Chapter 29

\(\cdot\) The orbital angular momentum of an electron has a magnitude of \(4.716 \times 10^{-34} \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}\) . What is the angular-momentum quantum number \(/\) for this electron?

Consider states with \(l = 3 .\) (a) In units of \(\hbar ,\) what is the largest possible value of \(L _ { z } ?\) (b) In units of \(\hbar ,\) what is the value of \(L ?\) Which is larger, \(L\) or the maximum possible \(L _ { z } ?\) (c) Assume a model in which \(\vec { \boldsymbol { L } }\) is described as a classical vector. For each allowed value of \(L _ { z } ,\) what angle does the vector \(\vec { L }\) make with the \(+ z\) axis?

An electron is in the hydrogen atom with \(n = 3 .\) (a) Find the possible values of \(L\) and \(L _ { z }\) for this electron, in units of \(\hbar .\) (b) For each value of \(L ,\) find all the possible angles between \(L\) and the \(z\) axis.

An electron is in the hydrogen atom with \(n = 5 .\) (a) Find the possible values of \(L\) and \(L _ { z }\) for this electron, in units of \hbar. (b) For each value of \(L ,\) find all the possible angles between \(L\) and the \(z\) axis. (d) What are the maximum and minimum values of the magnitude of the angle between \(L\) and the \(z\) axis?

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 \(\hbar\) and in SI units. (c) What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of \(\hbar\) and in SI units. (d) What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of \(\hbar\) and in SI units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the \(z\) direction to its orbital angular momentum in the \(z\) direction?

(a) How many different 3\(d\) states does hydrogen have? Make a list showing all of them. (b) How many different 3f states does it have?

Make a list of the four quantum numbers \(n , l , m _ { l } ,\) and \(s\) for each of the 12 electrons in the ground state of the magnesium atom.

(a) List the different possible combinations of quantum numbers \(/\) and \(m _ { l }\) for the \(n = 5\) shell. (b) How many electrons can be placed in the \(n = 5\) shell?

For bromine \(( Z = 35 ) ,\) make a list of the number of electrons in each subshell \(( 1 s , 2 s , 2 p ,\) etc. \() .\)

(a) Write out the electron configuration \(\left( 1 s ^ { 2 } 2 s ^ { 2 } ,\) etc. \right\()\) for Li and \(\mathrm { Na }\) . (b) How many electrons does each of these atoms have in its outer shell?

(a) Write out the ground-state electron configuration \(\left( 1 s ^ { 2 } , 2 s ^ { 2 } ,\) etc. \right\()\) for the carbon atom. (b) What element of next- larger \(Z\) has chemical properties similar to those of carbon? (See Example \(29.3 . )\) Give the ground-state electron con- figuration for this element.

(a) Write out the ground-state electron configuration \(\left( 1 s ^ { 2 } 2 s ^ { 2 } ,\) etc. \right\()\) for the beryllium atom. (b) What element of next- larger \(Z\) has chemical properties similar to those of beryllium? (See Example \(29.3 .\) ) Give the ground-state electron con- figuration of this element. (c) Use the procedure of part (b) to predict what element of next-larger \(Z\) than in (b) will have chemical properties similar to those of the element you found in part (b), and give its ground-state electron configuration.

Write out the electron configuration \(\left( 1 s ^ { 2 } 2 s ^ { 2 } ,\) etc. \right\()\) for Ne, Ar, and Kr. (b) How many electrons does each of these atoms have in its outer shell? (c) Predict the chemical behavior of these three atoms. Explain your reasoning.

Calculate, in units of \(\hbar ,\) 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 \(n \hbar\) postulated in the Bohr model. What trend do you see?

If the energy of the \(\mathrm { H } _ { 2 }\) covalent bond is \(- 4.48 \mathrm { eV } ,\) what wavelength of light is needed to break that molecule apart? In what part of the electromagnetic spectrum does this light lie?

(a) A molecule decreases its vibrational energy by 0.250 eV by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie? (b) An atom decreases its energy by 8.50 eV by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic spectrum does that wavelength of light lie? (c) A molecule decreases its rotational energy by \(3.20 \times\) \(10 ^ { - 3 }\) eV by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electro- magnetic spectrum does that wavelength of light lie?

