Solved: For an electron in a hydrogen atom, the z

In the planetary model of the atom, the radius of the nucleus (1  1015 m) is analogous to the radius of the sun (7  108 m). The electrons orbit the nucleus at a radial distance (1  1010 m) that is analogous to the radial distance (1.5  1011 m) at which the earth orbits the sun. Suppose that the dimensions of the sun and the earths orbit had the same proportions as those of an atomic nucleus and an electrons orbit. What then would be true about the distance between the earth and the sun? (a) It would be much greater than it actually is. (b) It would be much smaller than it actually is. (c) It would be roughly the same as it actually is. Reaso

Find (a) the longest and (b) the shortest wavelengths of the Balmer series.

The Bohr model does not apply when more than one electron orbits the nucleus because it does not account for the electrostatic force that one electron exerts on another. For instance, an electrically neutral lithium atom (Li) contains three electrons in orbit around a nucleus that includes three protons (Z  3), and Bohrs analysis is not applicable. However, the Bohr model can be used for the doubly charged positive ion of lithium (Li2) that results when two electrons are removed from the neutral atom, leaving only one electron to orbit the nucleus. Obtain the ionization energy that is needed to remove the remaining electron from Li2.

In the line spectrum of atomic hydrogen there is also a group of lines known as the Brackett series. These lines are produced when electrons, excited to high energy levels, make transitions to the n  4 level. Determine (a) the longest wavelength in this series and (b) the wavelength that corresponds to the transition from ni  6 to nf  4. (c) Refer to Figure 24.9 and identify the spectral region in which these lines are found.

Which one of the following statements is true? (a) An atom is less easily ionized when its outermost electron is in an excited state than when it is in the ground state. (b) An atom is more easily ionized when its outermost electron is in an excited state than when it is in the ground state. (c) The energy state (excited state or ground state) of the outermost electron in an atom has nothing to do with how easily the atom can be ionized.

An electron in the hydrogen atom is in the n  4 energy level. When this electron makes a transition to a lower energy level, is the wavelength of the photon emitted in (a) the Lyman series only, (b) the Balmer series only, (c) the Paschen series only, or (d) could it be in the Lyman, the Balmer, or the Paschen series?

A tube contains atomic hydrogen, and nearly all of the electrons in the atoms are in the ground state or n  1 energy level. Electromagnetic radiation with a continuous spectrum of wavelengths (including those in the Lyman, Balmer, and Paschen series) enters one end of the tube and leaves the other end. The exiting radiation is found to contain strong absorption lines. To which one or more of the series do the wavelengths of these absorption lines correspond? Assume that once an electron absorbs a photon and jumps to a higher energy level, it does not absorb yet another photon and jump to an even higher energy level.

Determine the number of possible states for the hydrogen atom when the principal quantum number is (a) n 1 and (b) n 2. R

Consider two hydrogen atoms. There are no external magnetic fields present, and the electron in each atom has the same energy. According to the Bohr model and to quantum mechanics, is it possible for the electrons in these atoms (a) to have zero orbital angular momentum and (b) to have different orbital angular momenta?

In the Bohr model of the hydrogen atom, the electron in the ground state (n  1) is in an orbit that has a radius of exactly 5.29  1011 m, so that the uncertainty in its radial position is y  0 m. According to the Heisenberg uncertainty principle, what does the fact that there is no uncertainty in the electrons radial position imply about the electrons radial speed? The uncertainty principle implies (a) nothing about the radial speed, (b) that the radial speed has only a small uncertainty, (c) that the radial speed has an infinitely large uncertainty.

In the Bohr model for the hydrogen atom, the closer the electron is to the nucleus, the smaller is the total energy of the electron. Is this also true in the quantum mechanical picture of the hydrogen atom?

In the quantum mechanical picture of the hydrogen atom, the orbital angular momentum of the electron may be zero in any of the possible energy states. For which energy state must the orbital angular momentum be zero?

