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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

(a) Briefly cite the main differences between ionic, covalent, and metallic bonding. (b) State the Pauli exclusion principle.

The net potential energy \(E_{N}\) between two adjacent ions is sometimes represented by the expression $$ E_{N}=-\frac{C}{r}+D \exp \left(-\frac{r}{\rho}\right) $$ in which \(r\) is the interionic separation and \(C\), \(D\), and \(\rho\) are constants whose values depend on the specific material. (a) Derive an expression for the bonding energy \(E_{0}\) in terms of the equilibrium interionic separation \(r_{0}\) and the constants \(D\) and \(\rho\) using the following procedure: 1\. Differentiate \(E_{N}\) with respect to \(r\) and set the resulting expression equal to zero. 2\. Solve for \(C\) in terms of \(D, \rho\), and \(r_{0}\) - 3\. Determine the expression for \(E_{0}\) by substitution for \(C\) in Equation \(2.12\). (b) Derive another expression for \(E_{0}\) in terms of \(r_{0}, C\), and \(\rho\) using a procedure analogous to the one outlined in part (a).

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

(a) How many grams are there in one amu of a material? (b) Mole, in the context of this book, is taken in units of gram-mole. On this basis, how many atoms are there in a pound-mole of a substance?

The net potential energy between two adjacent ions, \(E_{N}\), may be represented by the sum of Equations \(2.8\) and \(2.9\); that is, $$ E_{N}=-\frac{A}{r}+\frac{B}{r^{n}} $$ Calculate the bonding energy \(E_{0}\) in terms of the parameters \(A, B\), and \(n\) using the following procedure: 1\. Differentiate \(E_{N}\) with respect to \(r\), and then set the resulting expression equal to zero, because the curve of \(E_{N}\) versus \(r\) is a minimum at \(E_{0 \text { - }}\) 2\. Solve for \(r\) in terms of \(A, B\), and \(n\), which yields \(r_{0}\), the equilibrium interionic spacing. 3\. Determine the expression for \(E_{0}\) by substituting \(r_{0}\) into Equation \(2.11\).
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