Which of the following sets of quantum numbers are not...

Which of the following sets of quantum numbers are not allowed? For each incorrect set, state why it is incorrect. a. $n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2}$ b. $n=4, \ell=3, m_{\ell}=2, m_{s}=-\frac{1}{2}$ c. $n=4, \ell=1, m_{\ell}=1, m_{s}=+\frac{1}{2}$ d. $n=2, \ell=1, m_{\ell}=-1, m_{s}=-1$ e. $n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2}$ f. $n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2}$

Which of the following sets of quantum numbers are not allowed? For each incorrect set, state why it is incorrect.
a. $n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2}$
b. $n=4, \ell=3, m_{\ell}=2, m_{s}=-\frac{1}{2}$
c. $n=4, \ell=1, m_{\ell}=1, m_{s}=+\frac{1}{2}$
d. $n=2, \ell=1, m_{\ell}=-1, m_{s}=-1$
e. $n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2}$
f. $n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2}$

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by n, indicates the energy level of an electron in an atom and must be a positive integer (n = 1, 2, 3, ...). It sets the maximum allowed value for the azimuthal quantum number, meaning that the orbital’s angular momentum is bounded by n-1.

Azimuthal Quantum Number

The azimuthal (or angular momentum) quantum number, denoted by ?, determines the subshell or orbital shape and must be an integer between 0 and n-1. This condition ensures that the orbital’s angular momentum does not exceed what is physically allowed for a given energy level.

Magnetic Quantum Number

The magnetic quantum number, denoted by m?, specifies the orientation of the orbital in space and can take on integer values ranging from -? to +?. This range corresponds to the different possible orientations of the orbital’s angular momentum vector in the presence of a magnetic field.

Spin Quantum Number

The spin quantum number, denoted by ms, describes the intrinsic angular momentum or spin of the electron. Its allowed values are only +1/2 and -1/2, representing the two possible spin states of an electron. This quantization is fundamental to the electron’s behavior in quantum systems.

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