Consider two different hydrogen atoms. The electron in each atom is in a different excited state, so that each electron has a different total energy. Is it possible for the electrons to have the same orbital angular momentum L, according to (a) the Bohr model and (b) quantum mechanics?

The magnitude of the orbital angular momentum of the electron in a hydrogen atom is observed to increase. According to (a) the Bohr model and (b) quantum mechanics, does this necessarily mean that the total energy of the electron also increases?

Determine which of the energy levels in Figure 30.14 are occupied by the electrons in the ground state of hydrogen (1 electron), helium (2 electrons), lithium (3 electrons), beryllium (4 electrons), and boron (5 electrons).

Using the convention of letters to refer to the orbital quantum number, write down the ground-state configuration of the electrons in krypton (Z 36).

Can a 5g subshell contain (a) 22 electrons? (b) 17 electrons?

An electronic configuration for manganese (Z 25) is written as 1s 2 2s 2 2p6 3s 2 3p6 4s 2 3d4 4p1 . Does this configuration represent (a) the ground state or (b) an excited state? X

The highest-energy X-rays produced by an X-ray tube have a wavelength of 1.20 1010 m. What is the speed of the electrons in Figure 30.18 just before they strike the metal target? Ignore the effects of relativity.

Strictly speaking, the Bohr model does not apply to multiple-electron atoms, but it can be used to make estimates. Use the Bohr model to estimate the minimum energy that an incoming electron must have to knock a K-shell electron entirely out of an atom in a platinum (Z 78) target in an X-ray tube.

Use the Bohr model to estimate the wavelength of the K line in the X-ray spectrum of platinum (Z 78).

X-ray tube A and X-ray tube B use the same voltage to accelerate the electrons. However, tube A uses a copper target, whereas tube B uses a silver target. Which one of the following statements is true? (a) The cutoff wavelength is greater for tube A. (b) The cutoff wavelength is greater for tube B. (c) Both tubes have the same cutoff wavelengths.

Is it possible to adjust the electric potential V used to operate an X-ray tube so that Bremsstrahlung X-rays are created, but characteristic X-rays are not created? (a) Yes, if V is small enough. (b) Yes, if V is large enough. (c) No, regardless of the value of V.

Which one of the following statements is true? (a) The K wavelength can be smaller than the cutoff wavelength 0, assuming that both are produced by the same X-ray tube. (b) The K wavelength is produced when an electron undergoes a transition from the n  1 energy level to the n  2 energy level. (c) The K wavelength is always smaller than the K wavelength for a given metal target. Th

A certain laser is designed to operate continuously. Which one of the following statements is false? (a) The population inversion used in this laser involves a higher energy state and a lower energy state. (b) The population inversion used in this laser involves a metastable higher energy state. (c) The laser needs an external source of energy to operate. (d) The external energy source that the laser uses can be disconnected once the population inversion is established

Laser A produces green light. Laser B produces red light. Which laser utilizes energy levels that have a larger energy difference between them? (a) Laser A (b) Laser B (c) The energy difference between the levels is the same for both lasers.

A hydrogen atom (Z  1) is in the third excited state. It makes a transition to a different state, and a photon is either absorbed or emitted. Determine the quantum number nf of the final state and the energy of the photon when the photon is (a) emitted with the shortest possible wavelength, (b) emitted with the longest possible wavelength, and (c) absorbed with the longest possible wavelength.

What is the quantum number of the third excited state?

When an atom emits a photon, is the final quantum number nf of the atom greater than or less than the initial quantum number ni ?

When an atom absorbs a photon, is the final quantum number nf of the atom greater than or less than the initial quantum number ni ?

How is the wavelength of a photon related to its energy?

The K-shell and L-shell ionization energies of a metal are 8979 eV and 951 eV, respectively. (a) Assuming that there is a vacancy in the L shell, what must be the minimum voltage across an X-ray tube with a target made from this metal to produce K X-ray photons? (b) Determine the wavelength of a K photon. c

How is the K photon produced and how much energy does it have?

