OpenStax Chemistry: Atoms First 2e
3.3: Development of Quantum Theory
Chemistry: Atoms First 2e
3: Electronic Structure and Periodic Properties of Elements
3.3: Development of Quantum Theory
31. How are the Bohr model and the quantum mechanical model of the hydrogen atom similar?
How are they different?
Solution
Both models have a central positively charged nucleus with electrons moving about the nucleus
in accordance with the Coulomb electrostatic potential. The Bohr model assumes that the
electrons move in circular orbits that have quantized energies, angular momentum, and radii that
are specified by a single quantum number, n = 1,2,3,…, but this quantization is an ad hoc
assumption made by Bohr to incorporate quantization into an essentially classical mechanics
description of the atom. Bohr also assumed that electrons orbiting the nucleus normally do not
emit or absorb electromagnetic radiation, but do so when the electron switches to a different
orbit. In the quantum mechanical model, the electrons do not move in precise orbits (such orbits
violate the Heisenberg uncertainty principle) and, instead, a probabilistic interpretation of the
electron’s position at any given instant is used, with a mathematical function ψ called a wave
function that can be used to determine the electron’s spatial probability distribution. These wave
functions, or orbitals, are three-dimensional stationary waves that can be specified by three
quantum numbers that arise naturally from their underlying mathematics (no ad hoc assumptions
required): the principal quantum number, n (the same one used by Bohr), which specifies shells
such that orbitals having the same n all have the same energy and approximately the same spatial
extent; the angular momentum quantum number l, which is a measure of the orbital’s angular
momentum and corresponds to the orbitals’ general shapes, as well as specifying subshells such
that orbitals having the same l (and n) all have the same energy; and the orientation quantum
number m, which is a measure of the z component of the angular momentum and corresponds to
the orientations of the orbitals. The Bohr model gives the same expression for the energy as the
quantum mechanical expression and, hence, both properly account for hydrogen’s discrete
spectrum (an example of getting the right answers for the wrong reasons, something that many
chemistry students can sympathize with), but gives the wrong expression for the angular
momentum (Bohr orbits necessarily all have non-zero angular momentum, but some quantum
orbitals [s orbitals] can have zero angular momentum).
32. What are the allowed values for each of the four quantum numbers: n, l, ml, and ms?
Solution
n = 1... ; l = 0...n – 1; ml = –l, –1 + 1,...,l– 1, l;
33. Describe the properties of an electron associated with each of the following four quantum
numbers: n, l, ml, and ms.
Solution
n determines the general range for the value of energy and the probable distances that the
electron can be from the nucleus. l determines the shape of the orbital. m1 determines the
orientation of the orbitals of the same l value with respect to one another. ms determines the spin
of an electron.
34. Answer the following questions:
(a) Without using quantum numbers, describe the differences between the shells, subshells, and
orbitals of an atom.
(b) How do the quantum numbers of the shells, subshells, and orbitals of an atom differ?
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3.3: Development of Quantum Theory
Chemistry: Atoms First 2e
3: Electronic Structure and Periodic Properties of Elements
3.3: Development of Quantum Theory
31. How are the Bohr model and the quantum mechanical model of the hydrogen atom similar?
How are they different?
Solution
Both models have a central positively charged nucleus with electrons moving about the nucleus
in accordance with the Coulomb electrostatic potential. The Bohr model assumes that the
electrons move in circular orbits that have quantized energies, angular momentum, and radii that
are specified by a single quantum number, n = 1,2,3,…, but this quantization is an ad hoc
assumption made by Bohr to incorporate quantization into an essentially classical mechanics
description of the atom. Bohr also assumed that electrons orbiting the nucleus normally do not
emit or absorb electromagnetic radiation, but do so when the electron switches to a different
orbit. In the quantum mechanical model, the electrons do not move in precise orbits (such orbits
violate the Heisenberg uncertainty principle) and, instead, a probabilistic interpretation of the
electron’s position at any given instant is used, with a mathematical function ψ called a wave
function that can be used to determine the electron’s spatial probability distribution. These wave
functions, or orbitals, are three-dimensional stationary waves that can be specified by three
quantum numbers that arise naturally from their underlying mathematics (no ad hoc assumptions
required): the principal quantum number, n (the same one used by Bohr), which specifies shells
such that orbitals having the same n all have the same energy and approximately the same spatial
extent; the angular momentum quantum number l, which is a measure of the orbital’s angular
momentum and corresponds to the orbitals’ general shapes, as well as specifying subshells such
that orbitals having the same l (and n) all have the same energy; and the orientation quantum
number m, which is a measure of the z component of the angular momentum and corresponds to
the orientations of the orbitals. The Bohr model gives the same expression for the energy as the
quantum mechanical expression and, hence, both properly account for hydrogen’s discrete
spectrum (an example of getting the right answers for the wrong reasons, something that many
chemistry students can sympathize with), but gives the wrong expression for the angular
momentum (Bohr orbits necessarily all have non-zero angular momentum, but some quantum
orbitals [s orbitals] can have zero angular momentum).
32. What are the allowed values for each of the four quantum numbers: n, l, ml, and ms?
Solution
n = 1... ; l = 0...n – 1; ml = –l, –1 + 1,...,l– 1, l;
33. Describe the properties of an electron associated with each of the following four quantum
numbers: n, l, ml, and ms.
Solution
n determines the general range for the value of energy and the probable distances that the
electron can be from the nucleus. l determines the shape of the orbital. m1 determines the
orientation of the orbitals of the same l value with respect to one another. ms determines the spin
of an electron.
34. Answer the following questions:
(a) Without using quantum numbers, describe the differences between the shells, subshells, and
orbitals of an atom.
(b) How do the quantum numbers of the shells, subshells, and orbitals of an atom differ?
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