Answer
- \(5p\) has 3 orbitals.
- \(3d_{z^{2}}\) has 1 orbital.
- \(4d\) has 5 orbitals.
- \(n = 5\) has 25 orbitals.
- \(n = 4\) has 16 orbitals.
Work Step by Step
To determine the number of orbitals with the given designations, we need to understand the notation used.
In the notation \(n l\), \(n\) represents the principal quantum number, which indicates the energy level or shell of the orbital. The values of \(n\) can be any positive integer starting from 1.
The letter \(l\) represents the azimuthal quantum number, which determines the shape of the orbital. The values of \(l\) can range from 0 to \(n-1\). The corresponding letters for different values of \(l\) are as follows:
\(l = 0\) corresponds to the s orbital.
\(l = 1\) corresponds to the p orbital.
\(l = 2\) corresponds to the d orbital.
\(l = 3\) corresponds to the f orbital.
Now let's analyze each designation:
1. \(5p\):
There are 3 orbitals.
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. \(n = 5\):
$l=0$ (1 orbital)
$l=1$ (3 orbitals)
$l=2$ (5 orbitals)
$l=3$ (7 orbitals)
$l=4$ (9 orbitals)
There is a total of 25 orbitals.
5. \(n = 4\):
$l=0$ (1 orbital)
$l=1$ (3 orbitals)
$l=2$ (5 orbitals)
$l=3$ (7 orbitals)
There is a total of 16 orbitals.
