Question

For any sub-shell defined by '$l$' value how many values of magnetic quantum number ($m_l$) are possible? ________.

• A. $(2l)$
• B. $(2l-1)$
• C. $(2l+1)$
• D. $(l+1)$
• E. $(l-1)$

Hint:

$m_l$ values: $-l, \dots, 0, \dots, +l$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
This question pertains to the rules of quantum numbers that describe the properties of electrons in an atom. The azimuthal quantum number, \(l\), defines the shape of an orbital and corresponds to a specific sub-shell (e.g., s, p, d, f). The magnetic quantum number, \(m_l\), specifies the orientation of that orbital in space.
Step 2: Key Formula or Approach
For a given value of the azimuthal quantum number \(l\), the magnetic quantum number \(m_l\) can take any integer value from \(-l\) to \(+l\), including 0.
\[ m_l \in \{-l, -l+1, \dots, 0, \dots, l-1, l\} \] To find the total number of possible values, we need to count the number of integers in this range.
Step 3: Detailed Explanation
The range of integer values for \(m_l\) is from \(-l\) to \(+l\). We can count the total number of values as follows:
- There are \(l\) positive values (from 1, 2, ..., up to \(l\)).
- There is one value for zero (0).
- There are \(l\) negative values (from -1, -2, ..., down to \(-l\)).
Summing these up gives the total number of possible values:
\[ \text{Total values} = l + 1 + l = 2l + 1 \] This number corresponds to the number of orbitals within that sub-shell. For instance:
- For an s-subshell (\(l=0\)), the number of orbitals is \(2(0)+1 = 1\). (\(m_l = 0\))
- For a p-subshell (\(l=1\)), the number of orbitals is \(2(1)+1 = 3\). (\(m_l = -1, 0, +1\))
- For a d-subshell (\(l=2\)), the number of orbitals is \(2(2)+1 = 5\). (\(m_l = -2, -1, 0, +1, +2\))
Step 4: Final Answer
The number of possible values for the magnetic quantum number \(m_l\) is \((2l+1)\).