Final answer:
To find the total number of orbitals in a shell, calculate the square of the shell number (n²). Each shell can hold up to 2n² electrons because each orbital within a shell can hold two electrons. For instance, the fourth shell has 16 orbitals and can hold up to 32 electrons.
Step-by-step explanation:
Finding the Total Number of Orbitals in a Given Shell
To find the total number of orbitals in any given shell of an atom, you can use the shell number (n). The number of orbitals in any shell is equal to n². Since each orbital can hold up to two electrons, the maximum number of electrons that can fit in a shell is given by 2n².
For example, for the first shell (n = 1), the number of orbitals is 1², which equals 1. Therefore, there is only one s orbital where a maximum of two electrons can reside. As we move to higher shells like n = 4, we see that 4² equals 16, indicating there are 16 orbitals in the fourth shell. These orbitals include one 4s, three 4p, five 4d, and seven 4f orbitals.
The number of orbitals increases with the shell number because the number of subshells also increases. The value of n determines the number of subshells; for example, the fourth shell has four subshells: s, p, d, and f. The orbitals within these subshells are differentiated by the magnetic quantum number m₁.