Final answer:
The maximum number of electron states in the nth shell is derived using the Pauli exclusion principle and the rules for quantum numbers, summing over the electron states in each subshell based on their respective quantum numbers. For any shell n, this sum equates to 2n², confirming the statement.
Step-by-step explanation:
To show that the maximum number of electron states in the nth shell of an atom is 2n², we use the Pauli exclusion principle and the rules for quantum numbers. Every electron in an atom is described by four quantum numbers: the principal quantum number (n), angular momentum quantum number (l), magnetic quantum number (m_l), and spin quantum number (m_s). The principal quantum number n determines the shell, and n can be any positive integer. The angular momentum quantum number l can be any integer from 0 to n-1. Since l defines the subshell, for each value of l, there are 2(2l + 1) possible electron states due to the magnetic quantum number and spin quantum number.
For a given shell n, we can find the total number of electron states by summing up all the states from each subshell, which means summing over all possible values of l from 0 to n-1:
- When n=1, l=0, so there is only 1 subshell (the s-subshell) and the number of electron states is 2(2*0+1)=2.
- When n=2, l can be 0 or 1, resulting in an s-subshell and a p-subshell, and the total number of electron states becomes 2(2*0+1) + 2(2*1+1) = 2 + 6 = 8.
Generalizing this to any shell n, the sum of electron states in all subshells gives the maximum number of electron states in the nth shell, which turns out to be 2n², proving our initial statement.