Question

The electron configuration of a neutral atom is \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2}\). Write a complete set of quantum numbers for each of the electrons. Name the element.

Short answer

The element is Magnesium; quantum numbers describe electron positions.

Step-by-step solution

Understanding the Electron Configuration

The given electron configuration is \(1s^2\ 2s^2\ 2p^6\ 3s^2\). This notation tells us the distribution of electrons in the atomic orbitals, where numbers denote the principal energy level, letters (s, p, d) denote the type of orbital, and superscripts show the number of electrons in those orbitals.

Identifying Quantum Numbers

For each electron, we will determine a set of four quantum numbers: the principal quantum number \(n\), azimuthal quantum number \(l\), magnetic quantum number \(m_l\), and spin quantum number \(m_s\).

Determining Quantum Numbers for Each Electron

- For \(1s^2\): - Electrons 1 and 2: \(n = 1\), \(l = 0\), \(m_l = 0\), \(m_s = +\frac{1}{2}\) or \(-\frac{1}{2}\).- For \(2s^2\): - Electrons 3 and 4: \(n = 2\), \(l = 0\), \(m_l = 0\), \(m_s = +\frac{1}{2}\) or \(-\frac{1}{2}\).- For \(2p^6\): - Electrons 5 to 10: \(n = 2\), \(l = 1\), \(m_l = -1, 0, +1\), \(m_s = +\frac{1}{2}\) or \(-\frac{1}{2}\).- For \(3s^2\): - Electrons 11 and 12: \(n = 3\), \(l = 0\), \(m_l = 0\), \(m_s = +\frac{1}{2}\) or \(-\frac{1}{2}\).

Listing Quantum Numbers for All Electrons

Here is a complete set for each electron:1. \((1, 0, 0, +\frac{1}{2})\)2. \((1, 0, 0, -\frac{1}{2})\)3. \((2, 0, 0, +\frac{1}{2})\)4. \((2, 0, 0, -\frac{1}{2})\)5. \((2, 1, -1, +\frac{1}{2})\)6. \((2, 1, -1, -\frac{1}{2})\)7. \((2, 1, 0, +\frac{1}{2})\)8. \((2, 1, 0, -\frac{1}{2})\)9. \((2, 1, +1, +\frac{1}{2})\)10. \((2, 1, +1, -\frac{1}{2})\)11. \((3, 0, 0, +\frac{1}{2})\)12. \((3, 0, 0, -\frac{1}{2})\)

Identifying the Element

By counting the number of electrons in the configuration, which is 12, we can determine that the element is Magnesium (Mg).

Key concepts

Electron Configuration

Electron configuration is a method used to denote the arrangement of electrons in an atom. It provides a clear picture of how electrons are distributed across different atomic orbitals. Each electron in an atom occupies a position that can be described by a set of four quantum numbers. The electron configuration of an atom is written in the format of numbers and letters such as \(1s^2\ 2s^2\ 2p^6\ 3s^2\).
- **Numbers** (like 1, 2, 3) indicate the principal energy levels or shells.- **Letters** (s, p, d, f) denote the type of atomic orbital.- **Superscripts** show the number of electrons present in each sublevel.
Writing electron configurations helps to predict an element's chemical behavior and to understand its position in the periodic table. With a total of 12 electrons distributed among these orbitals, we can conclude that the electron configuration represents magnesium (Mg).

Magnesium

Magnesium is an alkaline earth metal found in group 2 of the periodic table. It is represented by the symbol Mg and has an atomic number of 12. This indicates that a neutral magnesium atom has 12 protons and, typically, 12 electrons.
- Magnesium is silvery-white and shines brightly when first cut.- It reacts with oxygen in the air, forming a layer of magnesium oxide that protects the metal beneath.
Magnesium plays an essential role in biology; it is crucial in enzymes that are fundamental to many biochemical reactions. In electron configurations, magnesium's arrangement of electrons is given by \(1s^2\ 2s^2\ 2p^6\ 3s^2\), confirming that it has a complete outer shell of electrons in its third energy level, making it relatively stable.

Principal Quantum Number

The principal quantum number, symbolized as \(n\), is an essential component of the quantum mechanical model of the atom. It indicates the main energy level or shell occupied by an electron. The value of \(n\) is always a positive integer (1, 2, 3,...). Larger values of \(n\) correspond to electrons that are farther from the nucleus and thus are higher in energy.
- For magnesium, the principal quantum numbers are 1, 2, and 3 corresponding to its electron configuration.
The principal quantum number, \(n\), also dictates the maximum number of electrons in a shell, given by the equation \(2n^2\). Understanding \(n\) is vital in predicting an atom's size and energy levels the electrons may occupy.

Spin Quantum Number

The spin quantum number, symbolized as \(m_s\), describes the intrinsic angular momentum of an electron, often referred to simply as "spin." This concept is crucial when assigning electronic configurations and completing quantum number sets. The value of \(m_s\) can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\), indicating the two possible spin states of an electron.
- Spin determines the magnetic properties of electrons and is vital in the study of atomic structure.- Electrons within the same orbital must have opposite spins, adhering to the Pauli Exclusion Principle.
When considering electron configurations of magnesium, the spin quantum number ensures that two electrons in each filled orbital are balanced and occupy the lowest possible energy state. This is an essential concept in quantum mechanics and helps explain the stability and behavior of atoms.