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