What are the maximum number of electrons in each shell?

Question

In my textbook, it says that the maximum number of electrons that can fit in any given shell is given by 2n². This would mean 2 electrons could fit in the first shell, 8 could fit in the second shell, 18 in the third shell, and 32 in the fourth shell.

However, I was previously taught that the maximum number of electrons in the first orbital is 2, 8 in the second orbital, 8 in the third shell, 18 in the fourth orbital, 18 in the fifth orbital, 32 in the sixth orbital. I am fairly sure that orbitals and shells are the same thing.

Which of these two methods is correct and should be used to find the number of electrons in an orbital?

I am in high school so please try to simplify your answer and use fairly basic terms.

Answer 1

Shells and orbitals are not the same. In terms of quantum numbers, electrons in different shells will have different values of principal quantum number n.

To answer your question...

In the first shell (n=1), we have:

• The 1s orbital

In the second shell (n=2), we have:

• The 2s orbital
• The 2p orbitals

In the third shell (n=3), we have:

• The 3s orbital
• The 3p orbitals
• The 3d orbitals

In the fourth shell (n=4), we have:

• The 4s orbital
• The 4p orbitals
• The 4d orbitals
• The 4f orbitals

So another kind of orbitals (s, p, d, f) becomes available as we go to a shell with higher n. The number in front of the letter signifies which shell the orbital(s) are in. So the 7s orbital will be in the 7th shell.

Now for the different kinds of orbitals Each kind of orbital has a different "shape", as you can see on the picture below. You can also see that:

• The s-kind has only one orbital
• The p-kind has three orbitals
• The d-kind has five orbitals
• The f-kind has seven orbitals

Each orbital can hold two electrons. One spin-up and one spin-down. This means that the 1s, 2s, 3s, 4s, etc., can each hold two electrons because they each have only one orbital.

The 2p, 3p, 4p, etc., can each hold six electrons because they each have three orbitals, that can hold two electrons each (3*2=6).

The 3d, 4d etc., can each hold ten electrons, because they each have five orbitals, and each orbital can hold two electrons (5*2=10).

Thus, to find the number of electrons possible per shell

First, we look at the n=1 shell (the first shell). It has:

• The 1s orbital

An s-orbital holds 2 electrons. Thus n=1 shell can hold two electrons.

The n=2 (second) shell has:

• The 2s orbital
• The 2p orbitals

s-orbitals can hold 2 electrons, the p-orbitals can hold 6 electrons. Thus, the second shell can have 8 electrons.

The n=3 (third) shell has:

• The 3s orbital
• The 3p orbitals
• The 3d orbitals

s-orbitals can hold 2 electrons, p-orbitals can hold 6, and d-orbitals can hold 10, for a total of 18 electrons.

Therefore, the formula $2n^2$ holds! What is the difference between your two methods?

There's an important distinction between "the number of electrons possible in a shell" and "the number of valence electrons possible for a period of elements".

There's space for $18 \text{e}^-$ in the 3rd shell: $3s + 3p + 3d = 2 + 6 + 10 = 18$, however, elements in the 3rd period only have up to 8 valence electrons. This is because the $3d$-orbitals aren't filled until we get to elements from the 4th period - ie. elements from the 3rd period don't fill the 3rd shell.

The orbitals are filled so that the ones of lowest energy are filled first. The energy is roughly like this:

$$1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s$$

An easy way to visualize this is like this:

Answer 2

The pattern of maximum possible electrons = $2n^2$ is correct.

Also, note that Brian's answer is good and takes a different approach.

Have you learned about quantum numbers yet?

If not...

Each shell (or energy level) has some number of subshells, which describe the types of atomic orbitals available to electrons in that subshell. For example, the $s$ subshell of any energy level consists of spherical orbitals. The $p$ subshell has dumbbell-shaped orbitals. The orbital shapes start to get weird after that. Each subshell contains a specified number of orbitals, and each orbital can hold two electrons. The types of subshells available to a shell and the number of orbitals in each subshell are mathematically defined by quantum numbers. Quantum numbers are parameters in the wave equation that describes each electron. The Pauli Exclusion Principle states that no two electrons in the same atom can have the exact same set of quantum numbers. A more thorough explanation using quantum numbers can be found below. However, the outcome is the following:

The subshells are as follows:

• The $s$ subshell has one orbital for a total of 2 electrons
• The $p$ subshell has three orbitals for a total of 6 electrons
• The $d$ subshell has five orbitals for a total of 10 electrons
• The $f$ subshell has seven orbitals for a total of 14 electrons
• The $g$ subshell has nine orbitals for a total of 18 electrons
• The $h$ subshell has eleven orbitals for a total of 22 electrons

etc.

