Explain why each of the following sets of quantum numbers...

Explain why each of the following sets of quantum numbers would not be permissible for an electron, according to the rules for quantum numbers. a. $n=1, l=0, m_{l}=0, m_{s}=+1$ b. $n=1, l=3, m_{l}=+3, m_{s}=+\frac{1}{2}$ c. $n=3, l=2, m_{l}=+3, m_{s}=-\frac{1}{2}$ d. $n=0, l=1, m_{l}=0, m_{s}=+\frac{1}{2}$ e. $n=2, l=1, m_{l}=-1, m_{s}=+\frac{3}{2}$

Explain why each of the following sets of quantum numbers would not be permissible for an electron, according to the rules for quantum numbers.
a. $n=1, l=0, m_{l}=0, m_{s}=+1$
b. $n=1, l=3, m_{l}=+3, m_{s}=+\frac{1}{2}$
c. $n=3, l=2, m_{l}=+3, m_{s}=-\frac{1}{2}$
d. $n=0, l=1, m_{l}=0, m_{s}=+\frac{1}{2}$
e. $n=2, l=1, m_{l}=-1, m_{s}=+\frac{3}{2}$

Key Concepts

Principal Quantum Number (n)

The principal quantum number denotes the energy level or shell of an electron in an atom and must be a positive integer (n ? 1). It determines the overall size and energy of the orbital, and other quantum numbers are defined in relation to n. In any valid electron configuration, n sets the upper limits for allowed values of other quantum numbers such as the angular momentum quantum number (l).

Azimuthal (Angular Momentum) Quantum Number (l)

The azimuthal quantum number defines the shape of an orbital and determines the orbital angular momentum of an electron. Its allowed values are integers ranging from 0 to n-1 for a given principal quantum number n. This restriction means that l cannot be greater than or equal to n, ensuring consistency in the structure of atomic orbitals.

Magnetic Quantum Number (m_l)

The magnetic quantum number describes the orientation of an orbital in space relative to an external magnetic field. It can take on integer values ranging from -l to +l for a given value of the angular momentum quantum number (l). This range limits the possible spatial orientations for orbitals, and any value of m_l outside this range for a given l is not permissible.

Spin Quantum Number (m_s)

The spin quantum number represents the intrinsic spin or angular momentum of an electron. It has only two allowed values, +1/2 and -1/2, reflecting the fundamental two-state nature of electron spin. Any value for m_s apart from these two, such as +1 or +3/2, is not allowed in standard quantum mechanics.

Quantum Number Selection Rules

The selection rules for quantum numbers establish the relationships and restrictions among them. These rules ensure that for any given principal quantum number (n), the angular momentum quantum number (l) is bounded (0 ? l ? n-1), the magnetic quantum number (m_l) is limited by l (-l ? m_l ? +l), and the spin quantum number (m_s) can only be +1/2 or -1/2. Violation of any of these rules, such as an m_l value exceeding l or an m_s value outside the allowed spin states, results in an impermissible electron configuration.

Transcript

Please give Ace some feedback