Question

How many orbitals are there in \(\mathrm{K}, \mathrm{L}, \mathrm{M}\) and \(\mathrm{N}\) shells? What are they?

Short answer

Answer: The K, L, M, and N shells have the following orbitals: K Shell (n = 1): 1 orbital (1s) L Shell (n = 2): 4 orbitals (2s, 2p_x, 2p_y, 2p_z) M Shell (n = 3): 9 orbitals (3s, 3p_x, 3p_y, 3p_z, 3d_{xy}, 3d_{yz}, 3d_{xz}, 3d_{x^2-y^2}, 3d_{z^2}) N Shell (n = 4): 16 orbitals (4s, 4p_x, 4p_y, 4p_z, 4d_{xy}, 4d_{yz}, 4d_{xz}, 4d_{x^2-y^2}, 4d_{z^2}, 4f_{xxx}, 4f_{xxy}, 4f_{xyy}, 4f_{yyy}, 4f_{yyz}, 4f_{xyz}, 4f_{xxz})

Step-by-step solution

Identify the Shell Numbers (n)

The given shells are K, L, M, and N. These shells correspond to the shell numbers (n) as follows: K = 1, L = 2, M = 3, N = 4.

Determine the Subshells and Orbitals

For each shell (n), there can be n subshells, which are labeled as 0, 1, 2, ..., n-1. The subshells correspond to the orbitals as 0 = s, 1 = p, 2 = d, and 3 = f. K Shell: Since n = 1, this shell has only one subshell (0), which is an s subshell. There is only one s orbital in this subshell. L Shell: Since n = 2, this shell has two subshells (0 and 1). These are s and p subshells. There is one s orbital in the s subshell and three p orbitals in the p subshell, totaling four orbitals. M Shell: Since n = 3, this shell has three subshells (0, 1, and 2). These are s, p, and d subshells. There is one s orbital in the s subshell, three p orbitals in the p subshell, and five d orbitals in the d subshell, totaling nine orbitals. N Shell: Since n = 4, this shell has four subshells (0, 1, 2, and 3). These are s, p, d, and f subshells. There is one s orbital in the s subshell, three p orbitals in the p subshell, five d orbitals in the d subshell, and seven f orbitals in the f subshell, totaling 16 orbitals.

Summarize the Results

The number of orbitals in each shell and their types are as follows: K Shell (n = 1): 1 orbital (1s) L Shell (n = 2): 4 orbitals (2s, 2p_x, 2p_y, 2p_z) M Shell (n = 3): 9 orbitals (3s, 3p_x, 3p_y, 3p_z, 3d_{xy}, 3d_{yz}, 3d_{xz}, 3d_{x^2-y^2}, 3d_{z^2}) N Shell (n = 4): 16 orbitals (4s, 4p_x, 4p_y, 4p_z, 4d_{xy}, 4d_{yz}, 4d_{xz}, 4d_{x^2-y^2}, 4d_{z^2}, 4f_{xxx}, 4f_{xxy}, 4f_{xyy}, 4f_{yyy}, 4f_{yyz}, 4f_{xyz}, 4f_{xxz})

Key concepts

Electron Shells

In the study of the atomic structure, it is essential to understand electron shells, which are layers of electrons surrounding an atom's nucleus. Each shell is designated by a principal quantum number, denoted as 'n'. The higher the value of 'n', the farther the shell is from the nucleus, and the higher the energy level.

Electron shells are like concentric circles around a nucleus, with the innermost shell being closest. They provide an organized structure for electrons to exist around an atom. The first shell, 'K', can hold up to 2 electrons; the second, 'L', up to 8; the 'M' shell can hold up to 18; and the 'N' shell can hold up to 32 electrons, following the 2n^2 rule. These capacities are due to the types and number of subshells that each shell contains, which in turn define the maximum number of electrons they can accommodate.

The importance of these shells lies in their influence on the chemical behavior of an atom. Electrons in shells further away from the nucleus can engage in reactions and bonding with other atoms, whereas those in closer shells are more tightly bound to the atom. Therefore, understanding electron shells is critical for grasping the fundamentals of atomic interactions and chemical reactions.

Subshells in Quantum Chemistry

Diving deeper into atomic structure, quantum chemistry introduces us to subshells. Subshells are subdivisions of electron shells, each characterized by different levels of energy and angular momentum, represented by the azimuthal quantum number 'l'. The values of 'l' range from 0 to (n-1), where 'n' is the principal quantum number.

Each subshell is identified by specific letters: 0 (s), 1 (p), 2 (d), and 3 (f). A complete understanding of subshells is necessary because they provide insight into the complex arrangement of electrons within an atom. Every subshell type has a characteristic shape and orientation in space, which has profound effects on an atom's bonding and properties.

• The 's' subshell is spherical and can hold a maximum of 2 electrons.
• The 'p' subshell is dumbbell-shaped and can accommodate up to 6 electrons across three orbitals.
• The 'd' subshell is more complex, having five orbitals and a capacity for 10 electrons.
• The 'f' subshell is even more complex, consisting of seven orbitals that can hold a total of 14 electrons.

Subshells dictate how electrons are stacked within a shell and play a pivotal role in the chemical and physical properties of elements, as different subshells have different energies and shapes, influencing how atoms interact with each other.

Orbital Types s p d f

To understand the orbitals within an atom, it is crucial to explore the various types. Orbitals are regions in space where there is a high probability of finding an electron. The four main types of orbitals — s, p, d, and f — correspond to the subshells and are distinguished by their shapes, which dictates how electrons are distributed around an atom's nucleus.

Each type of orbital has a characteristic shape:

• 's' orbitals: Spherical, with the 1s orbital being the most basic form.
• 'p' orbitals: A set of three orbitals oriented at right angles to each other, resembling dumbbells: px, py, and pz.
• 'd' orbitals: More complex with five orientations. These include the dxy, dyz, dxz, dx2-y2, and dz2 orbitals.
• 'f' orbitals: Even more complex with seven different orientations, which can be difficult to visualize but are critical in understanding heavy elements.

Each orbital can hold a maximum of two electrons, one with a spin 'up' and one with 'down'. The arrangement of electrons in these orbitals follows a set of rules — the Aufbau principle, the Pauli exclusion principle, and Hund's rule. These rules describe how electrons fill available orbitals based on energy levels and electron pairing. Understanding the types of orbitals and their arrangement is fundamental for predicting the behavior of atoms in chemical bonding and reactions.