Question

State which of the following are incorrect designations for orbitals according to the quantum theory: \(3 p, 4 s, 2 f,\) and \(1 p .\) Briefly explain your answers.

Short answer

'2 f' and '1 p' are incorrect designations; '3 p' and '4 s' are correct.

Step-by-step solution

Understand Quantum Numbers

Orbitals are labeled using a combination of numbers and letters that represent quantum numbers. The principal quantum number, denoted by \(n\), signifies the energy level and is represented by a positive integer (1, 2, 3,...). The azimuthal quantum number, \(l\), indicates the subshell and is represented by letters (s, p, d, f) corresponding to 0, 1, 2, 3, respectively. Ensure that the designated orbital satisfies these quantum number rules.

Examine Orbital '3 p'

For the orbital '3 p', \(n = 3\) and \(l = 1\). Since a 'p' orbital corresponds to \(l = 1\) and can exist for \(n \geq 2\), '3 p' is a valid designation.

Examine Orbital '4 s'

For the orbital '4 s', \(n = 4\) and \(l = 0\). Since an 's' orbital corresponds to \(l = 0\) and is allowed for any energy level starting from \(n = 1\), '4 s' is a valid designation.

Examine Orbital '2 f'

For the orbital '2 f', \(n = 2\) and \(l = 3\). Since an 'f' orbital, \(l = 3\), requires \(n \geq 4\) (because \(l\) can have a maximum value of \(n-1\)), '2 f' cannot exist. Therefore, '2 f' is an incorrect designation.

Examine Orbital '1 p'

For the orbital '1 p', \(n = 1\) and \(l = 1\). However, \(l\) can range only from 0 to \(n-1\), which means for \(n = 1\), \(l\) can only be 0, corresponding to an 's' orbital. Thus, '1 p' is an incorrect designation.

Key concepts

Orbitals

In the world of quantum mechanics, orbitals are fascinating regions around an atom's nucleus where electrons are most likely to be found. They are solutions to the Schrödinger equation for electrons in an atom. Orbitals are defined by a set of quantum numbers and represented visually as three-dimensional spaces. This concept helps us understand not just where electrons might be, but also how they organize and fill the space around a nucleus.

Different types of orbitals are denoted by letters such as 's,' 'p,' 'd,' and 'f.' Each of these letters corresponds to a different shape and energy of the orbital:

• 's' orbitals are spherical and the simplest kind.
• 'p' orbitals have a dumbbell shape and appear in sets of three.
• 'd' orbitals are more complex, with cloverleaf shapes, and come in sets of five.
• 'f' orbitals have even more complex shapes and appear in sets of seven.

It is important to remember that these shapes help chemists and physicists predict the chemical bonding and properties of atoms. Each type of orbital can hold a maximum of two electrons, which must have opposite spins. Understanding orbitals is key to mastering the intricacies of how atoms interact.

Principal Quantum Number

The principal quantum number (\(n\)) is a crucial part of the quantum number set that defines an electron's properties in an atom. It helps determine the overall size and energy level of an orbital.

This number is always a positive integer: 1, 2, 3, and so on. These integers correspond to the different energy levels of an atom, with higher numbers indicating orbitals that are farther from the nucleus and higher in energy.

• For example, when you say "3p," the '3' is the principal quantum number.
• It tells you that the electrons are in the third energy level.
• As you progress up to higher values (\(n = 4, 5, 6,...\), the orbitals become larger and more energetic.

The principal quantum number also dictates the maximum number of subshells within an energy level. For instance, a principal quantum number of 3 can have s, p, and d subshells. It's like the core factor that sets the stage for determining an atom's electron configuration.

Azimuthal Quantum Number

The azimuthal quantum number, often referred to as the angular momentum quantum number (\(l\)), further defines the shape of the orbital and its subshell type. It gives each orbital its specific geometric characteristics and is integral to how orbitals are sub-divided within an energy level identified by the principal quantum number.

The azimuthal quantum number can be any integer from 0 to \(n-1\), where \(n\) is the principal quantum number. This means each energy level has a set variety of subshells. Here is how it works:

• When \(l = 0\), the subshell is an 's' orbital.
• When \(l = 1\), the subshell is a 'p' orbital.
• When \(l = 2\), the subshell is a 'd' orbital.
• When \(l = 3\), the subshell is an 'f' orbital.

It's important to note that the maximum value of \(l\) must be one less than the principal quantum number. This is why an energy level of 4 (n = 4) can have up to an 'f' subshell, but something like a '2f' orbital is not possible because \(l = 3\) would require \(n\) to be at least 4. Understanding this helps us determine the number of different types of orbitals available at a given energy level.