Number of degenerate spatial orbitals with spin of a hydrogen like atom with principal quantum number n = 8 is 128

Explanation:
To understand the number of degenerate spatial orbitals with spin of a hydrogen-like atom, we need to consider the quantum numbers and their relationship to the number of orbitals.

Quantum Numbers:
In atomic physics, quantum numbers are used to describe the energy levels and properties of electrons within an atom. The quantum numbers include the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (m), and the spin quantum number (s).

1. Principal Quantum Number (n):
The principal quantum number (n) represents the energy level of the electron. It determines the size and distance of the electron from the nucleus. In this case, n = 8.

2. Azimuthal Quantum Number (l):
The azimuthal quantum number (l) specifies the shape of the orbital. It can have values ranging from 0 to (n-1). The total number of possible values for l is given by the formula 2l+1.

3. Magnetic Quantum Number (m):
The magnetic quantum number (m) describes the orientation of the orbital in space. It can have values ranging from -l to +l.

4. Spin Quantum Number (s):
The spin quantum number (s) describes the spin orientation of the electron. It can have two possible values: +1/2 (spin-up) or -1/2 (spin-down).

Calculating the number of degenerate spatial orbitals:

1. For a given value of n, the number of possible values for l is (2l+1).
- For n = 8, the possible values for l range from 0 to (8-1) = 7.
- Therefore, the total number of possible values for l is (2(0)+1) + (2(1)+1) + ... + (2(7)+1) = 1 + 3 + 5 + ... + 15 = 64.

2. Each value of l has a range of possible values for m, given by -l to +l.
- For each value of l, the total number of possible values for m is (2l+1).
- Therefore, the total number of possible values for m is (2(0)+1) + (2(1)+1) + ... + (2(7)+1) = 1 + 3 + 5 + ... + 15 = 64.

3. Since each spatial orbital can accommodate two electrons with opposite spins, the total number of degenerate spatial orbitals is twice the total number of possible values for m.
- Therefore, the total number of degenerate spatial orbitals is 2 * 64 = 128.

Conclusion:
The number of degenerate spatial orbitals with spin of a hydrogen-like atom with principal quantum number n = 8 is 128.