Which of the following electron configurations is correct for copper, (atomic number 29\() ?\) (a) \([\operatorname{Ar}] 3 d^{10} 4 s^{1}\) (b) \([\mathrm{Kr}] 3 d^{9} 4 s^{1}\) (c) [Ar] \(3 d^{9} 4 s^{2}\) (d) \([\mathrm{Kr}] 3 d^{10} 4 s^{1}\)

Quantum numbers are like the unique address of an electron within an atom and help in determining its location and behavior. There are four types of quantum numbers: principal (n), azimuthal (l), magnetic (m_l), and spin (m_s).

The principal quantum number (n) indicates the electron's energy level or shell, and it defines the distance of an electron from the nucleus. For copper with an atomic number of 29, the principal quantum number ranges from 1 to 4, signifying the presence of electrons in the first four shells.

The azimuthal quantum number (l) defines the shape of the orbital and can take integer values from 0 to (n-1). It helps us distinguish subshells: s (l=0), p (l=1), d (l=2), and f (l=3). In the case of copper, the azimuthal quantum numbers are important for identifying the s and d subshells it occupies.

The magnetic quantum number (m_l) indicates the orbital's orientation in space within a subshell and can range from -l to +l. Finally, the spin quantum number (m_s) specifies the direction of the electron's spin, which can be +1/2 or -1/2.

The concept of the noble gas core simplifies writing electron configurations by using the electron configuration of a noble gas as a reference point. Noble gases have complete valence electron shells, which gives them a particularly stable electron configuration.

For elements beyond helium in the periodic table, the electron configuration often starts with the symbol of the noble gas that precedes the element being considered, enclosed in square brackets. This represents the core electrons, akin to the noble gas's electronic structure, and helps focus on the valence electrons which define the chemical properties of the element.

For copper (Cu), the noble gas core is Argon (Ar), because Argon is the last noble gas encountered as we move up the periodic table before reaching copper. Thus, the electron configuration for copper starts with [Ar], followed by the electron configuration for the remaining electrons that are outside the noble gas core.

The periodic table is an organized chart that categorizes elements based on atomic number, electron configuration, and recurring chemical properties. Elements are arranged in a series of rows called periods and columns known as groups. The periodic table structure helps predict the types of chemical reactions that elements might undergo.

Copper, the element in question, sits in period 4 and group 11 of the periodic table. This position indicates that it will have characteristics common to transition metals, including the tendency to form colored compounds and have variable oxidation states.

The placement in the periodic table also provides clues about the electron configuration of an element. Since copper is in period 4, it has electrons in the fourth shell, and being in the d-block suggests it will have electrons in the d subshell. Understanding where copper sits on the periodic table—alongside its atomic number—guides us in constructing its electron configuration.

Subshells are subdivisions of electron shells, designated as s, p, d, or f, with different capacities for electrons. The s subshell can hold a maximum of two electrons and has a spherical shape, while the d subshell can carry up to ten electrons and has a more complex shape.

In the electron configuration for elements, the 4s subshell is typically filled before the 3d subshell due to the order of energy levels. However, there are exceptions to this rule, particularly among transition metals like copper, where a fully filled d subshell or a half-filled d subshell provides additional stability.

For copper, with an atomic number of 29, the more stable configuration is achieved by having a full set of ten electrons in the 3d subshell, which is energetically favorable over filling the 4s subshell. Thus, copper’s electron configuration ends with 3d¹⁰ 4s¹, deviating from the expected electronic configuration where the 4s subshell would be filled before starting to fill the 3d subshell.