What is the maximum number of electrons that can be accommodated in 3 1 L shell 2 n Shell 3 O shell?

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The correct answer is 2n2.

CONCEPT:

Pauli's exclusion principle states that no two electrons in an atom can have identical values for all four quantum numbers(n,l, ml and ms).
For example, subshell 1s can accommodate only 2 electrons.
For 1s orbital- n=1, l=0, ml= 0 and ms = +$\frac{1}{2}$ or –$\frac{1}{2}$
According to this principle, the number of electrons that can be accommodated in a shell is given by 2n2.
For example, K shell contains only one subshell 1s, so the maximum number of electrons occupied in K shell is 2.

EXPLANATION:

The maximum number of electrons that can be accommodated in a shell is indicated by the formula 2n2.
Number of electrons in one shell =2n2

Where n is the shell number. So the second orbital would be 2(22) = 8

Principle energy level (n)	Subshells	Number of orbitals per subshell	Maximum number of electrons in the shell(2n2)
1	s	1	2
2	s	1	8
p	3
3	s	1	18
p	3
d	5
4	s	1	32
p	3
d	5
f	7

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As you know, electrons are always moving. They spin very quickly around the nucleus of an atom. As the electrons zip around, they can move in any direction, as long as they stay in their shell. Any direction you can imagine — upwards, downwards, or sidewards — electrons can do it. Electrons are constantly spinning in those atomic shells and those shells, or orbitals, are specific distances from the nucleus. If you are an electron in the first shell, you are always closer to the nucleus than the electrons in the second shell.

Let's cover some basics of atomic shells: 1. The center of the atom is called the nucleus.

2. Electrons are found in areas called shells. A shell is sometimes called an energy level.

3. Shells are areas that surround the center of an atom. 4. Each of those shells has a name (K, L, M...).

There are a couple of ways that atomic shells are described. The most general terms are the basic regions where you find electrons. Chemists use an "n" value, or the letters K, L, M, N, O, P, and Q. The "K" shell is the one closest to the nucleus, and "Q" is the farthest away. For simple atoms, those "n" values usually match the row number on the periodic table and are also known as energy levels. The second description looks at how electrons act inside of the shells. There are certain patterns of movement. Chemists have described those patterns with the "l" value. The "l" values tell you what suborbital an electron is found in. You will see the lowercase letters s, p, d, f, g, and h for the suborbitals.

For example, the electron in a hydrogen (H) atom would have the values n=1 and l=0. The single electron would be found in the "K" shell and the "s" suborbital. If you go on to learn more about chemistry, you may see its description written as 1s1. Helium (He) is still in the K shell (top row), but it has two electrons. The first electron would be 1s1 and the second would be 1s2. What about lithium (Li) at atomic number three with three electrons? It would be described as 1s2 2s1. Why is that?

Not all shells and suborbitals hold the same number of electrons. For the first eighteen elements, there are some easy rules. The K shell only holds two electrons. The L shell only holds eight electrons. The M shell only holds eight electrons. The M shell can actually hold up to 18 electrons as you move to higher atomic numbers. The maximum number of electrons you will find in any shell is 32.

Suborbital Basics

We talked a little bit about s, p, d, f, g, and h suborbital descriptions. While the electrons are found in energy levels and regions around the nucleus, they can also be found in special areas within those energy levels. A guy named Schrödinger started realizing that all electrons weren't the same and they didn’t move in the same way. So, looking back at lithium we saw 1s2 2s1. Those values describe where you can find the three electrons. Two are in energy level one in suborbital s. The third electron is in energy level two and suborbital s. Are they both in the same suborbital s? No. The letter of the suborbital references the shapes of regions you will find electrons. Suborbital "s" is in a spherical shape. Suborbital "p" is shaped kind of like barbells or a figure eight. Then you have "d" with two possible shapes, and it just gets crazy from there. Just remember that those letters refer to regions where you are likely to find the electrons within their energy level.

One last example: silicon (Si) at atomic number 14. You have fourteen electrons. Written out the long way, it looks like 1s2 2s2 2p6 3s2 3p2. Do you see how the numbers add up to fourteen? Row one has a shell that can hold two electrons. That’s covered by 1s2. Row two of the periodic table corresponds to shell two, which can hold eight electrons. You can see those eight in 2s2 and 2p6. Finally, we have shell/row three. Since suborbitals can only hold so many electrons, you see them divided into "s" and "p". Silicon only has four electrons in the third shell. Suborbital "s" can hold two, and the other two are found in "p". When you get past argon (Ar) at atomic number 18, you will start finding the "d" suborbitals in the transition elements.
We've been telling you that electrons reside in specific shells or move in specific patterns in suborbitals. We can't really tell you exactly where an electron is at any moment in time. We can only approximate, or guess, where an electron is located. According to something called quantum theory, an electron can be found anywhere around the nucleus. Using advanced math, scientists are able to approximate the general location of electrons. These general areas are the shells and suborbitals.

Researchers Create “Designer Electrons” (Stanford Univ. Video)

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.