An acceptable set of quantum numbers for an electron in an atom must adhere to certain rules and constraints. The quantum numbers are as follows:
Principal Quantum Number (n): It specifies the energy level or shell in which the electron resides. The value of 'n' must be a positive integer (1, 2, 3, ...) and determines the overall size and energy of the orbital.
Azimuthal Quantum Number (ℓ): It determines the shape or subshell of the orbital. The value of ℓ must be an integer ranging from 0 to (n-1). The corresponding subshell shape is determined as follows:
- ℓ = 0 corresponds to an s orbital (spherical shape).
- ℓ = 1 corresponds to a p orbital (dumbbell shape).
- ℓ = 2 corresponds to a d orbital (cloverleaf shape).
- ℓ = 3 corresponds to an f orbital (complex shape), and so on.
Magnetic Quantum Number (mℓ): It specifies the orientation of the orbital within a subshell. The value of mℓ ranges from -ℓ to +ℓ, including zero. For example, if ℓ = 1 (p subshell), the possible values of mℓ are -1, 0, and 1, indicating the three different orientations of the p orbitals along the x, y, and z axes.
Spin Quantum Number (MS): It describes the spin of the electron. The value of MS can either be +1/2 (spin-up) or -1/2 (spin-down), representing the two possible spin states of an electron.
Therefore, an acceptable set of quantum numbers for an electron in an atom could be represented as (n, ℓ, mℓ, MS) such as (2, 1, 0, +1/2) or (3, 2, -1, -1/2), where the values of n, ℓ, mℓ, and MS satisfy the aforementioned rules and constraints.