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?
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:
etc. Each energy level (shell) has more subshells available to it:
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.
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:
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.
In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation. THE MAIN QUANTUM NUMBERS OVERALL DEFINE AVAILABLE ORBITAL SUBSHELLS #n# is the principal quantum number, indicating the quantized energy level, and corresponds to the period/row on the periodic table. #n = 1, 2, 3, . . . , N# where #N# is some large arbitrary integer in the real numbers. #l# is the angular momentum quantum number, which corresponds to the shape of the orbital (#s#, #p#, #d#, #f#, #g#, #h#, etc). It is equal to #0, 1, . . . , n-1#.
With #n = 2#, we have access to the #2s# and #2p# orbitals, since we assign #l = 0# for the #s# orbitals and #l = 1# for the #p# orbitals. (Similarly, we assign #l = 2# for the #d# orbitals, #l = 3# for the #f# orbitals, and so on alphabetically: #g#, #h#, etc.) Note that with #n = 2# we do not get #l = 2# because #n - 1 = 2 - 1 = 1# for a max #l# of #1#. THE MAGNETIC QUANTUM NUMBERS FURTHER DEFINE A SET OF (ORTHOGONAL) ORBITALS With #l = 0,1#, we introduce the magnetic quantum number #m_l#, which takes on the values #0, pm1, . . . , pml#. Therefore:
Furthermore, we have the spin quantum number #m_s# for each unique electron, which can be #pm"1/2"# and cannot be the same for two electrons with the same #m_l#. That is, since all electrons in the same orbital have the same #n#, #l#, and #m_l#, they can only have #m_s = "+1/2"# show up once and #m_s = "-1/2"# show up once.
TOTAL NUMBER OF ELECTRONS POSSIBLE Finally, with the #s# and #p# subshells combined, we have:
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