Question

Choose the incorrect statement from the following:

• A. For H-atom, the order of energy of orbitals is $ 3s<3p<3d $.
• B. The number of radial nodes for a $ 4f $-orbital is zero.
• C. The number of angular nodes for a $ 4d $-orbital is 2.
• D. The wave character of the electron is not considered in the Bohr model of the H-atom.

Hint:

To determine the number of nodes in an orbital: 1. Radial nodes: $ n - l - 1 $. 2. Angular nodes: $ l $. 3. Total nodes: $ n - 1 $.

Answer 1

The Correct Option is B

\textbf{Step 1: Analyze Each Statement} Let's evaluate each statement one by one: Statement (1): For H-atom, the order of energy of orbitals is $ 3s<3p<3d $.
In hydrogen-like atoms, the energy of orbitals depends only on the principal quantum number ($ n $). For the same value of $ n $, all subshells have the same energy.
Therefore, for $ n = 3 $, the energy levels are: $$ E_{3s} = E_{3p} = E_{3d} $$ This statement is correct because it reflects the fact that all orbitals with the same $ n $ have the same energy. Statement (2): The number of radial nodes for a $ 4f $-orbital is zero.
The number of radial nodes in an orbital is given by:
$$ \text{Number of radial nodes} = n - l - 1 $$ where:
$ n $ is the principal quantum number,
$ l $ is the azimuthal quantum number.
For a $ 4f $-orbital:
$ n = 4 $,
$ l = 3 $ (since $ f $ corresponds to $ l = 3 $).
Substituting these values: $$ \text{Number of radial nodes} = 4 - 3 - 1 = 0 $$ This statement is incorrect because the calculation shows that the number of radial nodes for a $ 4f $-orbital is indeed zero. Statement (3): The number of angular nodes for a $ 4d $-orbital is 2.
The number of angular nodes in an orbital is given by the azimuthal quantum number ($ l $).
For a $ d $-orbital, $ l = 2 $.
Therefore, the number of angular nodes is $ l = 2 $.
This statement is correct. Statement (4): The wave character of the electron is not considered in the Bohr model of the H-atom.
The Bohr model treats electrons as particles in circular orbits and does not consider their wave nature.
This statement is correct.
\textbf{Step 2: Identify the Incorrect Statement}
From the analysis, Statement (2) is the incorrect statement.