Solved: In relating Bohrs theory to the de Broglie

A beryllium atom is stripped of all but one of its electrons. What is the energy of the ground state? (a) 213.6 eV (b) 2218 eV (c) 2122 eV (d) 240.8 eV (e) 0

An electron in the n 5 5 energy level of hydrogen undergoes a transition to the n 5 3 energy level. What wavelength photon does the atom emit in this process? (a) 1.28 3 1026 m (b) 2.37 3 1026 m (c) 4.22 3 1027 m (d) 3.04 3 1026 m (e) 5.92 3 1025 m

Consider an atom having four distinct energy levels. If an electron is able to make transitions between any two levels, how many different wavelengths of radiation could the atom emit? (a) 2 (b) 3 (c) 4 (d) 5 (e) 6

Consider the n 5 3 energy level in a hydrogen atom. How many electrons can be placed in this level? (a) 1 (b) 2 (c) 8 (d) 9 (e) 18

The periodic table is based mainly on which of the following principles? (a) It is based on the uncertainty principle. (b) All electrons in an atom must have the same set of quantum numbers. (c) No two electrons in an atom can have the same set of quantum numbers. (d) All electrons in an atom are in orbitals having the same energy. (e) Energy is conserved in all interactions.

Which of the following electronic configurations are not allowed for an atom? Choose all correct answers. (a) 2s22p6 (b) 3s23p7 (c) 3d74s2 (d) 3d104s24p6 (e) 1s22s22d1

If an electron in an atom has the quantum numbers n 5 3, , 5 2, m, 5 1, and ms 5 12 , what state is it in? (a) 3s (b) 3p (c) 3d (d) 4d (e) 3f

If an electron had a spin of 32 , its spin quantum number could have the following four values: ms 5 132 , 112 , 212 , and 232 . If that were true, which one of the following elements with a filled shell would become the first of the noble gases? (a) He with 2 electrons (b) Li with 3 electrons (c) Be with 4 electrons (d) B with 5 electrons (e) C with 6 electrons

What can be concluded about a hydrogen atom with an electron in the d state? (a) The atom is ionized. (b) The orbital angular momentum of the atom is zero. (c) The orbital angular momentum of the atom is not zero. (d) The atom is in its ground state. (e) The principal quantum number is n 5 2.

In relating Bohrs theory to the de Broglie wavelength of electrons, why does the circumference of an electrons orbit become nine times greater when the electron moves from the n 5 1 level to the n 5 3 level? (a) There are nine times as many wavelengths in the new orbit. (b) The wavelength of the electron becomes nine times as long. (c) There are three times as many wavelengths, and each wavelength is three times as long. (d) The electron is moving nine times faster. (e) The atom is partly ionized.

When an electron collides with an atom, it can transfer all or some of its energy to the atom. A hydrogen atom is in its ground state. Incident on the atom are several electrons, each having a kinetic energy of 10.5 eV. What is the result? (a) The atom can be excited to a higher allowed state. (b) The atom is ionized. (c) The electrons pass by the atom without interaction.

Which of the following is not one of the basic assumptions of the Bohr model of hydrogen? (a) Only certain electron orbits are stable and allowed. (b) The electron moves in circular orbits about the proton under the influence of the Coulomb force. (c) The charge on the electron is quantized. (d) Radiation is emitted by the atom when the electron moves from a higher energy state to a lower energy state. (e) The angular momentum associated with the electrons orbital motion is quantized.

When an atom emits a photon, what happens? (a) One of its electrons leaves the atom. (b) The atom moves to a state of higher energy. (c) The atom moves to a state of lower energy. (d) One of its electrons collides with another particle. (e) None of those events occur.

The ionization energies for Li, Na, K, Rb, and Cs are 5.390, 5.138, 4.339, 4.176, and 3.893 eV, respectively. Explain why these values are to be expected in terms of the atomic structures.

Why is stimulated emission so important in the operation of a laser?

A particle of charge q and mass m, moving with a constant speed v, perpendicular to a constant magnetic field B, follows a circular path. If in this case the angular momentum about the center of this circle is quantized so that mvr 5 2nU, show that the allowed radii for the particle are rn 5 2nU qB where n 5 1, 2, 3, . . . .

