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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

Expert verified
Answer: The four quantum numbers are the principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (m_l), and spin quantum number (s). The principal quantum number (n) defines the main energy level of an electron; the azimuthal quantum number (l) determines the shape of the orbital and the angular momentum of an electron; the magnetic quantum number (m_l) defines the orientation of the orbital in space; and the spin quantum number (s) describes the intrinsic angular momentum or spin of an electron.

Step by step solution

01

1. Introduction to Quantum Numbers

Quantum numbers are parameters that describe the state of an electron in an atom. There are four quantum numbers that are used to fully describe the state of an electron: principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (m_l), and spin quantum number (s).
02

2. Principal Quantum Number (n)

The principal quantum number (n) defines the main energy level (shell) of an electron and determines the size of an orbital. The value of n ranges from 1 to ∞, but in practice, it usually ranges from 1 to 7. A higher value of n means a higher energy level and a greater distance from the nucleus.
03

3. Azimuthal Quantum Number (l)

The azimuthal quantum number (l) is also known as the angular momentum quantum number. It defines the shape of the orbital and the angular momentum of an electron. The value of l ranges from 0 to (n-1), where n is the principal quantum number. The value of l is related to the orbital subshell, with 0 corresponding to the s subshell, 1 to the p subshell, 2 to the d subshell, and 3 to the f subshell.
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4. Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) determines the orientation of the orbital in space. It defines the number of possible orientations that an orbital can have for a given value of l. The value of m_l ranges from -l to +l, including 0. For example, if l = 2 (d subshell), the possible values of m_l are -2, -1, 0, 1, and 2.
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5. Spin Quantum Number (s)

The spin quantum number (s) describes the intrinsic angular momentum of an electron, or its spin. Electrons have a fixed spin of 1/2 but can have two possible orientations: spin up (+1/2) and spin down (-1/2), represented by m_s. This results in two electrons being able to occupy each orbital, as long as they have opposite spins (which follows Pauli's Exclusion Principle). In conclusion, the four quantum numbers help to define the properties of electrons in an atom, including their energy levels, shapes, orientations of orbitals, and intrinsic spins. Understanding the role of each quantum number is vital in understanding the electronic configuration and behavior of atoms.

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Most popular questions from this chapter

(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?

Compute the percentage ionic character of the interatomic bond for each of the following compounds: \(\mathrm{MgO}, \mathrm{GaP}, \mathrm{CsF}, \mathrm{CdS},\) and \(\mathrm{FeO}\)

Potassium iodide (KI) exhibits predominantly ionic bonding. The \(\mathrm{K}^{+}\) and \(\mathrm{I}^{-}\) ions have electron structures that are identical to which two inert gases?

Without consulting Figure 2.6 or Table \(2.2,\) determine whether each of the electron configurations given below is an inert gas, a halogen, an alkali metal, an alkaline earth metal, or a transition metal. Justify your choices. (a) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{5}\) (b) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{7} 4 s^{2}\) (c) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6}\) (d) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{1}\) (e) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6} 4 d^{5} 5 s^{2}\) (f) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2}\)

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).
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