What must be the minimum voltage across the X-ray tube to produce a K photon?

What is meant by the phrases K-shell ionization energy and L-shell ionization energy?

What does the difference between the K-shell and L-shell ionization energies represent?

Consider applying the Bohr model to a neutral helium atom (Z  2). The model takes into account a number of factors. Which one of the following does it not take into account? (a) The quantization of the orbital angular momentum of an electron (b) The centripetal acceleration of an electron (c) The electric potential energy of an electron (d) The electrostatic repulsion between electrons (e) The electrostatic attraction between the nucleus and an electron

According to the Bohr model, what determines the shortest wavelength in a given series of wavelengths emitted by the atom? (a) The quantum number ni that identifies the higher energy level from which the electron falls into a lower energy level (b) The quantum number nf that identifies the lower energy level into which the electron falls from a higher energy level (c) The ratio nf /ni, where nf is the quantum number that identifies the lower energy level into which the electron falls and ni is the quantum number that identifies the higher level from which the electron falls (d) The sum nf ni of two quantum numbers, where nf identifies the lower energy level into which the electron falls and ni identifies the higher level from which the electron falls (e) The difference nf ni of two quantum numbers, where nf identifies the lower energy level into which the electron falls and ni identifies the higher level from which the electron falls S

According to quantum mechanics, only one of the following combinations of the principal quantum number n and the orbital quantum number is possible for the electron in a hydrogen atom. Which combination is it? (a) n  3,  3 (b) n  2,  3 (c) n  1,  2 (d) n  0,  0 (e) n  3,  1 8. Wh

Which one of the following statements is false? (a) The orbits in the Bohr model have precise sizes, whereas in the quantum mechanical picture of the hydrogen atom they do not. (b) In the absence of external magnetic fields, both the Bohr model and quantum mechanics predict the same total energy for the electron in the hydrogen atom. (c) The spin angular momentum of the electron plays a role in both the Bohr model and the quantum mechanical picture of the hydrogen atom. (d) The magnitude of the orbital angular momentum cannot be zero in the Bohr model, but it can be zero in the quantum mechanical picture of the hydrogen atom.

Each of the following answers indicates the quantum mechanical states of two electrons, A and B. Which pair of states could not describe two of the electrons in a multiple-electron atom? (a) n m A41 B31 (b) n m A32 B31 (c) n m A20 0 B21 (d) n m A53 B41 0 (e) n m A32 B32 1 2 2 1 2 2 ms 1 2 1 2 1 ms 1 2 1 1 2 ms 1 2 1 1 2 1 ms 1 2 1 1 2 1 ms Focus on Concepts c30TheN

Consider the 5f and 6h subshells in a multiple-electron atom. Which of these subshells can contain 19 electrons? (a) Only the 6h subshell (b) Only the 5f subshell (c) Both subshells (d) Neither subshell

Silver (Z  47), copper (Z  29), and platinum (Z  78) can be used as the target in an X-ray tube. Rank in descending order (largest first) the energies needed for impinging electrons to knock a K-shell electron completely out of an atom in each of these targets. (a) Silver, copper, platinum (b) Silver, platinum, copper (c) Platinum, silver, copper (d) Platinum, copper, silver (e) Copper, silver, platinum 16. The voltage applied across an X-ray tube is doubled. What happens to the cutoff wavelength in the spectrum of wavelengths emitted by the tubes metal target? (a) It also doubles. (b) It decreases by a factor of two. (c) It increases by a factor of four. (d) It decreases by a factor of four. (e) Nothing happens to it.

