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.
Check whether quantum numbers are allowed
For a given principal quantum number n, the angular quantum number l runs from 0 through n - 1. For a chosen l, the magnetic quantum number ml runs through every integer from -l to +l; the spin quantum number is +1/2 or -1/2. These constraints let you reject an impossible orbital label before considering energy or electron filling.
Check n, then l, then ml, then spin in that order.
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.
Worked example 3
Is the set n = 3, l = 2, ml = -2, ms = +1/2 allowed? What orbital does it describe?
Try it first: The allowed l values for n = 3 are 0, 1, and 2.
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.
After the lesson · Question bank
Independent practice
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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.