Show that the speed of the electron in the nth Bohr orbit

An electron in the n _ 5 energy level of hydrogen undergoes a transition to the n _ 3 energy level. What wavelength photon does the atom emit in this process? (a) 1.28 _ 10_6 m (b) 2.37 _ 10_6 m (c) 4.22 _ 10_7 m (d) 3.04 _ 10_6 m (e) 5.92 _ 10_5 m

A beryllium atom is stripped of all but one of its electrons. What is the energy of the ground state? (a) _13.6 eV (b) _218 eV (c) _122 eV (d) _40.8 eV (e) 0

An electron in a hydrogen atom is in the n _ 3 energy level. How many different quantum states are available to it? (a) 1 (b) 2 (c) 8 (d) 9 (e) 18

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

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.

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

Which of the following electronic confi gurations are not allowed? (a) 2s22p6 (b) 3s23p7 (c) 3d74s2 (d) 3d104s24p6 (e) 1s22s22d1

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 _ 2.

If an electron had a spin of 32 , its spin quantum number could have the following four values: ms _ 32 , 12 , _12 , and _32 . If that were true, which one of the following elements with a fi lled shell would become the fi rst 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

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 _ 1 level to the n _ 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.

In the hydrogen atom, the quantum number n can increase without limit. Because of this fact, does the frequency of possible spectral lines from hydrogen also increase without limit?

Does the light emitted by a neon sign constitute a continuous spectrum or only a few colors? Defend your answer.

In an x-ray tube, if the energy with which the electrons strike the metal target is increased, the wavelengths of the characteristic x-rays do not change. Why not?

Must an atom fi rst be ionized before it can emit light? Discuss.

Is it possible for a spectrum from an x-ray tube to show the continuous spectrum of x-rays without the presence of the characteristic x-rays?

Suppose the electron in the hydrogen atom obeyed classical mechanics rather than quantum mechanics. Why should such a hypothetical atom emit a continuous spectrum rather than the observed line spectrum?

When a hologram is produced, the system (including light source, object, beam splitter, and so on) must be held motionless within a quarter of the lights wavelength. Why?

If matter has a wave nature, why is that not observable in our daily experience?

Discuss some consequences of the exclusion principle.

Can the electron in the ground state of hydrogen absorb a photon of energy less than 13.6 eV? Can it absorb a photon of energy greater than 13.6 eV? Explain.

Why do lithium, potassium, and sodium exhibit similar chemical properties?

List some ways in which quantum mechanics altered our view of the atom pictured by the Bohr theory.

It is easy to understand how two electrons (one with spin up, one with spin down) can fi ll the 1s shell for a helium atom. How is it possible that eight more electrons can fi t into the 2s, 2p level to complete the 1s2s22p6 shell for a neon atom?

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?

The wavelengths of the Lyman series for hydrogen are given by 1 l 5 RH a1 2 1 n2b n 5 2, 3, 4, . . . (a) Calculate the wavelengths of the fi rst three lines in this series. (b) Identify the region of the electromagnetic spectrum in which these lines appear.

The wavelengths of the Paschen series for hydrogen are given by 1 l 5 RH a 1 32 2 1 n2b n 5 4, 5, 6, . . . (a) Calculate the wavelengths of the fi rst three lines in this series. (b) Identity the region of the electromagnetic spectrum in which these lines appear.

The size of the atom in Rutherfords model is about 1.0 _ 10_10 m. (a) Determine the attractive electrostatic force between an electron and a proton separated by this distance. (b) Determine (in eV) the electrostatic potential energy of the atom.

The size of the nucleus in Rutherfords model of the atom is about 1.0 fm _ 1.0 _ 10_15 m. (a) Determine the repulsive electrostatic force between two protons separated by this distance. (b) Determine (in MeV) the electrostatic potential energy of the pair of protons.

The size of the atom in Rutherfords model is about 1.0 _ 10_10 m. (a) Determine the speed of an electron moving about the proton using the attractive electrostatic force between an electron and a proton separated by this distance. (b) Does this speed suggest that Einsteinian relativity must be considered in studying the atom? (c) Compute the de Broglie wavelength of the electron as it moves about the proton. (d) Does this wavelength suggest that wave effects, such as diffraction and interference, must be considered in studying the atom?

In a Rutherford scattering experiment, an a-particle (charge _ 2e) heads directly toward a gold nucleus (charge _ 79e). The a-particle had a kinetic energy of 5.0 MeV when very far (r _) from the nucleus. Assuming the gold nucleus to be fi xed in space, determine the distance of closest approach. Hint: Use conservation of energy with PE _ keq1q2/r.