Consider two energy levels that characterize the atoms of a material used in a laser. A population inversion between these two levels __________. (a) has the lower energy level more populated than it normally is and the higher energy level less populated than it normally is (b) is the same thing as a metastable state (c) requires no external source of energy to be sustained (d) has the higher energy level more populated than it normally is and the lower energy level less populated than it normally is

The nucleus of the hydrogen atom has a radius of about 1  1015 m. The electron is normally at a distance of about 5.3  1011 m from the nucleus. Assuming that the hydrogen atom is a sphere with a radius of 5.3  1011 m, find (a) the volume of the atom, (b) the volume of the nucleus, and (c) the percentage of the volume of the atom that is occupied by the nucleus. 2

The nucleus of a hydrogen atom is a single proton, which has a radius of about 1.0  1015 m. The single electron in a hydrogen atom normally orbits the nucleus at a distance of 5.3  1011 m. What is the ratio of the density of the hydrogen nucleus to the density of the complete hydrogen atom? 3

Review Conceptual Example 1 and use the information therein as an aid in working this problem. Suppose that youre building a scale model of the hydrogen atom, and the nucleus is represented by a ball that has a radius of 3.2 cm (somewhat smaller than a baseball). How many miles away (1 mi  1.61  105 cm) should the electron be placed?

In a Rutherford scattering experiment a target nucleus has a diameter of 1.4  1014 m. The incoming particle has a mass of 6.64  1027 kg. What is the kinetic energy of an particle that has a de Broglie wavelength equal to the diameter of the target nucleus? Ignore relativistic effects. * 5

There are Z protons in the nucleus of an atom, where Z is the atomic number of the element. An particle carries a charge of 2e. In a scattering experiment, an particle, heading directly toward a nucleus in a metal foil, will come to a halt when all the particles kinetic energy is converted to electric potential energy. In such a situation, how close will an particle with a kinetic energy of 5.0  1013 J come to a gold nucleus (Z  79)? * 6.

The nucleus of a copper atom contains 29 protons and has a radius of 4.8  1015 m. How much work (in electron volts) is done by the electric force as a proton is brought from infinity, where it is at rest, to the surface of a copper nucleus?

For a doubly ionized lithium atom Li2 (Z  3), what is the principal quantum number of the state in which the electron has the same total energy as a ground-state electron has in the hydrogen atom?

A singly ionized helium atom (He) has only one electron in orbit about the nucleus. What is the radius of the ion when it is in the second excited state?

Using the Bohr model, determine the ratio of the energy of the nth orbit of a triply ionized beryllium atom (Be3, Z  4) to the energy of the nth orbit of a hydrogen atom (H).

The electron in a hydrogen atom is in the first excited state, when the electron acquires an additional 2.86 eV of energy. What is the quantum number n of the state into which the electron moves?

Find the energy (in joules) of the photon that is emitted when the electron in a hydrogen atom undergoes a transition from the n  7 energy level to produce a line in the Paschen series.

(a) What is the ionization energy of a hydrogen atom that is in the n  4 excited state? (b) For a hydrogen atom, determine the ratio of the ionization energy for the n  4 excited state to the ionization energy for the ground state.

A hydrogen atom is in the ground state. It absorbs energy and makes a transition to the n  3 excited state. The atom returns to the ground state by emitting two photons. What are their wavelengths?

In the hydrogen atom, what is the total energy (in electron volts) of an electron that is in an orbit that has a radius of 4.761  1010 m?

Consider the Bohr energy expression (Equation 30.13) as it applies to singly ionized helium He (Z  2) and doubly ionized lithium Li2 (Z  3). This expression predicts equal electron energies for these two species for certain values of the quantum number n (the quantum number is different for each species). For quantum numbers less than or equal to 9, what are the lowest three energies (in electron volts) for which the helium energy level is equal to the lithium energy level? P

A sodium atom (Z  11) contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply, because the neutral atom contains 11 electrons instead of a single electron. However, we can apply the model to the outermost electron as an approximation, provided that we use an effective value Zeffective rather than 11 for the number of protons in the nucleus. (a) The ionization energy for the outermost electron in a sodium atom is 5.1 eV. Use the Bohr model with Z  Zeffective to calculate a value for Zeffective. (b) Using Z  11 and Z  Zeffective, determine the corresponding two values for the radius of the outermost Bohr orbit

A wavelength of 410.2 nm is emitted by the hydrogen atoms in a high-voltage discharge tube. What are the initial and final values of the quantum number n for the energy level transition that produces this wavelength?

