Question

How many orbitals in an atom can have each of the following designations: (a) \(5 f\) (b) \(4 p_{i}\) (c) \(5 d ;\) (d) \(n=2 ?\)

Short answer

(a) 7 orbitals, (b) 3 orbitals, (c) 5 orbitals, (d) 4 orbitals

Step-by-step solution

Understanding the Quantum Numbers

Quantum numbers are used to describe the properties of atomic orbitals and the properties of electrons in orbitals. The key quantum numbers include the principal quantum number (), the azimuthal quantum number (), and the magnetic quantum number (). In this exercise, we need to identify the number of orbitals for different sublevels.

Number of Orbitals for 5f

The designation 5f indicates the principal quantum number is 5 and the azimuthal quantum number for the f subshell is 3 ( = 3). For an f subshell, the possible values of the magnetic quantum number () range from -3 to 3. Therefore, there are a total of 2l + 1 orbitals: (2 * 3 + 1 = 7) orbitals.

Number of Orbitals for 4p

The designation 4p indicates the principal quantum number is 4 and the azimuthal quantum number for the p subshell is 1 ( = 1). For a p subshell, the possible values of the magnetic quantum number () range from -1 to 1. Therefore, there are a total of 2l + 1 orbitals: (2 * 1 + 1 = 3) orbitals.

Number of Orbitals for 5d

The designation 5d indicates the principal quantum number is 5 and the azimuthal quantum number for the d subshell is 2 ( = 2). For a d subshell, the possible values of the magnetic quantum number () range from -2 to 2. Therefore, there are a total of 2l + 1 orbitals: (2 * 2 + 1 = 5) orbitals.

Number of Orbitals for n=2

When the principal quantum number is 2 ( = 2), we have two possible subshells: 2s and 2p. The 2s subshell ( = 0) has 1 orbital and the 2p subshell ( = 1) has 3 orbitals (as calculated in Step 3). Summing these, we get: (1 + 3 = 4) orbitals.

Key concepts

Principal Quantum Number

The principal quantum number, denoted by the symbol \(n\), is a key concept in quantum mechanics. It determines the overall size and energy of an electron's orbital in an atom. The value of \(n\) can be any positive integer (1, 2, 3, ...). As \(n\) increases, the orbital becomes larger and the electron spends more time further away from the nucleus. Higher \(n\) values also mean higher energy levels.

For example, in the problem, when you see '5f', the '5' represents the principal quantum number. This indicates that the electron is in the fifth energy level, which is further from the nucleus and generally has higher energy compared to levels with smaller \(n\) values.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized as \(l\), is another crucial concept. It defines the shape of an orbital and is dependent on the principal quantum number, \(n\). The possible values of \(l\) range from 0 to \(n-1\). Each value of \(l\) corresponds to a different subshell or orbital type:

• \(l = 0\) represents an s subshell
• \(l = 1\) represents a p subshell
• \(l = 2\) represents a d subshell
• \(l = 3\) represents an f subshell

For instance, in '4p', the 'p' denotes a p subshell with \(l = 1\), while in '5d', the 'd' denotes a d subshell with \(l = 2\). This quantum number influences the angular momentum and shape of the electron cloud.

Magnetic Quantum Number

The magnetic quantum number, represented as \(m_l\), determines the orientation of the orbital in space. Its possible values depend on the azimuthal quantum number \(l\). Specifically, \(m_l\) ranges from \(-l\) to \(+l\) in whole numbers.

For example, if \(l = 1\) (which corresponds to a p subshell), the magnetic quantum number \(m_l\) can be -1, 0, or +1. This results in three possible orientations for the p orbitals.

In the exercise, for 5f (where \(l = 3\)), \(m_l\) can range from -3 to +3, giving a total of 7 possible orbitals because there are 7 possible values for \(m_l\) (-3, -2, -1, 0, +1, +2, +3).

Atomic Orbitals

Atomic orbitals are regions around the nucleus where electrons are likely to be found. They are defined by the principal, azimuthal, and magnetic quantum numbers. Each combination of these numbers corresponds to a specific orbital.

For example:

• An orbital with \(n = 2\) and \(l = 0\) is a 2s orbital.
• An orbital with \(n = 4\) and \(l = 1\) is a 4p orbital.

The shape and orientation of these orbitals are influenced by the quantum numbers \(l\) and \(m_l\). Understanding the different types of atomic orbitals (s, p, d, and f) helps in visualizing where electrons are likely to be found in an atom.

Electron Configuration

Electron configuration refers to the way electrons are distributed among the orbitals of an atom. It follows a specific order based on the increasing energy levels and sublevels. The Aufbau principle, Pauli exclusion principle, and Hund's rule guide this distribution:

• The Aufbau principle states that electrons fill lower-energy orbitals first before moving to higher-energy orbitals.
• The Pauli exclusion principle states that no two electrons can have the same set of all four quantum numbers, meaning an orbital can hold at most two electrons with opposite spins.
• Hund's rule states that electrons will fill degenerate orbitals (orbitals with the same energy) singly first before pairing up.

For example, a 2p subshell with 3 orbitals can hold a maximum of 6 electrons, each orbital could have up to 2 electrons (one with spin-up and one with spin-down). Understanding electron configurations is essential for predicting chemical properties and reactivity.