A hydrogen atom is in its fi rst excited state (n _ 2). Using the Bohr theory of the atom, calculate (a) the radius of the orbit, (b) the linear momentum of the electron, (c) the angular momentum of the electron, (d) the kinetic energy, (e) the potential energy, and (f) the total energy.

For a hydrogen atom in its ground state, use the Bohr model to compute (a) the orbital speed of the electron, (b) the kinetic energy of the electron, and (c) the electrical potential energy of the atom.

Show that the speed of the electron in the nth Bohr orbit in hydrogen is given by vn 5 ke e 2 nU

A photon is emitted when a hydrogen atom undergoes a transition from the n _ 5 state to the n _ 3 state. Calculate (a) the wavelength, (b) the frequency, and (c) the energy (in electron volts) of the emitted photon.

A hydrogen atom emits a photon of wavelength 656 nm. From what energy orbit to what lower-energy orbit did the electron jump?

Following are four possible transitions for a hydrogen atom I. ni _ 2; nf _ 5 II. ni _ 5; nf _ 3 III. ni _ 7; nf _ 4 IV. ni _ 4; nf _ 7 (a) Which transition will emit the shortest-wavelength photon? (b) For which transition will the atom gain the most energy? (c) For which transition(s) does the atom lose energy?

What is the energy of a photon that, when absorbed by a hydrogen atom, could cause an electronic transition from (a) the n _ 2 state to the n _ 5 state and (b) the n _ 4 state to the n _ 6 state?

A hydrogen atom initially in its ground state (n _ 1) absorbs a photon and ends up in the state for which n _ 3. (a) What is the energy of the absorbed photon? (b) If the atom eventually returns to the ground state, what photon energies could the atom emit?

How much energy is required to ionize a hydrogen atom when it is in (a) the ground state and (b) the n _ 3 state?

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

(a) If an electron makes a transition from the n _ 4 Bohr orbit to the n _ 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 _ 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? To which series does it belong?

Two hydrogen atoms, both initially in the ground state, undergo a head-on collision. If both atoms are to be excited to the n _ 2 level in this collision, what is the minimum speed each atom can have before the collision?

(a) Calculate the angular momentum of the Moon due to its orbital motion about Earth. In your calculation use 3.84 _ 108 m as the average EarthMoon distance and 2.36 _ 106 s as the period of the Moon in its orbit. (b) If the angular momentum of the Moon obeys Bohrs quantization rule (L _ n_), 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 _ 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 r 5 n2U2 Zme ke e 2 What is the radius of the fi rst Bohr orbit in (a) He, (b) Li2, and (c) Be3?

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 _ 4. (c) Find the energy corresponding to n _ 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?

Determine the wavelength of an electron in the third excited orbit of the hydrogen atom, with n _ 4.

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

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

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

The r-meson has a charge of _e, 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 confi guration of the ground state for nitrogen (Z _ 7). (b) Write out the values for the possible set of quantum numbers n, _, m_, and ms for the electrons in nitrogen.

Two electrons in the same atom have n _ 3 and _ _ 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?

How many different sets of quantum numbers are possible for an electron for which (a) n _ 1, (b) n _ 2, (c) n _ 3, (d) n _ 4, and (e) n _ 5? Check your results to show that they agree with the general rule that the number of different sets of quantum numbers is equal to 2n2.

Zirconium (Z _ 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 confi guration in the ground state of zirconium?

The K-shell ionization energy of copper is 8 979 eV. The L-shell ionization energy is 951 eV. Determine the wavelength of the K_ emission line of copper. What must the minimum voltage be on an x-ray tube with a copper target to see the K_ line?

The K_ x-ray is emitted when an electron undergoes a transition from the L shell (n _ 2) to the K shell (n _ 1). Use the method illustrated in Example 28.4 to calculate the wavelength of the K_ x-ray from a nickel target (Z _ 28).

When an electron drops from the M shell (n _ 3) to a vacancy in the K shell (n _ 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. ADDITIONAL PROBLEMS

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 _ 1 state to the n _ 2 state? (b) If the electrons gain this energy by collision between hydrogen atoms in a high-temperature gas, fi nd the minimum temperature of the heated hydrogen gas. The thermal energy of the heated atoms is given by 3kBT/2, where kB 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, fi nd (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 _ 2 state to the n _ 1 state without emitting a photon. Instead, the excess energy is transferred to an outer electron (one in the n _ 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 _ 2 into the vacant n _ 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 _ 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.)

(a) Construct an energy level diagram for the He ion, for which Z _ 2. (b) What is the ionization energy for He?

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 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 _ 1, the frequency of the emitted light is given by f 5 2p2mke 2e 4 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.