A hydrogen atom emits a photon that has momentum with a magnitude of 5.452  1027 kg m/s. This photon is emitted because the electron in the atom falls from a higher energy level into the n  1 level. What is the quantum number of the level from which the electron falls? Use a value of 6.626  1034 Js for Plancks constant.

For atomic hydrogen, the Paschen series of lines occurs when nf  3, whereas the Brackett series occurs when nf  4 in Equation 30.14. Using this equation, show that the ranges of wavelengths in these two series overlap.

Doubly ionized lithium Li2 (Z  3) and triply ionized beryllium Be 3 (Z  4) each emit a line spectrum. For a certain series of lines in the lithium spectrum, the shortest wavelength is 40.5 nm. For the same series of lines in the beryllium spectrum, what is the shortest wavelength? *

(a) Derive an expression for the speed of the electron in the nth Bohr orbit, in terms of Z, n, and the constants k, e, and h. For the hydrogen atom, determine the speed in (b) the n  1 orbit and (c) the n  2 orbit. (d) Generally, when speeds are less than one-tenth the speed of light, the effects of special relativity can be ignored. Are the speeds found in (b) and (c) consistent with ignoring relativistic effects in the Bohr model?

In the Bohr model of hydrogen, the electron moves in a circular orbit around the nucleus. Determine the angular speed of the electron, in revolutions per second, when it is in (a) the ground state and (b) the first excited state.

A hydrogen atom is in its second excited state. Determine, according to quantum mechanics, (a) the total energy (in eV) of the atom, (b) the magnitude of the maximum angular momentum the electron can have in this state, and (c) the maximum value that the z component Lz of the angular momentum can have.

The table lists quantum numbers for five states of the hydrogen atom. Which (if any) of them are not possible? For those that are not possible, explain why.

The orbital quantum number for the electron in a hydrogen atom is  5. What is the smallest possible value (the most negative) for the total energy of this electron? Give your answer in electron volts.

It is known that the possible values for the magnetic quantum number m are 4, 3, 2, 1, 0, 1, 2, 3, and 4. Determine the orbital quantum number and the smallest possible value of the principle quantum number n. 27. ssm T

The maximum value for the magnetic quantum number in state A is m  2, while in state B it is m  1. What is the ratio LA/LB of the magnitudes of the orbital angular momenta of an electron in these two states? *

The electron in a certain hydrogen atom has an angular momentum of 8.948  1034 Js. What is the largest possible magnitude for the z component of the angular momentum of this electron? For accuracy, use h  6.626  1034 Js.

For an electron in a hydrogen atom, the z component of the angular momentum has a maximum value of Lz  4.22  1034 Js. Find the three smallest possible values (the most negative) for the total energy (in electron volts) that this atom could have.

An electron is in the n  5 state. What is the smallest possible value for the angle between the z component of the orbital angular momentum and the orbital angular momentum?

Two of the three electrons in a lithium atom have quantum numbers of n  1,  0, m  0, and n  1,  0, m  0, What quantum numbers can the third electron have if the atom is in (a) its ground state and (b) its first excited state? 32.

Following the style used in Table 30.3, determine the electronic configuration of the ground state for yttrium Y (Z  39). Refer to Figure 30.16 to see the order in which the subshells fill.

Figure 30.16 was constructed using the Pauli exclusion principle and indicates that the n  1 shell holds 2 electrons, the n  2 shell holds 8 electrons, and the n  3 shell holds 18 electrons. These numbers can be obtained by adding the numbers given in the figure for the subshells contained within a given shell. How many electrons can be put into the n  5 shell, which is only partly shown in the figure?

