To determine the orbital for which the value \(\( n + 1 + m + s \)\) is maximum, we need to evaluate this expression for each given orbital: 3p, 4d, 5p, and 5s. Here's the step-by-step solution:
1. Assign Quantum Numbers:
- For the 3p orbital:
- \(\( n = 3 \)\)
- \(\( l = 1 \)\)
- For the 4d orbital:
- \(\( n = 4 \)\)
- \(\( l = 2 \)\)
- For the 5p orbital:
- \(\( n = 5 \)\)
- \(\( l = 1 \)\)
- For the 5s orbital:
- \(\( n = 5 \)\)
- \(\( l = 0 \)\)
2. Calculate \(\( n + l \)\) for Each Orbital:
- For 3p:
- \(\( n + l = 3 + 1 = 4 \)\)
- For 4d:
- \(\( n + l = 4 + 2 = 6 \)\)
- For 5p:
- \(\( n + l = 5 + 1 = 6 \)\)
- For 5s:
- \(\( n + l = 5 + 0 = 5 \)\)
3. Calculate \(\( m \)\) Values:
- The possible \(\( m \)\) values range from \(\(-l\)\) to \(\(+l\)\), where \(\( m \)\) represents the magnetic quantum number.
- For 3p (\(\( l = 1 \)\)):
- Maximum \(\( m \)\): \(\( +1 \)\)
- Minimum \(\( m \)\): \(\(-1 \)\)
- For 4d (\(\( l = 2 \)\)):
- Maximum \(\( m \)\): \(\( +2 \)\)
- Minimum \(\( m \)\): \(\(-2 \)\)
- For 5p (\(\( l = 1 \)\)):
- Maximum \(\( m \)\): \(\( +1 \)\)
- Minimum \(\( m \)\): \(\(-1 \)\)
- For 5s (\(\( l = 0 \)\)):
- Maximum \(\( m \)\): \(\( 0 \)\)
- Minimum \(\( m \)\): \(\( 0 \)\)
4. Electron Spin Quantum Number \(\( s \)\):
- The electron spin quantum number \(\( s \)\) can be either \(\( +\frac{1}{2} \)\) or \(\( -\frac{1}{2} \)\).
- To maximize the expression, use \(\( s = +\frac{1}{2} \)\).
5. Evaluate \(\( n + 1 + m + s \)\) for Maximum \(\( m \)\):
- For 3p:
- \(\( 4 + 1 + 1 + \frac{1}{2} = 6.5 \)\)
- For 4d:
- \(\( 6 + 1 + 2 + \frac{1}{2} = 9.5 \)\)
- For 5p:
- \(\( 6 + 1 + 1 + \frac{1}{2} = 8.5 \)\)
- For 5s:
- \(\( 5 + 1 + 0 + \frac{1}{2} = 6.5 \)\)
6. Evaluate \(\( n + 1 + m + s \)\) for Minimum \(\( m \)\):
- For 3p:
- \(\( 4 + 1 + (-1) + \frac{1}{2} = 4.5 \)\)
- For 4d:
- \(\( 6 + 1 + (-2) + \frac{1}{2} = 5.5 \)\)
- For 5p:
- \(\( 6 + 1 + (-1) + \frac{1}{2} = 6.5 \)\)
- For 5s:
- \(\( 5 + 1 + 0 + \frac{1}{2} = 6.5 \)\)
7. Compare All Values:
- The values for \(\( n + 1 + m + s \)\) include:
- \(\( 3p: \)\) 6.5 (max), 4.5 (min)
- \(\( 4d: \)\) 9.5 (max), 5.5 (min)
- \(\( 5p: \)\) 8.5 (max), 6.5 (min)
- \(\( 5s: \)\) 6.5 (max), 6.5 (min)
- The maximum value among these is 9.5.
8. Conclusion:
- The orbital for which the value \(\( n + 1 + m + s \)\) is maximum is the 4d orbital.
Therefore, the correct answer is:
(C) 4d
