Question

How many orbitals in an atom can have each of the following designations: (a) \(1 s ;(b) 4 d\) (c) \(3 p ;\) (d) \(n=3 ?\)

Short answer

a) 1, b) 5, c) 3, d) 9

Step-by-step solution

Identify the number of orbitals in the 1s subshell

The 's' subshell can hold only one orbital. Therefore, the 1s designation only has one orbital.

Determine the number of orbitals in the 4d subshell

The 'd' subshell can hold a total of five orbitals. Therefore, the 4d designation will have five orbitals.

Identify the number of orbitals in the 3p subshell

The 'p' subshell can hold a total of three orbitals. Hence, the 3p designation has three orbitals.

Determine the number of orbitals for a given n-level, n=3

For any given principal quantum number n, the total number of orbitals can be calculated using the formula: 'number of orbitals = n^2’. Therefore, for n=3, the total number of orbitals is 3^2 = 9 orbitals.

Key concepts

Electron Configuration

Electron configuration is the distribution of electrons in an atom's orbitals. It follows a specific order based on increasing energy levels. The Aufbau principle tells us that electrons occupy the lowest energy orbital available. We also use the Pauli Exclusion Principle and Hund's Rule to fill up these orbitals. The notation is written as a series of numbers and letters, representing the energy level, subshell type, and the number of electrons in that subshell. For example, the configuration for oxygen is written as 1s² 2s² 2p⁴. This tells us there are 2 electrons in the 1s subshell, 2 in the 2s, and 4 in the 2p.

Understanding electron configurations helps us predict the chemical behavior of an element. These configurations influence how atoms interact, form bonds, and the overall reactivity in chemical reactions.

Quantum Numbers

Quantum numbers are essential to describe the position and energy of an electron in an atom. There are four types:

• Principal Quantum Number (n): Indicates the main energy level and distance of the electron from the nucleus. Its values are positive integers (1, 2, 3, ...).
• Azimuthal Quantum Number (l): Defines the shape of the orbital, can range from 0 to (n-1). Each value corresponds with a different subshell type (s, p, d, f).
• Magnetic Quantum Number (m_l): Determines the orientation of the orbital around the nucleus and ranges from -l to +l.
• Spin Quantum Number (m_s): Indicates the direction of an electron's spin, either +1/2 or -1/2.

Together, these numbers give a unique address to each electron, like coordinates in a spatial grid.

Subshells

Subshells are subdivisions of electron shells (principal energy levels). Each subshell is denoted by a combination of the principal quantum number (n) and the azimuthal quantum number (l):

• s subshell: l = 0, spherical shape, holds 1 orbital.
• p subshell: l = 1, dumbbell shape, holds 3 orbitals.
• d subshell: l = 2, clover shape, holds 5 orbitals.
• f subshell: l = 3, complex shape, holds 7 orbitals.

The number of orbitals in each subshell determines the maximum number of electrons it can hold, calculated as 2 times the number of orbitals. So, an s subshell holds 2 electrons, p holds 6, d holds 10, and f holds 14. Understanding which subshells are filled helps explain how elements behave and the structure of the periodic table.

Principal Quantum Number

The principal quantum number, represented by n, is key to understanding an atom's electron structure. As an integer value (1, 2, 3, ...), it indicates:

• The main energy level of an electron.
• The size of the electron cloud, with higher n values meaning the electron is further from the nucleus.
• The possible number of orbitals in that level, calculated as n².
• The energy difference between levels, with higher values having higher energy.

For example, if n = 3, there are 9 orbitals available (3²). These levels form the basis for the arrangement of electrons around the nucleus.