength than the dot of material 2 does. Based on this model, what is a possible explanation? a. The mass
> When a given dot with side length L makes a transition from its first excited state to its ground state, the dot emits green (550 nm) light. If a dot with side length 1.1L is used instead, what wavelength is emitted in the same transition, according to t
> Why do the transition elements (Z = 21 to 30) all have similar chemical properties?
> According to this model, which statement is true about the energy-level spacing of dots of different sizes? a. Smaller dots have equally spaced levels, but larger dots have energy levels that get farther apart as the energy increases. b. Larger dots ha
> Suppose that positron–electron annihilations occur on the line 3 cm from the center of the line connecting two detectors. Will the resultant photons be counted as having arrived at these detectors simultaneously? a. No, because the time difference betwe
> What is the energy of each photon produced by positron– electron annihilation? a. 1/2 mev2, where v is the speed of the emitted positron; b. mev2; c. 1/2 mec2; d. mec2.
> If the annihilation photons come from a part of the body that is separated from the detector by 20 cm of tissue, what percentage of the photons that originally travelled toward the detector remains after they have passed through the tissue? a. 1.4%; b.
> If the voltage rather than the current is kept constant, what happens as the temperature increases from 25°C to 150°C? a. At first the current increases, then it decreases. b. The current increases. c. The current decreases, eventually approaching zer
> A particle moving in one dimension (the x­axis) is described by the wave function where b = 2.00 m-1, A > 0, and the +x-axis points toward the right. a. Determine A so that the wave function is normalized. b. Sketch the graph of the wav
> Let ψ1 and ψ2 be two solutions of Eq. (40.23) with energies E1 and E2, respectively, where E1 ≠ E2 . Is ψ = Aψ1 + Bψ2, where A and B are nonzero constants, a solution to Eq. (40.23)? Explain your answer.
> Compute Ψ 2 for Ψ = Ψ sin ωt, where Ψ is time independent and ω is a real constant. Is this a wave function for a stationary state? Why or why not?
> Consider a wave function given by Ψ(x) = A sin kx, where k = 2π/λ and A is a real constant. a. For what values of x is there the highest probability of finding the particle described by this wave function? Explain. b. For which values of x is the proba
> A particle is described by a wave function Ψ(x) = Ae-ax2, where A and a are real, positive constants. If the value of a is increased, what effect does this have on a. the particle’s uncertainty in position and b. the particle’s uncertainty in momentum?
> Do gravitational forces play a significant role in atomic structure? Explain.
> Consider the free-particle wave function of Example 40.1. Let k2 = 3k1 = 3k. At t = 0 the probability distribution function Ψ (x, t) 2 has a maximum at x = 0. a. What is the smallest positive value of x for which the probability distribution function
> A free particle moving in one dimension has wave function Ψ(x, t) = A[ei(kx-ωt) – e i(2kx-4ωt) where k and v are positive real constants. a. At t = 0 what are the two smallest positive values of x for which the probability function Ψ(x, t) 2 is a maxi
> An electron is moving as a free particle in the –x-direction with momentum that has magnitude 4.50*10-24 kg.m/s. What is the onedimensional timedependent wave function of the electron?
> What particle (a particle, electron, or positron) is emitted in the following radioactive decays? a. 14 27
> The atomic mass of 14C is 14.003242 u. Show that the β- decay of 14C is energetically possible, and calculate the energy released in the decay.
> 238U decays spontaneously by α emission to 234Th. Calculate a. the total energy released by this process and b. the recoil velocity of the 234Th nucleus. The atomic masses are 238.050788 u for 238U and 234.043601 u for 234Th.
> What nuclide is produced in the following radioactive decays? a. α decay of 94 239
> a. Is the decay n p + β- + ve energetically possible? If not, explain why not. If so, calculate the total energy released. b. Is the decay p n + β+ + ve energetically possible? If not, explain why not. If so, calculate the total energy rel
> Use Eq. (43.11) to calculate the binding energy per nucleon for the nuclei 36 86
> Calculate the mass defect, the binding energy (in MeV), and the binding energy per nucleon of a. the nitrogen nucleus, 7 14
> Table 41.3 shows that for the ground state of the potassium atom, the outermost electron is in a 4s state. What does this tell you about the relative energies of the 3d and 4s levels for this atom? Explain. From Table 41.3: TABLE 41.3 Ground-State
> As Eq. (40.21) indicates, the time-dependent wave function for a stationary state is a complex number having a real part and an imaginary part. How can this function have any physical meaning, since part of it is imaginary? From Eq. (40.21): Time-d
> How many protons and how many neutrons are there in a nucleus of the most common isotope of a. silicon, 14 28
> Two atoms of cesium (Cs) can form a Cs2 molecule. The equilibrium distance between the nuclei in a Cs2 molecule is 0.447 nm. Calculate the moment of inertia about an axis through the center of mass of the two nuclei and perpendicular to the line joining
> The water molecule has an l = 1 rotational level 1.01 * 10-5 eV above the l = 0 ground level. Calculate the wavelength and frequency of the photon absorbed by water when it undergoes a rotationallevel transition from l = 0 to l = 1. The magnetron oscill
> The H2 molecule has a moment of inertia of 4.6 * 10-48 kg . m2. What is the wavelength l of the photon absorbed when H2 makes a transition from the l = 3 to the l = 4 rotational level?
