How do you determine the number of orbitals?

Energy levels and orbitals help describe the electronic structure of an atom. They designate how electrons are arranged within atoms, and the description of such energies is derived from quantum theory.

Quantum Theory

Quantum theory postulates that atoms can only exist in certain energy states. If an atom, or an electron by correlation, changes state, it absorbs or emits an amount of energy equal to the energy difference between the states.

The energy emitted or absorbed is quantized; it is energy characterized by definite amounts. These allowed energy states can be described by sets of numbers called quantum numbers.

Quantum Numbers

An electron’s arrangement in an atom can be described by four quantum numbers: n, l, m_l_ and ms. These relate to energy level, electron subshells, orbital direction and spin, respectively.

First Quantum Number: Energy Level

The first quantum number is designated by n and is the principal energy level.

The principal energy level definition tells the observer the size of the orbital and determines energy. An increase in n is an increase in energy, and this also means the electron is farther away from the nucleus.

The first quantum number can only take integral values, beginning with 1; n = 1, 2, 3, 4 ... Each energy level corresponds to a letter as well: n = 1 (K), 2 (L), 3 (M), 4 (N) ...

First Quantum Number: Orbital and Electron Calculations

To calculate the amount of orbitals from the principal quantum number, use n2. There are n2 orbitals for each energy level. For n = 1, there is 12 or one orbital. For n = 2, there are 22 or four orbitals. For n = 3 there are nine orbitals, for n = 4 there are 16 orbitals, for n = 5 there are 52 = 25 orbitals, and so on.

To calculate the maximum number of electrons in each energy level, the formula 2n2 can be used, where n is the principal energy level (first quantum number). For example, energy level 1, 2(1)2 calculates to two possible electrons that will fit into the first energy level.

Second Quantum Number: Electron Subshells

The second quantum number denotes sublevels and is designated by the letter l. This quantum number denotes electron subshells and the general shape of the electron cloud.

The first two quantum numbers are related. For any given n, l can take on any integral starting with 0 to a maximum of (n – 1); l = 0, 1, 2, 3 ...

The quantum levels, l = 0, 1, 2, 3 correspond to the electron subshells s, p, d, f, respectively. The shape of s is spherical, p is figure-eight shaped, and the d and f orbitals have a more intricate design, mostly involving clover-shaped orbitals.

Each electron subshell can contain a certain amount of electrons, s = 2, p = 6, d = 10 and f = 14.

Third Quantum Number: Orbital Direction

The third quantum number m_l_, denotes how the electron cloud is directed in space.

This quantum number can have any integral value, including 0, between l and –l (the second quantum number), or, m_l = _l ... 2, 1, 0, -1, -2 ... -l

For l = 0, there is only 1 m_l value, also 0. This contains only one orbital. For a p orbital, ml_ = 1, 0, -1. This corresponds to the three p orbitals in three different directions, px, py, pz, corresponding to the three-dimensional x,y and z axis.

Fourth Quantum Number: Electron Spin

The fourth quantum number designates clockwise or counterclockwise spin.

An electron is a charged particle spinning on an axis and therefore has magnetic properties. This quantum number is not related to n, l, ml, and can have only two possible values: +1/2 or -1/2.

The addition of the fourth quantum number allows electrons to fill into orbitals without breaking the Pauli exclusion principle. This states that no two electrons can have the same set of four quantum numbers.

Using Quantum Numbers to Calculate Orbitals

Recall that finding the amount of orbitals in an energy level can be derived by the formula n2. For the energy level 3, n = (3)2 or nine orbitals.

A more completed calculation can be made using the information from the quantum numbers above. For n = 3, the values of l can be added. For l = 0, there is only one orbital, ml = 0. For l = 1, there are three values (ml = −1, 0 or +1). For l = 2, there are five possible values (ml = −2, −1, 0, +1 or +2). So adding the possibilities gives 1 + 3 + 5 = 9 orbitals in total.

Video Transcript

What is the relationship between the principal quantum number 𝑛 and the total number of orbitals?

Quantum numbers are a set of four numbers that help us describe an electron in an atom. The first quantum number is the principal quantum number, which is the topic of this question. This quantum number tells us the energy level or shell the electron occupies, as small as the size of an atomic orbital. The principal quantum number can be any integer from one to seven.

The remaining three quantum numbers further describe an electron in an atom. The next quantum number is the subsidiary quantum number, which describes the orbital type, that is, whether it’s an s-, p-, d-, or f-type atomic orbital. 𝑚 sub 𝐿 is the magnetic quantum number, which describes the orientation of the atomic orbital in space. And finally we have the spin quantum number. This quantum number determines the spin state of an electron in an atomic orbital.

In this question, we need to determine the relationship between the principal quantum number and the total number of orbitals. As the principal quantum number increases, the number of orbitals increases. In the first shell, there is a single s orbital. In the second shell, there is an s orbital as well as three p orbitals. In the third shell, there is an s orbital, three p orbitals, and five d orbitals. So there’s one orbital in the first shell, four orbitals in the second, and nine orbitals in the third. So what’s the relationship between these numbers?

Well, three squared is equal to nine, two squared is equal to four, and one squared is also equal to one. So it seems the relationship between the principal quantum number 𝑛 and the total number of orbitals is 𝑛 squared.

How many orbitals are there in an atom?

How do you know how many orbitals are in each sublevel?

How many orbitals does 4 have?

How many orbitals can n 4 have?