Question

State the maximum number of electrons that can occupy each of the following orbitals: (a) \(1 s\) (b) \(2 p\) (c) \(3 d\) (d) \(4 f\)

Short answer

(a) 2, (b) 6, (c) 10, (d) 14 electrons.

Step-by-step solution

Understanding Orbital Designations

Orbitals in an atom are designated by their principal quantum number and the type of orbital (s, p, d, f). The principal quantum number defines the energy level, while the letter indicates the shape and the number of subshells.

Electron Capacity of Each Orbital Type

Each type of orbital can hold a different maximum number of electrons: - An 's' orbital, with one possible orientation, can hold 2 electrons. - A 'p' orbital, with three orientations (px, py, pz), can hold 6 electrons total. - A 'd' orbital, with five orientations, can hold 10 electrons. - An 'f' orbital, with seven orientations, can hold 14 electrons.

Application to Each Orbital

Based on the maximum electron capacity for each type of orbital: - (a) 1s orbital can hold a maximum of 2 electrons. - (b) 2p orbital can hold a maximum of 6 electrons. - (c) 3d orbital can hold a maximum of 10 electrons. - (d) 4f orbital can hold a maximum of 14 electrons.

Key concepts

Atomic Orbitals

Atoms have specific regions where electrons may be found, known as atomic orbitals. These orbitals are not spherical but have unique shapes that reflect the probability distribution of electrons. Each orbital is characterized by a principal quantum number and a letter that designates its type.

• 's' orbitals are spherical and are the simplest, filling first as they are closest to the nucleus.
• 'p' orbitals are dumbbell-shaped and come in sets of three.
• 'd' orbitals have more complex, cloverleaf shapes and include five different orientations.
• 'f' orbitals are even more intricate with seven possible orientations.

The shape and types of orbitals affect how electrons are arranged around an atom. Understanding these properties is key to explaining the chemical behavior of elements.

Quantum Numbers

Quantum numbers are essential to determining the position and energy of an electron within an atom. They are like an address for electrons.

Principal Quantum Number (n)

This number indicates the electron's energy level or shell and is always a positive integer. Larger values of 'n' mean electrons are further from the nucleus and have more energy.

Angular Momentum Quantum Number (l)

This quantum number determines the shape of the orbital. The values range from 0 to (n-1). For example, an 's' orbital has an l value of 0, 'p' has 1, 'd' has 2, and 'f' has 3.

Magnetic Quantum Number (m_l)

This specifies the orientation of the orbital in a magnetic field and its possible values are integers between -l and +l. Therefore, 'p' orbitals which have l=1 can have m_l values of -1, 0, and +1.

Spin Quantum Number (m_s)

Each electron has a spin of either +1/2 or -1/2, reflecting its intrinsic angular momentum.
Together these numbers provide a unique description of an electron's state, critical for visualizing atomic structure.

Electron Capacity

Electron capacity defines how many electrons an orbital can hold. This is influenced by the number of orientations each type of orbital has, directed by quantum mechanics.

• 's' orbitals can hold up to 2 electrons, corresponding to one pair of opposite spins.
• 'p' orbitals contain three different orientations, allowing a total capacity of 6 electrons.
• 'd' orbitals, with five orientations, can accommodate 10 electrons.
• 'f' orbitals, having seven orientations, support 14 electrons.

The electron capacity not only highlights how electrons fill the atomic orbitals but also affects chemical bonding and element properties. By knowing the capacity limit, chemists can predict and explain how atoms interact in complex molecules.