Answer
See here:
- \(5p\) has 1 orbital.
- \(3d_{z^{2}}\) has 1 orbital.
- \(4d\) has 5 orbitals.
- \(n = 5\) has 50 orbitals.
- \(n = 4\) has 32 orbitals.
Work Step by Step
Let's analyze each designation:
1. \(5p\):
For the \(5p\) designation, \(n = 5\) and \(l = 1\). Since \(l = 1\), there is only one p orbital. Therefore, there is only one orbital with the designation \(5p\).
2. \(3d_{z^{2}}\):
For the \(3d_{z^{2}}\) designation, \(n = 3\) and \(l = 2\). Since \(l = 2\), there are five d orbitals. However, the subscript \(z^{2}\) indicates a specific orientation of the d orbital along the z-axis. This means that only one of the five d orbitals has this specific orientation. Therefore, there is only one orbital with the designation \(3d_{z^{2}}\).
3. \(4d\):
For the \(4d\) designation, \(n = 4\) and \(l = 2\). Since \(l = 2\), there are five d orbitals. Therefore, there are five orbitals with the designation \(4d\).
4. \(n = 5\):
The designation \(n = 5\) represents all the orbitals in the fifth energy level. To determine the number of orbitals, we need to know the maximum number of orbitals in a given energy level. The maximum number of orbitals in an energy level is given by \(2n^2\). For \(n = 5\), the maximum number of orbitals is \(2 \times 5^2 = 50\). Therefore, there are 50 orbitals with the designation \(n = 5\).
5. \(n = 4\):
Similarly, the designation \(n = 4\) represents all the orbitals in the fourth energy level. The maximum number of orbitals in this energy level is \(2 \times 4^2 = 32\). Therefore, there are 32 orbitals with the designation \(n = 4\).
