Question

given the following sets of quantum numbers for n i ml ms, which one of these sets is not a possible set for an electron in an atom?
n l ml ms
a. 3 2 2 -½
b. 2 1 -1 ½
c. 4 3 2 ½
d. 4 3 -2 -½
e. 5 2 3 ½
hint: n = period (row), l = n - 1, ml = +/- l, 0

Explanation:

• For quantum numbers:
• The principal quantum number \(n\) (shell number) can be \(n = 1,2,3,\cdots\).
• The angular - momentum quantum number \(l\) has values \(l=0,1,\cdots,n - 1\).
• The magnetic quantum number \(m_l\) has values \(m_l=-l,-l + 1,\cdots,0,\cdots,l-1,l\).
• The spin quantum number \(m_s=\pm\frac{1}{2}\).
• Option A: \(n = 3\), \(l=2\) (\(l=n - 1=3 - 1 = 2\)), \(m_l = 2\) (\(-2\leq m_l\leq2\)), \(m_s=-\frac{1}{2}\). This set is valid.
• Option B: \(n = 2\), \(l = 1\) (\(l=n - 1=2 - 1=1\)), \(m_l=-1\) (\(-1\leq m_l\leq1\)), \(m_s=\frac{1}{2}\). This set is valid.
• Option C: \(n = 4\), \(l = 3\) (\(l=n - 1=4 - 1 = 3\)), \(m_l = 2\) (\(-3\leq m_l\leq3\)), \(m_s=\frac{1}{2}\). This set is valid.
• Option D: \(n = 4\), \(l = 3\) (\(l=n - 1=4 - 1 = 3\)), \(m_l=-2\) (\(-3\leq m_l\leq3\)), \(m_s=-\frac{1}{2}\). This set is valid.
• Option E: \(n = 5\), \(l = 2\) (\(l=n - 1=5 - 1=4

eq2\) is incorrect as \(l\) should satisfy \(l = 0,1,\cdots,n - 1\)). Also, \(m_l = 3\) and \(l = 2\), but \(m_l\) must satisfy \(-l\leq m_l\leq l\) (\(-2\leq m_l\leq2\)). So this set is not valid.

Answer:

E. 5 2 3 ½