(a) If an electron makes a transition from the n 5 4 Bohr orbit to the n 5 2 orbit, determine the wavelength of the photon created in the process. (b) Assuming that the atom was initially at rest, determine the recoil speed of the hydrogen atom when this photon is emitted.

Consider a large number of hydrogen atoms, with electrons all initially in the n 5 4 state. (a) How many different wavelengths would be observed in the emission spectrum of these atoms? (b) What is the longest wavelength that could be observed? (c) To which series does the wavelength found in (b) belong?

A photon with energy 2.28 eV is absorbed by a hydrogen atom. Find (a) the minimum n for a hydrogen atom that can be ionized by such a photon and (b) the speed of the electron released from the state in part (a) when it is far from the nucleus.

(a) Calculate the angular momentum of the Moon due to its orbital motion about Earth. In your calculation use 3.84 3 108 m as the average EarthMoon distance and 2.36 3 106 s as the period of the Moon in its orbit. (b) If the angular momentum of the Moon obeys Bohrs quantization rule (L 5 nU), determine the value of the quantum number n. (c) By what fraction would the EarthMoon radius have to be increased to increase the quantum number by 1?

An electron is in the second excited orbit of hydrogen, corresponding to n 5 3. Find (a) the radius of the orbit and (b) the wavelength of the electron in this orbit.

(a) Write an expression relating the kinetic energy KE of the electron and the potential energy PE in the Bohr model of the hydrogen atom. (b) Suppose a hydrogen atom absorbs a photon of energy E, resulting in the transfer of the electron to a higher-energy level. Express the resulting change in the potential energy of the system in terms of E. (c) What is the change in the electrons kinetic energy during this process?

The orbital radii of a hydrogen-like atom is given by the equation rn 5 n2U2 Zmekee2 What is the radius of the first Bohr orbit in (a) He1, (b) Li21, and (c) Be31?

Consider a Bohr model of doubly ionized lithium. (a) Write an expression similar to Equation 28.14 for the energy levels of the sole remaining electron. (b) Find the energy corresponding to n 5 4. (c) Find the energy corresponding to n 5 2. (d) Calculate the energy of the photon emitted when the electron transits from the fourth energy level to the second energy level. Express the answer both in electron volts and in joules. (e) Find the frequency and wavelength of the emitted photon. (f) In what part of the spectrum is the emitted light?

A general expression for the energy levels of oneelectron atoms and ions is En 5 2 mke 2q1 2q2 2 2U2n2 Here m is the reduced mass of the atom, given by m 5 m1m2/(m1 1 m2), where m1 is the mass of the electron and m2 is the mass of the nucleus; ke is the Coulomb constant; and q1 and q2 are the charges of the electron and the nucleus, respectively. The wavelength for the n 5 3 to n 5 2 transition of the hydrogen atom is 656.3 nm (visible red light). What are the wavelengths for this same transition in (a) positronium, which consists of an electron and a positron, and (b) singly ionized helium? Note: A positron is a positively charged electron.

Using the concept of standing waves, de Broglie was able to derive Bohrs stationary orbit postulate. He assumed a confined electron could exist only in states where its de Broglie waves form standing-wave patterns, as in Figure 28.6. Consider a particle confined in a box of length L to be equivalent to a string of length L and fixed at both ends. Apply de Broglies concept to show that (a) the linear momentum of this particle is quantized with p 5 mv 5 nh/2L and (b) the allowed states correspond to particle energies of En 5 n2E0, where E0 5 h2/(8mL2).

List the possible sets of quantum numbers for electrons in the 3d subshell.

When the principal quantum number is n 5 4, how many different values of (a) , and (b) m, are possible?

The r-meson has a charge of 2e, a spin quantum number of 1, and a mass 1 507 times that of the electron. If the electrons in atoms were replaced by r-mesons, list the possible sets of quantum numbers for r-mesons in the 3d subshell.

(a) Write out the electronic configuration of the ground state for nitrogen (Z 5 7). (b) Write out the values for the possible set of quantum numbers n, ,, m,, and ms for the electrons in nitrogen.

A certain element has its outermost electron in a 3p subshell. It has valence 13 because it has three more electrons than a certain noble gas. What element is it?