Which of the following subshell configurations are not allowed? For those that are not allowed, give the reason why. (a) 3s 1 (b) 2d2 (c) 3s 4 (d) 4p8 (e) 5f 12

When an electron makes a transition between energy levels of an atom, there are no restrictions on the initial and final values of the principal quantum number n. According to quantum mechanics, however, there is a rule that restricts the initial and final values of the orbital quantum number . This rule is called a selection rule and states that  1. In other words, when an electron makes a transition between energy levels, the value of can only increase or decrease by one. The value of may not remain the same nor may it increase or decrease by more than one. According to this rule, which of the following energy level transitions are allowed? (a) 2s 1s (b) 2p 1s (c) 4p 2p (d) 4s 2p (e) 3d 3s * 3

In the ground state, the outermost shell (n  1) of helium (He) is filled with electrons, as is the outermost shell (n  2) of neon (Ne). The full outermost shells of these two elements distinguish them as the first two so-called noble gases. Suppose that the spin quantum number ms had three possible values, rather than two. If that were the case, which elements would be (a) the first and (b) the second noble gases? Assume that the possible values for the other three quantum numbers are unchanged, and that the Pauli exclusion principle still applies.

By using the Bohr model, decide which element is likely to emit a K X-ray with a wavelength of 4.5  109 m. 3

What is the minimum potential difference that must be applied to an X-ray tube to knock a K-shell electron completely out of an atom in a copper (Z  29) target? Use the Bohr model as needed.

An X-ray tube is being operated at a potential difference of 52.0 kV. What is the Bremsstrahlung wavelength that corresponds to 35.0% of the kinetic energy with which an electron collides with the metal target in the tube?

In the X-ray spectrum of niobium (Z  41), a K peak is observed at a wavelength of 7.462  1011 m. (a) Determine the magnitude of the difference between the observed wavelength of the K X-ray for niobium and that predicted by the Bohr model. (b) Express the magnitude of this difference as a percentage of the observed wavelength. 41

When a certain element is bombarded with high-energy electrons, K X-rays that have an energy of 9890 eV are emitted. Determine the atomic number Z of the element, and identify the element. Use the Bohr model as necessary.

The Bohr model, although not strictly applicable, can be used to estimate the minimum energy Emin that an incoming electron must have in an X-ray tube in order to knock a K-shell electron entirely out of an atom in the metal target. The K X-ray wavelength of metal A is 2.0 times the K X-ray wavelength of metal B. What is the ratio of Emin, A for metal A to Emin, B for metal B? *

An X-ray tube contains a silver (Z  47) target. The high voltage in this tube is increased from zero. Using the Bohr model, find the value of the voltage at which the K X-ray just appears in the X-ray spectrum.

Multiple-Concept Example 9 reviews the concepts that are important in this problem. An electron, traveling at a speed of 6.00  107 m/s, strikes the target of an X-ray tube. Upon impact, the electron decelerates to one-quarter of its original speed, an X-ray photon being emitted in the process. What is the wavelength of the photon?

A laser is used in eye surgery to weld a detached retina back into place. The wavelength of the laser beam is 514 nm, and the power is 1.5 W. During surgery, the laser beam is turned on for 0.050 s. During this time, how many photons are emitted by the laser?

The ultraviolet excimer laser used in the PRK technique (see Section 30.9) has a wavelength of 193 nm. A carbon dioxide laser produces a wavelength of 1.06  105 m. What is the minimum number of photons that the carbon dioxide laser must produce to deliver at least as much or more energy to a target as does a single photon from the excimer laser?

A pulsed laser emits light in a series of short pulses, each having a duration of 25.0 ms. The average power of each pulse is 5.00 mW, and the wavelength of the light is 633 nm. Find the number of photons in each pulse.

The drawing shows three energy levels of a laser that are involved in the lasing action. These levels are analogous to the levels in the Ne atoms of a He-Ne laser. The E2 level is a metastable level, and the E0 level is the ground state. The difference between the energy levels of the laser is shown in the drawing. (a) What energy (in eV per electron) must an external source provide to start the lasing action? (b) What is the wavelength of the laser light? (c) In what region of the electromagnetic spectrum (see Figure 24.9) does the laser light lie?

