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The Pauli exclusion principle states that no two electrons in the same system can exist simultaneously in the same state. The state of an electron is determined by three things:

Its energy level N

Its angular momentum L

and its spin S

The electron can only have a spin of +1/2 or -1/2, so there are only two states that an electron can be in for a given L and N.

The electron's angular momentum L depends on the state N that it is in. L must be an integer whose magnitude is less than N, so it can have (2N - 1) different states for a given N. (For example, if N = 2, L can equal -2, -1, 0, 1, 2).

The energy level N is also given as a (nonzero) positive integer (N can equal 1, 2, 3, 4, etc.). In an element's ground state, EVERY SINGLE electron state must be filled. So an element with five electrons in the fourth energy level N, energy levels 1, 2, and 3 must be filled, and energy level 4 must have five electrons in it.

The number of electrons in energy level 1 is the number of spin states time the number of angular momentum states for the level N, that is 2 spin states times (2N-1) angular momentum states = 2

For energy level N = 1, the number of electrons can be 2 * (2N-1) = 2

For energy level N = 2, 2 * (2N-1) = 6

For N = 3, 2 * (2N-1) = 10.

So the first three energy levels, when filled, contain 2 + 6 +10 = 18 electrons. The fourth energy level has 5 electrons according to the question, so there must be 18 + 5 = 23 electrons in total.

The element with 23 electrons, or an "atomic number" of 23 is Vanadium.

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