> A hypothetical NH molecule makes a rotational-level transition from l = 3 to l = 1 and gives off a photon of wavelength 1.780 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 hydr
> During each of these processes, a photon of light is given up. In each process, what wavelength of light is given up, and in what part of the electromagnetic spectrum is that wavelength? a. A molecule decreases its vibrational energy by 0.198 eV; b. a
> A p-n junction has a saturation current of 6.40 mA. a. At a temperature of 300 K, what voltage is needed to produce a positive current of 40.0 mA? b. For a voltage equal to the negative of the value calculated in part (a), what is the negative current
> a. A forward-bias voltage of 15.0 mV produces a positive current of 9.25 mA through a p-n junction at 300 K. What does the positive current become if the forward-bias voltage is reduced to 10.0 mV? b. For reverse-bias voltages of -15.0 mV and -10.0 mV,
> For a certain p-n junction diode, the saturation current at room temperature (20°C) is 0.950 mA. What is the resistance of this diode when the voltage across it is a. 85.0 mV and b. -50.0 mV ?
> The central-field approximation is more accurate for alkali metals than for transition metals such as iron, nickel, or copper. Why?
> For the H2 molecule the equilibrium spacing of the two protons is 0.074 nm. The mass of a hydrogen atom is 1.67 * 10-27 kg. Calculate the wavelength of the photon emitted in the rotational transition l = 2 to l = 1.
> At a temperature of 290 K, a certain p-n junction has a saturation current IS = 0.500 mA. a. Find the current at this temperature when the voltage is i. 1.00 mV, ii. -1.00 mV, iii. 100 mV, and iv. -100 mV. b. Is there a region of applied voltage w
> 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 eV of energy, what is t
> 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 K, the probability is 4.4 * 10-4 that an electron state is occupied at the bottom of the conduc
> Pure germanium has a band gap of 0.67 eV. The Fermi energy is in the middle of the gap. a. For temperatures of 250 K, 300 K, and 350 K, calculate the probability f(E) that a state at the bottom of the conduction band is occupied. b. For each temperatur
> 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?
> At the Fermi temperature TF, EF = kTF (see Exercise 42.22). When T = TF, what is the probability that a state with energy E = 2EF is occupied?
> Silver has a Fermi energy of 5.48 eV. Calculate the electron contribution to the molar heat capacity at constant volume of silver, CV, at 300 K. Express your result a. as a multiple of R and b. as a fraction of the actual value for silver, CV = 25.3 J/
> The Fermi energy of sodium is 3.23 eV. a. Find the average energy Eav of the electrons at absolute zero. b. What is the speed of an electron that has energy Eav? c. At what Kelvin temperature T is kT equal to EF? (This is called the Fermi temperature
> Calculate the density of states g(E) for the freeelectron model of a metal if E = 7.0 eV and V = 1.0 cm3. Express your answer in units of states per electron volt.
> Particle A is described by the wave function Ψ(x, y, z). Particle B is described by the wave function Ψ(x, y, z)eiɸ, where ɸ is a real constant. How does the probability of finding particle A within a volume dV around a certain point in space compare wit
> Calculate vrms for free electrons with average kinetic energy 3/2 kT at a temperature of 300 K. How does your result compare to the speed of an electron with a kinetic energy equal to the Fermi energy of copper, calculated in Example 42.7? Why is there s
> a. Calculate the electric potential energy for a K+ ion and a Br- ion separated by a distance of 0.29 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 eV. Atomic b
> 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 * 10-4 nm. How many electrons can be excited from the top of the valence band to the bottom of the cond
> 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
> The maximum wavelength of light that a certain silicon photocell can detect is 1.11 µm. 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.
> Potassium bromide (KBr) has a density of 2.75 * 103 kg/m3 and the same crystal structure as NaCl. The mass of a potassium atom is 6.49 * 10-26 kg, and the mass of a bromine atom is 1.33 * 10-25 kg. a. Calculate the average spacing between adjacent atoms
> The spacing of adjacent atoms in a crystal of sodium chloride is 0.282 nm. The mass of a sodium atom is 3.82 * 10-26 kg, and the mass of a chlorine atom is 5.89 * 10-26 kg. Calculate the density of sodium chloride.
