Final answer:
The total number of atomic states, including different spin states, within an atomic shell with principal quantum number n is given by the formula 2n², as each orbital defined by n can accommodate two electrons with opposite spins.
Step-by-step explanation:
The question asks us to find the total number of atomic states for electrons in a shell of an atom with a principal quantum number n. According to quantum mechanics, each shell (indexed by n) can have n different sublevels. For any given value of n, the number of orbitals is equal to n². Each of these orbitals can hold two electrons with opposite spins due to the Pauli exclusion principle. Therefore, the total number of electron states in the nth shell, considering both the orbital and spin quantum numbers, is 2n².
As an example, for n = 1, there are a total of 2 * 1² = 2 states. For n = 2, this becomes 2 * 2² = 8 states, and so on. Thus, the pattern suggests that for any value of n, the total number of atomic states including different spin states in a shell is 2n².