Each energy level (shell) has more subshells available to it:

• The first shell only has the $s$ subshell $\implies$ 2 electrons
• The second shell has the $s$ and $p$ subshells $\implies$ 2 + 6 = 8 electrons
• The third shell has the $s$, $p$, and $d$ subshells $\implies$ 2 + 6 + 10 = 18 electrons
• The fourth shell has the $s$, $p$, $d$, and $f$ subshells $\implies$ 2 + 6 + 10 + 14 = 32 electrons
• The fifth shell has the $s$, $p$, $d$, $f$, and $g$ subshells $\implies$ 2 + 6 + 10 + 14 + 18 = 50 electrons
• The sixth shell has the $s$, $p$, $d$, $f$, $g$, and $h$ subshells $\implies$ 2 + 6 + 10 + 14 + 18 + 22 = 72 electrons

The pattern is thus: $2, 8, 18, 32, 50, 72, ...$ or $2n^2$

In practice, no known atoms have electrons in the $g$ or $h$ subshells, but the quantum mechanical model predicts their existence.

Using quantum numbers to explain why the shells have the subshells they do and why the subshells have the number of orbitals they do.

Electrons in atoms are defined by 4 quantum numbers. The Pauli Exclusion Principle means that no two electrons can share the same quantum numbers.

The quantum numbers:

• $n$, the principle quantum number defines the shell. The values of $n$ are integers: $n=1,2,3,...$
• $\ell$, the orbital angular momentum quantum number defines the subshell. This quantum number defines the shape of the orbitals (probability densities) that the electrons reside in. The values of $\ell$ are integers dependent on the value of $n$: $\ell = 0,1,2,...,n-1$
• $m_{\ell}$, the magnetic quantum number defines the orientation of the orbital in space. This quantum number also determines the number of orbitals per subshell. The values of $m_\ell$ are integers and depend on the value of $\ell$: $m_\ell = -\ell,...,-1,0,1,...,+\ell$
• $m_s$, the spin angular momentum quantum number defines the spin state of each electron. Since there are only two allowed values of spin, thus there can only be two electrons per orbital. The values of $m_s$ are $m_s=\pm \frac{1}{2}$

For the first shell, $n=1$, so only one value of $\ell$ is allowed: $\ell=0$, which is the $s$ subshell. For $\ell=0$ only $m_\ell=0$ is allowed. Thus the $s$ subshell has only 1 orbital. The first shell has 1 subshell, which has 1 orbital with 2 electrons total.

For the second shell, $n=2$, so the allowed values of $\ell$ are: $\ell=0$, which is the $s$ subshell, and $\ell=1$, which is the $p$ subshell. For $\ell=1$, $m_\ell$ has three possible values: $m_\ell=-1,0,+1$. Thus the $p$ subshell has three orbitals. The second shell has 2 subshells: the $s$ subshell, which has 1 orbital with 2 electrons, and the $p$ subshell, which has 3 orbitals with 6 electrons, for a total of 4 orbitals and 8 electrons.

For the third shell, $n=3$, so the allowed values of $\ell$ are: $\ell=0$, which is the $s$ subshell, $\ell=1$, which is the $p$ subshell, and $\ell=2$, which is the $d$ subshell. For $\ell=2$, $m_\ell$ has five possible values: $m_\ell=-2,-1,0,+1,+2$. Thus the $d$ subshell has five orbitals. The third shell has 3 subshells: the $s$ subshell, which has 1 orbital with 2 electrons, the $p$ subshell, which has 3 orbitals with 6 electrons, and the $d$ subshell, which has 5 orbitals with 10 electrons, for a total of 9 orbitals and 18 electrons.

For the fourth shell, $n=4$, so the allowed values of $\ell$ are: $\ell=0$, which is the $s$ subshell, $\ell=1$, which is the $p$ subshell, $\ell=2$, which is the $d$ subshell, and $\ell=3$, which is the $f$ subshell. For $\ell=3$, $m_\ell$ has seven possible values: $m_\ell=-3,-2,-1,0,+1,+2,-3$. Thus the $f$ subshell has seven orbitals. The fourth shell has 4 subshells: the $s$ subshell, which has 1 orbital with 2 electrons, the $p$ subshell, which has 3 orbitals with 6 electrons, the $d$ subshell, which has 5 orbitals with 10 electrons, and the $f$ subshell, which has 7 orbitals with 14 electrons, for a total of 16 orbitals and 32 electrons.

Answer 3

The first shell can carry up to two electrons, the second shell can carry up to eight electrons.

The third shell can carry up 18 electrons, but it is more stable by carrying only eight electrons. There is a formula for obtaining the maximum number of electrons for each shell which is given by $2n^2~\ldots$ where n is the position of a certain shell.