Principal Quantum Number (n)
The principal quantum number, denoted by n, defines the energy level or shell of an atom. It determines the overall size, energy, and number of subshells present within that shell. In this question, n = 7 indicates that the electron states being considered belong to the seventh energy level, which includes a series of subshells corresponding to different angular momentum quantum numbers.
Azimuthal Quantum Number (l) and Subshell Designation
The azimuthal quantum number, l, defines the shape of an orbital and ranges from 0 to n-1 for a given principal quantum number. Each value of l is associated with a specific subshell (for instance, l = 0 corresponds to an s subshell, l = 1 to a p subshell, l = 2 to a d subshell, etc.). For n = 7, l can take values from 0 to 6, thereby producing subshells such as 7s, 7p, 7d, 7f, 7g, 7h, and 7i.
Magnetic Quantum Number (m?) and Orbital Degeneracy
The magnetic quantum number, m?, describes the orientation of an orbital and can have integer values between -l and +l. This results in 2l + 1 orbitals in each subshell. The number of orbitals directly determines the degeneracy of the subshell, meaning how many distinct spatial states are available for electrons.
Electron Spin and its Contribution to Electron States
Each orbital can accommodate two electrons with opposite spins, as dictated by the spin quantum number. Thus, when counting electron states, the factor of two is applied to the number of orbitals (2l + 1), leading to a total of 2(2l + 1) electron states per subshell.
Counting Electron States in a Shell
To compute the total number of electron states in a given shell, one calculates the number of states for each subshell individually and then sums them. For a shell with principal quantum number n, where l ranges from 0 to n-1, the number of electron states in a given subshell is 2(2l + 1), and the total number of states in the shell is the sum of these values. In the example provided, when this summation is performed for n = 7, it yields a total of 98 electron states.