Question

what is the relationship between n, ℓ, and the total number of nodes?
the number of planar nodes = ℓ.
the number of spherical nodes = n - ℓ - 1.

Explanation:

Step1: Recall the formula for total nodes

The total number of nodes (planar + spherical) is given by \( n - 1\), where \( n\) is the principal quantum number.

Step2: Analyze each orbital

• For \( 1s\): \( n = 1\), total nodes \(=1 - 1=0\) (matches with given values \( l = 0\), planar nodes \(=0\), spherical nodes \(=0\)).
• For \( 2s\): \( n = 2\), total nodes \(=2 - 1 = 1\) (matches with given \( l = 0\), planar nodes \(=0\), spherical nodes \(=1\)).
• For \( 2p_x\): \( n = 2\), total nodes \(=2 - 1=1\) (matches with given \( l = 1\), planar nodes \(=1\), spherical nodes \(=0\)).
• For \( 3s\): \( n = 3\), total nodes \(=3 - 1 = 2\) (matches with given \( l = 0\), planar nodes \(=0\), spherical nodes \(=2\)).
• For \( 3p_x\): \( n = 3\), total nodes \(=3 - 1=2\) (matches with given \( l = 1\), planar nodes \(=1\), spherical nodes \(=1\)).
• For \( 3d_{xy}\): \( n = 3\), total nodes \(=3 - 1 = 2\) (matches with given \( l = 2\), planar nodes \(=2\), spherical nodes \(=0\)).

Answer:

The total number of nodes (planar + spherical) is \( n - 1\).