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Chapter 2: Problem 5

Relative to electrons and electron states, what does each of the four quantum numbers specify?

Short Answer

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The four quantum numbers related to electrons and electron states are: 1. Principal Quantum Number (n): Specifies the energy level of an electron in an atom and the size of the orbital. It is a positive integer with values like n = 1, 2, 3, etc. Higher 'n' values indicate higher energy and electron distance from the nucleus. 2. Azimuthal Quantum Number (l): Defines the shape of the electron's orbital and depends on the value of the principal quantum number (n). It ranges from 0 to (n-1), with each value representing a unique orbital shape (e.g., l = 0 for s orbitals, l = 1 for p orbitals). 3. Magnetic Quantum Number (m_l): Gives information about the orientation of the orbital in space and depends on the value of the azimuthal quantum number (l). It ranges from -l to +l, including 0, with different values corresponding to unique orbital orientations. 4. Spin Quantum Number (m_s): Describes the intrinsic angular momentum or "spin" of the electron. There are two possible values: +1/2 (spin up) and -1/2 (spin down). The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of quantum numbers, ensuring electrons have unique spin states.

Step by step solution

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1. Principal Quantum Number (n)

The principal quantum number (n) specifies the energy level of an electron in an atom and the size of the orbital. It is a positive integer with values like n = 1, 2, 3, etc. As 'n' increases, the electron is further away from the nucleus and has higher energy.
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2. Azimuthal Quantum Number (l)

The azimuthal quantum number (l) defines the shape of the electron's orbital. It depends on the value of the principal quantum number (n) and can have integer values from 0 to (n-1). For example, if n = 2, then l can be either 0 or 1. Generally, l = 0 represents s orbitals (spherical shaped), l = 1 represents p orbitals (dumbbell-shaped), l = 2 represents d orbitals (double dumbbell-shaped), and l = 3 represents f orbitals (complex shapes).
03

3. Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) gives information about the orientation of the orbital in space. It depends on the value of the azimuthal quantum number (l) and can have integer values ranging from -l to +l, including 0. For example, if l = 1 (p orbital), then m_l can be -1, 0, or +1, which means there are three possible orientations for the p orbital.
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4. Spin Quantum Number (m_s)

The spin quantum number (m_s) describes the intrinsic angular momentum or "spin" of the electron. There are two possible values for m_s: +1/2 (spin up) and -1/2 (spin down). This means that each electron in an orbital has one of two possible spin states. This is also the reason behind the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of quantum numbers; they must differ in at least one quantum number.

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Most popular questions from this chapter

Chromium has four naturally occurring isotopes: \(4.34 \%\) of \({ }^{50} \mathrm{Cr}\), with an atomic weight of \(49.9460\) amu; \(83.79 \%\) of \({ }^{52} \mathrm{Cr}\), with an atomic weight of \(51.9405 \mathrm{amu} ; 9.50 \%\) of \({ }^{53} \mathrm{Cr}\), with an atomic weight of \(52.9407 \mathrm{amu} ;\) and \(2.37 \%\) of \({ }^{54} \mathrm{Cr}\), with an atomic weight of \(53.9389\) amu. On the basis of these data, confirm that the average atomic weight of \(\mathrm{Cr}\) is \(51.9963 \mathrm{amu}\)

Cite the difference between atomic mass and atomic weight.

(a) Cite two important quantum-mechanical concepts associated with the Bohr model of the atom. (b) Cite two important additional refinements that resulted from the wave- mechanical atomic model.

Calculate the force of attraction between a \(\mathrm{K}^{+}\) and an \(\mathrm{O}^{2-}\) ion whose centers are separated by a distance of \(1.5 \mathrm{~nm}\).

For \(\mathrm{a} \mathrm{K}^{+}-\mathrm{Cl}^{-}\)ion pair, attractive and repulsive energies \(E_{A}\) and \(E_{R}\), respectively, depend on the distance between the ions \(r\), according to $$ \begin{aligned} E_{A} &=-\frac{1.436}{r} \\ E_{R} &=\frac{5.86 \times 10^{-6}}{r^{9}} \end{aligned} $$ For these expressions, energies are expressed in electron volts per \(\mathrm{K}^{+}-\mathrm{Cl}^{-}\)pair, and \(r\) is the distance in nanometers. The net energy \(E_{N}\) is just the sum of the preceding two expressions. (a) Superimpose on a single plot \(E_{N}, E_{R}\), and \(E_{A}\) versus \(r\) up to \(1.0 \mathrm{~nm}\). (b) On the basis of this plot, determine (i) the equilibrium spacing \(r_{0}\) between the \(\mathrm{K}^{+}\)and \(\mathrm{Cl}^{-}\)ions, and (ii) the magnitude of the bonding energy \(E_{0}\) between the two ions. (c) Mathematically determine the \(r_{0}\) and \(E_{0}\) values using the solutions to Problem \(2.14\) and compare these with the graphical results from part (b).
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