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?
> 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
> 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. th
> 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
> 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?
> a. Write out the ground-state electron configuration (1s2, 2s2, ….) for the beryllium atom. b. What element of next larger Z has chemical properties similar to those of beryllium? Give the ground-state electron configuration of this element. c. Use th
> Compare the allowed energy levels for the hydrogen atom, the particle in a box, and the harmonic oscillator. What are the values of the quantum number n for the ground level and the second excited level of each system?
> For germanium (Ge, Z = 32), make a list of the number of electrons in each subshell (1s, 2s, 2p, ….). Use the allowed values of the quantum numbers along with the exclusion principle; do not refer to Table 41.3.
> Make a list of the four quantum numbers n, l, ml, and ms for each of the 10 electrons in the ground state of the neon atom. Do not refer to Table 41.2 or 41.3.
> A hydrogen atom in a particular orbital angular momentum state is found to have j quantum numbers 7/2 and 9/2. a. What is the letter that labels the value of l for the state? b. If n = 5, what is the energy difference between the j = 7/2 and j = 9/2 le
> Calculate the energy difference between the ms = 1/2 (“spin up”) and ms = - 1/2 (“spin down”) levels of a hydrogen atom in the 1s state when it is placed in a 1.45-T magnetic field in the negative z-direction. Which level, ms = 1/2 or ms = - 1/2 , has th
> The hyperfine interaction in a hydrogen atom between the magnetic dipole moment of the proton and the spin magnetic dipole moment of the electron splits the ground level into two levels separated by 5.9 * 10-6 eV. a. Calculate the wavelength and frequen
> a. If you treat an electron as a classical spherical object with a radius of 1.0 * 10-17 m, what angular speed is necessary to produce a spin angular momentum of magnitude 3/4 ħ? b. Use v = rω and the result of part (a) to calculate the speed v of a
> A hydrogen atom in the n = 1, ms = - 1/2 state is placed in a magnetic field with a magnitude of 1.60 T in the +z- direction. a. Find the magnetic interaction energy (in electron volts) of the electron with the field. b. Is there any orbital magnetic d
> A hydrogen atom in the 5g state is placed in a magnetic field of 0.600 T that is in the z-direction. a. Into how many levels is this state split by the interaction of the atom’s orbital magnetic dipole moment with the magnetic field? b. What is the ene
> A hydrogen atom undergoes a transition from a 2p state to the 1s ground state. In the absence of a magnetic field, the energy of the photon emitted is 122 nm. The atom is then placed in a strong magnetic field in the z-direction. Ignore spin effects; con
> Model a hydrogen atom as an electron in a cubical box with side length L. Set the value of L so that the volume of the box equals the volume of a sphere of radius a = 5.29 * 10-11 m, the Bohr radius. Calculate the energy separation between the ground and
> In Fig. 40.28, how does the probability of finding a particle in the center half of the region -A From Fig. 40.28: 40.28 Newtonian and quantum- mechanical probability distribution func- tions for a harmonic oscillator for the state n = 10. The Newt
> A hydrogen atom in a 3p state is placed in a uniform external magnetic field B. Consider the interaction of the magnetic field with the atom’s orbital magnetic dipole moment. a. What field magnitude B is required to split the 3p state into multiple leve
> A hydrogen atom is in a d state. In the absence of an external magnetic field, the states with different ml values have (approximately) the same energy. Consider the interaction of the magnetic field with the atom’s orbital magnetic dipole moment. a. Ca
> Show that Φ(ϕ)= eimlϕ = Φ(ϕ + 2π) (that is, show that Φ(ϕ) is periodic with period 2π) if and only if ml is restricted to the values 0, ±1, ±2,……. (Hint: Euler’s formula states that eif = cos ϕ + i sin ϕ.)
> a. What is the probability that an electron in the 1s state of a hydrogen atom will be found at a distance less than a/2 from the nucleus? b. Use the results of part (a) and of Example 41.4 to calculate the probability that the electron will be found at
> a. How many different 5g states does hydrogen have? b. Which of the states in part (a) has the largest angle between L and the z-axis, and what is that angle? c. Which of the states in part (a) has the smallest angle between L and the z-axis, and what
> a. Make a chart showing all possible sets of quantum numbers l and ml for the states of the electron in the hydrogen atom when n = 4. How many combinations are there? b. What are the energies of these states?
> Calculate, in units of ħ, the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of 2, 20, and 200. Compare each with the value of nħ postulated in the Bohr model. What trend do
> A hydrogen atom is in a state that has Lz = 2ħ. In the semiclassical vector model, the angular momentum vector L for this state makes an angle θL = 63.4° with the +z-axis. a. What is the l quantum number for this state? b. What is the smallest possible
> In a particular state of the hydrogen atom, the angle between the angular momentum vector L and the z-axis is θ = 26.6°. If this is the smallest angle for this particular value of the orbital quantum number l, what is l?
> Consider states with angular momentum quantum number l = 2. a. In units of ħ, what is the largest possible value of Lz? b. In units of ħ, what is the value of L? Which is larger: L or the maximum possible Lz? c. For each allowed
> The probability distributions for the harmonic-oscillator wave functions (see Figs. 40.27 and 40.28) begin to resemble the classical (Newtonian) probability distribution when the quantum number n becomes large. Would the distributions become the same as
> For a particle in a three-dimensional cubical box, what is the degeneracy (number of different quantum states with the same energy) of the energy levels a. 3π2ħ2/2mL2 and b. 9π2ħ2/2mL2?
> For the sodium atom of Example 40.8, find a. the ground-state energy; b. the wavelength of a photon emitted when the n = 4 to n = 3 transition occurs; c. the energy difference for any ∆n = 1 transition.
> For the ground­level harmonic oscillator wave function ψ(x) given in Eq. (40.47), |ψ|2 has a maximum at x = 0. a. Compute the ratio of |ψ|2 at x = +A to |ψ|2 at x = 0, where A is given by Eq. (40.48) with n = 0 for the ground level
> In Section 40.5 it is shown that for the ground level of a harmonic oscillator, ∆x ∆px = ħ/2. Do a similar analysis for an excited level that has quantum number n. How does the uncertainty product ∆x ∆px depend on n?
> While undergoing a transition from the n = 1 to the n = 2 energy level, a harmonic oscillator absorbs a photon of wavelength 6.50 µm. What is the wavelength of the absorbed photon when this oscillator undergoes a transition a. from the n = 2 to the n =
> The groundstate energy of a harmonic oscillator is 5.60 eV. If the oscillator undergoes a transition from its n = 3 to n = 2 level by emitting a photon, what is the wavelength of the photon?
> A harmonic oscillator absorbs a photon of wavelength 6.35 µm when it undergoes a transition from the ground state to the first excited state. What is the groundstate energy, in electron volts, of the oscillator?
> Chemists use infrared absorption spectra to identify chemicals in a sample. In one sample, a chemist finds that light of wavelength 5.8 µm is absorbed when a molecule makes a transition from its ground harmonic oscillator level to its first excited level
> Show that ψ(x) given by Eq. (40.47) is a solution to Eq. (40.44) with energy E0 = ħω/2. From Eq. (40.47) From Eq. (40.44)
> A wooden block with mass 0.250 kg is oscillating on the end of a spring that has force constant 110 N/m. Calculate the groundlevel energy and the energy separation between adjacent levels. Express your results in joules and in electron volts. Are quantu
