Question

which of the given combinations of quantum numbers is not possible
a n=1, l=0 m=0, s=+1/2
b n=3, l=2, m=-2, s=-1/2
c n=1, l=2 m=0, s=-1/2
d n=2, l=1, m=0, s=-1/2
continue
question 8 of 9

Explanation:

Step1: Recall quantum number rules

The principal quantum number \( n \) is a positive integer (\( n = 1, 2, 3, \dots \)). The azimuthal quantum number \( l \) satisfies \( 0 \leq l < n \), i.e., \( l \) can take values from \( 0 \) to \( n - 1 \). The magnetic quantum number \( m \) satisfies \( -l \leq m \leq l \), and the spin quantum number \( s \) is either \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

Step2: Analyze Option A

For \( n = 1 \), \( l \) can be \( 0 \) (since \( l < n \)). For \( l = 0 \), \( m \) must be \( 0 \) (since \( -0 \leq m \leq 0 \)). \( s = +\frac{1}{2} \) is valid. So this combination is possible.

Step3: Analyze Option B

For \( n = 3 \), \( l \) can be \( 0, 1, 2 \) (since \( l < 3 \)). Here \( l = 2 \). For \( l = 2 \), \( m \) can range from \( -2 \) to \( 2 \), so \( m = -2 \) is valid. \( s = -\frac{1}{2} \) is valid. So this combination is possible.

Step4: Analyze Option C

For \( n = 1 \), \( l \) must satisfy \( l < n \), so \( l \) can only be \( 0 \) (since \( n = 1 \), \( l < 1 \) implies \( l = 0 \)). But here \( l = 2 \), which violates the rule \( l < n \). So this combination is not possible.

Step5: Analyze Option D

For \( n = 2 \), \( l \) can be \( 0, 1 \) (since \( l < 2 \)). Here \( l = 1 \). For \( l = 1 \), \( m \) can range from \( -1 \) to \( 1 \), so \( m = 0 \) is valid. \( s = -\frac{1}{2} \) is valid. So this combination is possible.

Answer:

C. \( n = 1, l = 2, m = 0, s = -\frac{1}{2} \)