Question

Give the maximum number of electrons in an atom that can have these quantum numbers: a. \(n=0, \ell=0, m_{\ell}=0\) b. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-\frac{1}{2}\) c. \(n=3, m_{s}=+\frac{1}{2}\) d. \(n=2, \ell=2\) e. \(n=1, \ell=0, m_{\ell}=0\)

Short answer

(a) 0 electrons (b) 1 electron (c) 9 electrons (d) 0 electrons (e) 2 electrons

Step-by-step solution

(a) n=0, l=0, ml=0

Quantum numbers have specific roles and ranges. The principal quantum number n starts from the positive integers (1, 2, 3, ...). Since n=0 is not a valid principal quantum number, there can be no electrons with this configuration.

(b) n=2, l=1, ml=-1, ms=-1/2

The allowed values of l range from 0 to n-1. In this case, l=1 is allowed for n=2. The ml values range from -l to +l, so ml=-1 is allowed for l=1. ms takes values of ±1/2, so ms=-1/2 is allowed as well. This configuration is valid, and since the electron has a defined spin, there can be only one electron with this configuration.

(c) n=3, ms=+1/2

When n=3, the allowed l values are 0, 1, and 2. For each l, the allowed ml values are: - l=0: ml=0 - l=1: ml=-1, 0, +1 - l=2: ml=-2, -1, 0, +1, +2 Considering the ms value of +1/2, we have two possible electrons for each ml value (except for the one with ms=-1/2). Thus, there can be a maximum of 9 electrons with these quantum numbers.

(d) n=2, l=2

The allowed values of l range from 0 to n-1. For n=2, l values can be 0 or 1 only. Since l=2 is not a valid value for n=2, there can be no electrons with this configuration.

(e) n=1, l=0, ml=0

The allowed values for l range from 0 to n-1. In this case, l=0 is allowed for n=1. For l=0, the only allowed ml value is 0. There can be two electrons for this valid quantum number configuration based on the possible ms values of ±1/2.

Key concepts

Electron Configuration

Electron configuration describes the distribution of electrons in an atom’s orbitals. Understanding this helps figure out how electrons are arranged in different shells and subshells, defined by four types of quantum numbers.

• The principal quantum number \( n \) determines the shell.
• The azimuthal quantum number \( \ell \), sometimes called the angular momentum quantum number, defines the subshell shape.
• The magnetic quantum number \( m_\ell \) decides the orientation of an electron's orbital within a subshell.
• The spin quantum number \( m_s \) explains the electron's spin direction.

Breaking down these aspects through various quantum numbers helps predict where electrons can be, giving insight into the chemical properties and reactivity of elements. Electrons fill orbitals according to Pauli's exclusion principle and Hund’s rule, where each orbital holds up to two electrons with opposite spins before filling others in the same subshell.

Principal Quantum Number

The principal quantum number, \( n \), is essential in describing an electron's position in an atom. It tells us the main energy level or shell occupied by an electron.

• \( n \) can take positive integer values such as 1, 2, 3, and so forth.
• This number signifies the size of the electron orbitals: the larger the \( n \), the bigger the orbital and the higher the energy level.

This number directly impacts the number of electrons a shell can hold, determined by \( 2n^2 \). For example, with \( n = 3 \), the shell can accommodate up to 18 electrons (\( 2 imes 3^2 = 18 \)). Knowing \( n \) helps identify the possible orbitals and impacts electron configurations throughout the atom.

Electron Spin

Electron spin is a fundamental property akin to the electron’s own angular momentum. The spin quantum number, \( m_s \), explains how an electron can behave like a tiny magnet.

• \( m_s \) can either be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
• This indicates the two possible spin states: up (+1/2) or down (-1/2).

Understanding electron spin is vital because it influences how electrons are arranged within an atom. The Pauli exclusion principle highlights that no two electrons in an atom can have identical quantum numbers, meaning each electron in an orbital must have opposite spins. This concept ensures the diversity in the arrangement of electrons in atoms, affecting magnetic properties and chemical bonding.

Magnetic Quantum Number

The magnetic quantum number, \( m_\ell \), identifies the orientation of an orbital within its subshell. This quantum number provides further definition after the azimuthal quantum number \( \ell \).

• \( m_\ell \) ranges from \(-\ell\) to \(+\ell\).
• For a given \( \ell \), this creates \(2\ell + 1\) possible orientations of the orbital.

For example, if \( \ell = 1 \) (\( p \)-orbital), then \( m_\ell \) can be -1, 0, or +1, depicting three orientations corresponding to the three \( p \)-orbitals in a subshell. The concept of magnetic quantum number aids in visualizing the 3D spaces where electrons are likely to be found, helping predict bonding and geometry in molecules.