An ionic bond. (a) Calculate the electric potential energy for a \(\mathrm { K } ^ { + }\) ion and a Br \(^ { - }\) ion separated by a distance of \(0.29 \mathrm { nm } ,\) 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 bromine has an electron affinity of 3.5 eV. Use these data and the results of part (a) to estimate the binding energy of the KBr molecule. Do you expect the actual binding energy to be higher or lower than your estimate? Explain your reasoning.

The spacing of adjacent atoms in a NaCl crystal is 0.282\(\mathrm { nm }\) and the masses of the atoms are \(3.82 \times 10 ^ { - 26 } \mathrm { kg } ( \mathrm { Na } )\) and \(5.89 \times 10 ^ { - 26 } \mathrm { kg } ( \mathrm { Cl } ) .\) Use this information to calculate the density of sodium chloride.

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 this photon lie? (b) Explain why pure diamond is transparent and colorless. (Hint: Will photons of visible light that strike a diamond be absorbed or transmitted?) (c) Most gem diamonds have a yellow color. Explain how impurities in the diamond can cause this color.

The gap between valence and conduction bands in silicon is 1.12\(\mathrm { eV } .\) A nickel nucleus in an excited state emits a gamma-ray photon with wavelength \(9.31 \times 10 ^ { - 4 } \mathrm { nm }\) . How many electrons can be excited from the top of the valence band to the bottom of the conduction band by the absorption of this gamma ray?

Sketch a qualitative (no numbers) graph of the resistance as a function of temperature for (a) an ordinary conductor, such as Cu, including temperatures approaching \(0 \mathrm { K } ;\) (b) a superconductor; include temperatures above and below the critical temperature, and let the temperature approach 0\(\mathrm { K }\) .

For magnesium, the first ionization potential is 7.6\(\mathrm { eV }\) ; the second (the additional energy required to remove a second electron) is almost twice this, \(15 \mathrm { eV } ,\) and the third ionization potential is much larger, about 80 eV. Why do these numbers keep increasing?

The dissociation energy of the hydrogen molecule (i.e., the energy required to separate the two atoms) is 4.48 eV. In the gas phase (treated as an ideal gas), at what temperature is the average translational kinetic energy of a molecule equal to this energy?

The maximum wavelength of light that a certain silicon photocell can detect is 1.11\(\mu \mathrm { m }\) . (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. (Hint: Will visible light that strikes silicon be transmitted or absorbed?

Use the electron configurations of He, Ne, and Ar to explain why these atoms normally do not combine chemically with other atoms.

Use the electron configurations of \(\mathrm { H }\) and \(\mathrm { O }\) to explain why these atoms combine chemically in a two-to-one ratio to form water.

Use the electron configurations of Si and O to explain why these atoms combine chemically in a one-to-two ratio to form sand.

Consider an electron in hydrogen having total energy \(- 0.5440 \mathrm { eV } .\) (a) What are the possible values of its orbital angular momentum (in terms of \(\hbar\) ? (b) What wavelength of light would it take to excite this electron to the next higher shell? Is this photon visible to humans?

The energy of the van der Waals bond, which is responsible for a number of the characteristics of water, is about 0.50 eV. (a) At what temperature would the average translational kinetic energy of water molecules be equal to this energy? (b) At that temperature, would water be liquid or gas? Under ordinary everyday conditions, do van der Waals forces play a role in the behavior of water?

(a) What is the lowest possible energy (in electron volts) of an electron in hydrogen if its orbital angular momentum is \(\sqrt { 12 } \hbar ?\) (b) What are the largest and smallest values of the \(z\) component of the orbital angular momentum (in terms of \(\hat { h }\) ) for the electron in part (a)? (c) What are the largest and smallest values of the spin angular momentum (in terms of \(\hbar\) ) for the electron in part (a)?(d) What are the largest and smallest values of the orbital angular momentum (in terms of \(\hbar\) ) for an electron in the \(M\) shell of hydrogen?

An electron in hydrogen is in the 5\(f\) state. (a) Find the largest possible value of the \(z\) component of its angular momentum. (b) Show that for the electron in part (a), the corresponding \(x\) and \(y\) components of its angular momentum satisfy the equation \(\sqrt { L _ { x } ^ { 2 } + L _ { y } ^ { 2 } } = \hbar \sqrt { 3 }\)