Question

how many orbitals are described by each of the below combinations of quantum numbers?
$n = 3, ell=2$
orbitals
$n = 4, ell = 2, m_{ell}=2$
orbitals

Explanation:

Step1: Recall the relationship between $\ell$ and number of orbitals

The magnetic quantum number $m_{\ell}$ can take values from $-\ell$ to $\ell$. The number of values of $m_{\ell}$ gives the number of orbitals for a given $\ell$.

Step2: Calculate number of orbitals for $n = 3,\ell=2$

For $\ell = 2$, $m_{\ell}=- 2,-1,0,1,2$. So the number of orbitals is $2\ell + 1=2\times2 + 1=5$.

Step3: Analyze the case of $n = 4,\ell = 2,m_{\ell}=2$

When $m_{\ell}$ is fixed, there is only 1 orbital corresponding to that specific $m_{\ell}$ value.

Answer:

5
1