What values are assigned to quantum numbers \(n, l, m\) for : (a) \(3 \mathrm{~s}\), (b) \(4 p_{z}\) (c) \(4 d_{x^{2}-y^{2}}\) (d) \(5 d_{z^{2}}\)

The principal quantum number, denoted by the symbol ‘n’, is an integral number that determines the energy level and size of the electron cloud in an atom. It plays a dominant role in the electronic configuration of an atom. The principal quantum number starts from 1, increasing outward as you move away from the nucleus, adding more energy levels. Each value of ‘n’ corresponds to a particular 'shell'.

As the value of ‘n’ increases, the electrons are located further from the nucleus, meaning they are less tightly bound and require less energy to remove from the atom. In solving textbook exercises, identifying the principal quantum number is straightforward: it’s the integer that precedes the orbital letter (s, p, d, f), representing the row number in the periodic table the element is in. For example, in a '3s' orbital, the principal quantum number is 3, signifying the third energy level.

The azimuthal quantum number, symbolized by ‘l’, also known as the angular momentum quantum number, dictates the shape of the atomic orbital. It is subordinate to the principal quantum number, meaning its value is dependent on 'n'. The value of ‘l’ can range from 0 to n-1. Crucially, each value of ‘l’ is associated with a particular type of orbital: 's' for 0, 'p' for 1, 'd' for 2, and 'f' for 3.

For instance, in a ‘4p’ orbital, the principal quantum number is 4, and since it is a ‘p’ orbital, the azimuthal quantum number is 1 (one less than n). The shape of the orbital becomes more complex as ‘l’ increases. These shapes are not only important for visualizing electron placement but also for understanding molecular bonding and reactions.

Diving deeper into electron location, the magnetic quantum number, labeled ‘m’ or 'ml', indicates the orientation of an atomic orbital in space relative to the other orbitals within the same subshell. It is defined for a given value of the azimuthal quantum number ‘l’ and can take on integer values in the range from -l to +l, including zero. This quantum number helps in splitting subshells into individual orbitals.

For example, in a ‘p’ subshell where l=1, ‘m’ can be -1, 0, or 1, corresponding to the px, py, and pz orbitals, respectively. Each of these orbitals is oriented differently in three-dimensional space. For exercises involving the identification of ‘m’, you associate the orbital type with its spatial orientation: px (m = -1), py (m = 0), pz (m = 1). However, orientation can be complex for d and f orbitals and might require visualization tools or models for deeper comprehension.

Electronic configuration describes the distribution of electrons in an atom's orbitals, by providing a clear representation of how the orbitals are filled according to the Pauli Exclusion Principle and Hund's Rule. It is a systematic way to break down an atom’s electronic structure into its constituent energy levels and shapes.

The configuration is often communicated using the quantum numbers and the orbitals' labels: starting from the lowest energy level, electrons fill available states in a sequence determined by the increasing energy of n and l. For students, mastering electronic configuration is essential as it lays the groundwork for understanding chemical bonding, the arrangement of the periodic table, and properties like atomic and ionic radius, electronegativity, and ionization energy.

Atomic orbitals are the regions in space around the nucleus where the electrons are likely to be found most of the time. They are mathematically derived from quantum mechanics, represented by wave functions that correspond to specific energy levels and shapes – designated by the quantum numbers. The orbitals are designated by their shapes—spherical (s), dumbbell (p), cloverleaf (d), and complex (f).

Understanding orbitals help in mapping out the electron probability zones, providing key insights into electron behavior within these zones. Each orbital can hold a maximum of two electrons with opposite spins. In solving chemical problems, students should be aware of the filling order of these orbitals as they become vital in predicting atomic behavior during chemical reactions and bonds.