How is the number of orbitals determined when only the principal quantum number (n) is given?

The number of orbitals in a given energy level can be determined by considering the values of the angular momentum quantum number (l), which ranges from 0 to (n-1) for a principal quantum number (n). For each value of l, there are (2l + 1) orbitals associated with that angular momentum quantum number.

Thus, when n is provided, the total number of orbitals can be calculated by summing the orbitals contributed by each possible value of l:

• For l = 0 (s orbital), there is 1 orbital.
• For l = 1 (p orbitals), there are 3 orbitals.
• For l = 2 (d orbitals), there are 5 orbitals.
• For l = 3 (f orbitals), there are 7 orbitals.

The total number of orbitals for a given principal quantum number n can be represented using the formula for the total number of angular momentum levels up to (n - 1): 1 (for l=0) + 3 (for l=1) + 5 (for l=2) + ... + (2(n-1) + 1).

Using the formula for the sum of