A laser peripheral iridotomy is a procedure for treating an eye condition known as narrow-angle glaucoma, in which pressure buildup in the eye can lead to loss of vision. A neodymium YAG laser (wavelength  1064 nm) is used in the procedure to punch a tiny hole in the peripheral iris, thereby relieving the pressure buildup. In one application the laser delivers 4.1  103 J of energy to the iris in creating the hole. How many photons does the laser deliver?

Fusion is the process by which the sun produces energy. One experimental technique for creating controlled fusion utilizes a solidstate laser that emits a wavelength of 1060 nm and can produce a power of 1.0  1014 W for a pulse duration of 1.1  1011 s. In contrast, the helium/neon laser used in a bar-code scanner at the checkout counter emits a wavelength of 633 nm and produces a power of about 1.0  103 W. How long (in days) would the helium/neon laser have to operate to produce the same number of photons that the solid-state laser produces in 1.1  1011 s?

(a) What is the minimum energy (in electron volts) that is required to remove the electron from the ground state of a singly ionized helium atom (He, Z  2)? (b) What is the ionization energy for He? 53

Write down the fourteen sets of the four quantum numbers that correspond to the electrons in a completely filled 4f subshell.

Molybdenum has an atomic number of Z  42. Using the Bohr model, estimate the wavelength of the K X-ray.

In the line spectrum of atomic hydrogen there is also a group of lines known as the Pfund series. These lines are produced when electrons, excited to high energy levels, make transitions to the n  5 level. Determine (a) the longest wavelength and (b) the shortest wavelength in this series. (c) Refer to Figure 24.9 and identify the region of the electromagnetic spectrum in which these lines are found.

The voltage across an X-ray tube is 35.0 kV. Suppose that the molybdenum (Z  42) target in the X-ray tube is replaced by a silver (Z  47) target. Determine (a) the tubes cutoff wavelength and (b) the wavelengths of the K X-ray photons emitted by the molybdenum and silver targets.

Review Conceptual Example 6 as background for this problem. For the hydrogen atom, the Bohr model and quantum mechanics both give the same value for the energy of the nth state. However, they do not give the same value for the orbital angular momentum L. (a) For n  1, determine the values of L [in units of h/(2)] predicted by the Bohr model and quantum mechanics. (b) Repeat part (a) for n  3, noting that quantum mechanics permits more than one value of when the electron is in the n  3 state.

The energy of the n  2 Bohr orbit is 30.6 eV for an unidentified ionized atom in which only one electron moves about the nucleus. What is the radius of the n  5 orbit for this species?

The Bohr model can be applied to singly ionized helium He (Z  2). Using this model, consider the series of lines that is produced when the electron makes a transition from higher energy levels into the nf  4 level. Some of the lines in this series lie in the visible region of the spectrum (380750 nm). What are the values of ni for the energy levels from which the electron makes the transitions corresponding to these lines?

Consider a particle of mass m that can exist only between x  0 m and x  L on the x axis. We could say that this particle is confined to a box of length L. In this situation, imagine the standing de Broglie wwaves that can fit into the box. For example, the drawing shows the first three possibilities. Note in this picture that there are either one, two, or three half-wavelengths that fit into the distance L. Use Equation 29.8 for the de Broglie wavelength of a particle and derive an expression for the allowed energies (only kinetic energy) that the particle can have. This expression involves m, L, Plancks constant, and a quantum number n that can have only the values 1, 2, 3, . . . .

A certain species of ionized atoms produces an emission line spectrum according to the Bohr model, but the number of protons Z in the nucleus is unknown. A group of lines in the spectrum forms a series in which the shortest wavelength is 22.79 nm and the longest wavelength is 41.02 nm. Find the next-to-the-longest wavelength in the series of lines.

Referring to Figure 30.16 for the order in which the subshells fill and following the style used in Table 30.3, determine the ground-state electronic configuration for cadmium Cd (Z  48).