Worked example 1
How many orbitals and how many electrons can the n = 3 shell hold, and which subshells does it contain?
Try it first: List the allowed values of l first.
0 of 4 steps revealed.
BondingElectron orbitalsContent level: Core 18 min
Orbitals, Subshells and Quantum Numbers
What you'll be able to do: Describe an orbital with the four quantum numbers and count how many orbitals and electrons each subshell holds.
Introduction
Before atoms can bond, you need a picture of where their electrons actually live. Quantum numbers are the address system for that.
These are recommended, not required. You can start this lesson at any time.
Learning objectives
- State what each of the four quantum numbers describes
- Give the allowed values of l and ml for a shell
- Count orbitals and maximum electrons in s, p, d and f subshells
- Identify sets of quantum numbers that are not allowed
- Relate orbital shape and nodes to l and n
Lesson
The four quantum numbers
The principal quantum number n (1, 2, 3, ...) sets the shell: larger n means higher energy and a larger average distance from the nucleus. The angular momentum quantum number l runs from 0 to n - 1 and names the subshell: l = 0 is s, l = 1 is p, l = 2 is d, l = 3 is f. The magnetic quantum number ml runs from -l to +l and picks one orbital inside the subshell. The spin quantum number ms is +1/2 or -1/2.
Because l stops at n - 1, there is no 1p, no 2d and no 3f orbital.
Counting orbitals and electrons
A subshell with a given l holds 2l + 1 orbitals: s has 1, p has 3, d has 5, f has 7. Every orbital holds at most two electrons, so s holds 2, p holds 6, d holds 10 and f holds 14. A whole shell n holds n squared orbitals and 2n squared electrons, so shell 3 has 9 orbitals and 18 electrons.
Orbitals inside the same subshell are degenerate in an isolated atom, meaning they have equal energy.
Shapes and nodes
An s orbital is spherical, a p orbital is a two-lobed dumbbell along one axis, and most d orbitals are four-lobed cloverleaves. A node is a surface where the probability of finding the electron is zero. An orbital has l angular nodes (flat planes or cones) and n - l - 1 radial nodes, for a total of n - 1 nodes.
Key ideas
Definition
Orbital
A region described by one set of n, l and ml values that can hold at most two electrons of opposite spin.
Rule
Allowed values
l goes from 0 to n - 1, and ml goes from -l to +l in whole steps.
Rule
Subshell capacity
A subshell has 2l + 1 orbitals and holds 2(2l + 1) electrons.
Key concept
The 4d subshell
n = 4 and l = 2, giving 5 orbitals and a maximum of 10 electrons.
Equation
Orbitals in a subshell
orbitals = 2l + 1
Equation
Electrons in a shell
electrons = 2n²
Equation
Total nodes
nodes = n - 1
Worked examples
Worked example 2
How many radial and angular nodes does a 3p orbital have?
Try it first: Identify n and l for a 3p orbital.
0 of 3 steps revealed.
Common mistakes
Writing configurations that use a 2d or 1p subshell.
Why it's wrong: l cannot reach the value needed for d until n = 3, or for p until n = 2.
Check instead: Confirm that l is no larger than n - 1 before using a subshell label.
Thinking a p subshell holds three electrons.
Why it's wrong: A p subshell has three orbitals, and each orbital holds two electrons.
Check instead: Multiply the orbital count by two to get the electron capacity.
Treating ms as if it can be any value.
Why it's wrong: Spin is quantized and only +1/2 or -1/2 are allowed.
Check instead: Reject any quantum number set with a spin value other than plus or minus one half.
Practice this skill
No practice questions are available for this topic yet. You can still practice the whole unit.
What you should now know
Each electron in an atom is described by four quantum numbers: n sets the shell and size, l sets the subshell and shape, ml sets the orientation, and ms sets the spin. A subshell with quantum number l contains 2l + 1 orbitals, each orbital holds two electrons, and shell n holds n squared orbitals in total.
- n sets the shell, l the subshell shape, ml the orientation and ms the spin
- l is limited to values from 0 up to n - 1
- A subshell has 2l + 1 orbitals and twice that many electrons
- A shell holds n squared orbitals and 2n squared electrons
- An orbital has l angular nodes and n - l - 1 radial nodes
Sources and further reading
This lesson is original Chem Help content. No external sources were adapted.