> The vibrational and rotational energies of the CO molecule are given by Eq. (42.9). Calculate the wavelength of the photon absorbed by CO in each of these vibration­rotation transitions: a. n = 0, l = 2 → n = 1, l = 3; b. n
> When a hypothetical diatomic molecule having atoms 0.8860 nm apart undergoes a rotational transition from the l = 2 state to the next lower state, it gives up a photon having energy 8.841 * 10-4 eV. When the molecule undergoes a vibrational transition fr
> If a sodium chloride (NaCl) molecule could undergo an n n - 1 vibrational transition with no change in rotational quantum number, a photon with wavelength 20.0 µm would be emitted. The mass of a sodium atom is 3.82 * 10-26 kg, and the mass of a chlo
> a. A particle in a box has ave function Ψ (x, t) = Ψ2(x)e-iE2t/ħ, where Ψn and En are given by Eqs. (40.35) and (40.31), respectively. If the energy of the particle is measured, what is the result? b. If instead the particle has wave function Ψ(x, t)=(1
> A lithium atom has mass 1.17 * 10-26 kg, and a hydrogen atom has mass 1.67 * 10-27 kg. The equilibrium separation between the two nuclei in the LiH molecule is 0.159 nm. a. What is the difference in energy between the l = 3 and l = 4 rotational levels?
> The average kinetic energy of an idealgas atom or molecule is 3/2 kT, where T is the Kelvin temperature (Chapter 18). The rotational inertia of the H2 molecule is 4.6 * 10-48 kg . m2. What is the value of T for which 3/2 kT equals the energy separation
> If the energy of the H2 covalent bond is -4.48 eV, what wavelength of light is needed to break that molecule apart? In what part of the electromagnetic spectrum does this light lie?
> The orbital angular momentum of an electron has a magnitude of 4.716 * 10-34 kg.m2/s. What is the angular momentum quantum number l for this electron?
> An electron is in the hydrogen atom with n = 5. a. Find the possible values of L and Lz for this electron, in units of ħ. b. For each value of L, find all the possible angles between L and the z-axis. c. What are the maximum and minimum values of the
> Consider an electron in the N shell. a. What is the smallest orbital angular momentum it could have? b. What is the largest orbital angular momentum it could have? Express your answers in terms of ħ and in SI units. c. What is the largest orbital angul
> What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length 1.00 * 10-14 m, the approximate diameter of a nucleus?
> A particle is in the three dimensional cubical box of Section 41.1. For the state nX = 2, nY = 2, nZ = 1, for what planes (in addition to the walls of the box) is the probability distribution function zero? Compare this number of planes to the correspond
> For each of the following states of a particle in a three dimensional cubical box, at what points is the probability distribution function a maximum: a. nX = 1, nY = 1, nZ = 1 and b. nX = 2, nY = 2, nZ = 1?
> The energies for an electron in the K, L, and M shells of the tungsten atom are -69,500 eV, -12,000 eV, and -2200 eV, respectively. Calculate the wavelengths of the Kα and Kβ x rays of tungsten.
> Sketch the wave function for the potential-energy well shown in Fig. Q40.26 when E1 is less than U0 and when E3 is greater than U0. Figure Q40.26 U(x) Uo ol A В 8.
> Calculate the frequency, energy (in keV), and wavelength of the Kα x ray for the elements a. calcium (Ca, Z = 20); b. cobalt (Co, Z = 27); c. cadmium (Cd, Z = 48).
> A Kα x ray emitted from a sample has an energy of 7.46 keV. Of which element is the sample made?
> Estimate the energy of the highest-l state for a. the L shell of Be+ and b. the N shell of Ca+.
> a. The energy of the 2s state of lithium is -5.391 eV. Calculate the value of Zeff for this state. b. The energy of the 4s state of potassium is -4.339 eV. Calculate the value of Zeff for this state. c. Compare Zeff for the 2s state of lithium, the 3s
> a. The doubly charged ion N2+ is formed by removing two electrons from a nitrogen atom. What is the ground-state electron configuration for the N2+ ion? b. Estimate the energy of the least strongly bound level in the L shell of N2+. c. The doubly charg
> The energies of the 4s, 4p, and 4d states of potassium are given in Example 41.10. Calculate Zeff for each state. What trend do your results show? How can you explain this trend? From Example 41.10 EXAMPLE 41.10 ENERGIES FOR A VALENCE ELECTRON The
> The 5s electron in rubidium (Rb) sees an effective charge of 2.771e. Calculate the ionization energy of this electron.
> a. Write out the ground-state electron configuration (1s2, 2s2, …) for the carbon atom. b. What element of next larger Z has chemical properties similar to those of carbon? Give the ground-state electron configuration for this element.
> A photon is emitted when an electron in a three dimensional cubical box of side length 8.00 * 10-11 m makes a transition from the nX = 2, nY = 2, nZ = 1 state to the nX = 1, nY = 1, nZ = 1 state. What is the wavelength of this photon?
