The maximum number of electrons that are possible at a given energy level depends on the number of at the energy level.

Explanation:

Each orbital may contain a maximum of two electrons.

The energy level is the principal quantum number: 1, 2, 3, 4, 5, 6, 7.

The principal quantum number limits the kind of orbital (ℓ ) and the number of orbitals (mℓ ).

I. Principal quantum number, n = 1

  Angular or orbital quantum number : ℓ = 0 ⇒ ortibal s

  Magnetic quantum number:  mℓ = 0

                                 ⇒ number of orbitals 1 ⇒ number of electrons: 2

II. Principal quantun number, n = 2

  Angular or orbital quantum number : ℓ = 0 and 1 ⇒ ortibal s and p

  Magnetic quantum number:  

                                            for ℓ:  mℓ = 0: 1 s ortibal

                                            for ℓ = 1: mℓ = -1, 0, and +1:  3 p ortitals

                                 ⇒ number of orbitals 4 ⇒ number of electrons: 8

III. Principal quantun number, n = 3

    Angular or orbital quantum number : ℓ = 0, 1, and 2 ⇒ ortibal s, p and d

    Magnetic quantum number:  

                                            for ℓ:  mℓ = 0: 1 s ortibal

                                            for ℓ = 1: mℓ = -1, 0, and +1:  3 p ortitals

                                            for ℓ = 2: mℓ = -2, -1, 0, +1, and +2:  5 d ortitals

                                 ⇒ number of orbitals 9 ⇒ number of electrons: 18

IV. Principal quantun number, n = 4

    Angular or orbital quantum number : ℓ = 0, 1, 2, and 3 ⇒ ortibal s, p, d and f

    Magnetic quantum number:  

                                            for ℓ:  mℓ = 0: 1 s ortibal

                                            for ℓ = 1: mℓ = -1, 0, and +1:  3 p ortitals

                                            for ℓ = 2: mℓ = -2, -1, 0, +1, and +2:  5 d ortitals

                                            for ℓ = 3: mℓ = ±3, ±2, ±1, and 0: 7 f ortitals

                                 ⇒ number of orbitals 16 ⇒ number of electrons: 32.

Thus, you have:

energy level, n    maximum number of     maximum number of

                             orbitals                           electrons

      1                           1                                      2

      2                          4                                     8

      3                          9                                    18

      4                          16                                   32

Those numbers follow a rule: n² and 2n². You can verify that the previous numbers are in accordance with those formulas.

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