Question

The incorrect set of quantum numbers \((n,l,m,s)\) for an electron in \(3p\) orbital is:

• A. \(3,\ 1,\ -1,\ \dfrac{1}{2}\)
• B. \(3,\ 1,\ -2,\ -\dfrac{1}{2}\)
• C. \(3,\ 1,\ 1,\ \dfrac{1}{2}\)
• D. \(3,\ 1,\ +1,\ -\dfrac{1}{2}\)

Hint:

For any orbital, \[ m=-l \text{ to } +l \] For a \(p\)-orbital where \(l=1\), \[ m=-1,\ 0,\ +1 \] only.

Answer 1

The Correct Option is B

Step 1: Identify the quantum numbers for 3p orbital.
For a $3p$ orbital, the principal quantum number is $n = 3$ (third shell) and the azimuthal quantum number is $l = 1$ (since $p$ corresponds to $l = 1$).
Step 2: Determine allowed values of the magnetic quantum number.
The magnetic quantum number $m_l$ can take integer values from $-l$ to $+l$. Since $l = 1$, the allowed values are \[ m_l = -1,\ 0,\ +1 \] This gives three orientations for the $p$ orbital in space.
Step 3: Determine allowed values of the spin quantum number.
The spin quantum number $m_s$ can only be $+\frac{1}{2}$ or $-\frac{1}{2}$. These represent the two possible spin states of an electron.
Step 4: Check option (1): $(3, 1, -1, +\frac{1}{2})$.
Here $n = 3$, $l = 1$, $m_l = -1$ (allowed), $m_s = +\frac{1}{2}$ (allowed). This is a valid set.
Step 5: Check option (2): $(3, 1, -2, -\frac{1}{2})$.
Here $n = 3$, $l = 1$, but $m_l = -2$. Since $l = 1$, the magnetic quantum number can only be $-1, 0,$ or $+1$. The value $m_l = -2$ is not allowed because it falls outside the range $[-l, +l]$. This set is INCORRECT.
Step 6: Confirm and state the answer.
Options (3) and (4) have valid quantum numbers. Only option (2) has an invalid magnetic quantum number $m_l = -2$ for a $3p$ orbital. \[ \boxed{(3,\ 1,\ -2,\ -\tfrac{1}{2})\ \text{is incorrect}} \]