Two electrons in the same atom have n 5 3 and , 5 1. (a) List the quantum numbers for the possible states of the atom. (b) How many states would be possible if the exclusion principle did not apply to the atom?

Zirconium (Z 5 40) has two electrons in an incomplete d subshell. (a) What are the values of n and , for each electron? (b) What are all possible values of m, and ms? (c) What is the electron configuration in the ground state of zirconium?

A tungsten target is struck by electrons that have been accelerated from rest through a 40.0-kV potential difference. Find the shortest wavelength of the radiation emitted.

A bismuth target is struck by electrons, and x-rays are emitted. Estimate (a) the M- to L-shell transitional energy for bismuth and (b) the wavelength of the x-ray emitted when an electron falls from the M shell to the L shell.

When an electron drops from the M shell (n 5 3) to a vacancy in the K shell (n 5 1), the measured wavelength of the emitted x-ray is found to be 0.101 nm. Identify the element.

The K series of the discrete spectrum of tungsten contains wavelengths of 0.018 5 nm, 0.020 9 nm, and 0.021 5 nm. The K-shell ionization energy is 69.5 keV. Determine the ionization energies of the L, M, and N shells.

In a hydrogen atom, what is the principal quantum number of the electron orbit with a radius closest to 1.0 mm?

(a) How much energy is required to cause an electron in hydrogen to move from the n 5 1 state to the n 5 2 state? (b) If the electrons gain this energy by collision between hydrogen atoms in a high-temperature gas, find the minimum temperature of the heated hydrogen gas. The thermal energy of the heated atoms is given by 3k BT/2, where k B is the Boltzmann constant.

A pulsed ruby laser emits light at 694.3 nm. For a 14.0-ps pulse containing 3.00 J of energy, find (a) the physical length of the pulse as it travels through space and (b) the number of photons in it. (c) If the beam has a circular cross section 0.600 cm in diameter, what is the number of photons per cubic millimeter?

An electron in chromium moves from the n 5 2 state to the n 5 1 state without emitting a photon. Instead, the excess energy is transferred to an outer electron (one in the n 5 4 state), which is then ejected by the atom. In this Auger (pronounced ohjay) process, the ejected electron is referred to as an Auger electron. (a) Find the change in energy associated with the transition from n 5 2 into the vacant n 5 1 state using Bohr theory. Assume only one electron in the K shell is shielding part of the nuclear charge. (b) Find the energy needed to ionize an n 5 4 electron, assuming 22 electrons shield the nucleus. (c) Find the kinetic energy of the ejected (Auger) electron. (All answers should be in electron volts.)

As the Earth moves around the Sun, its orbits are quantized. (a) Follow the steps of Bohrs analysis of the hydrogen atom to show that the allowed radii of the Earths orbit are given by rn 5 n2U2 GMSME 2 where n is an integer quantum number, MS is the mass of the Sun, and ME is the mass of the Earth. (b) Calcuser62060 late the numerical value of n for the SunEarth system. (c) Find the distance between the orbit for quantum number n and the next orbit out from the Sun corresponding to the quantum number n 1 1. (d) Discuss the significance of your results from parts (b) and (c).

A laser used in eye surgery emits a 3.00-mJ pulse in 1.00 ns, focused to a spot 30.0 mm in diameter on the retina. (a) Find (in SI units) the power per unit area at the retina. (This quantity is called the irradiance.) (b) What energy is delivered per pulse to an area of molecular size (say, a circular area 0.600 nm in diameter)?

An electron has a de Broglie wavelength equal to the diameter of a hydrogen atom in its ground state. (a) What is the kinetic energy of the electron? (b) How does this energy compare with the magnitude of the ground-state energy of the hydrogen atom?

Use Bohrs model of the hydrogen atom to show that when the atom makes a transition from the state n to the state n 2 1, the frequency of the emitted light is given by f 5 2p2mke 2e4 h3 c 2n 2 1 1n 2 122n2 d

Suppose the ionization energy of an atom is 4.100 eV. In this same atom, we observe emission lines that have wavelengths of 310.0 nm, 400.0 nm, and 1 378 nm. Use this information to construct the energy level diagram with the least number of levels. Assume that the higher energy levels are closer together.