Question

Consider the following schematic plots of orbital wavefunction (\(\psi_r\)) against distance (\(r\)) from the nucleus.

The figure representing two radial nodes in the orbital is

• A. D
• B. A
• C. B
• D. C

Hint:

To count radial nodes from a graph of \(\psi_r\) vs \(r\): - Count the number of times the curve completely cuts through the \(\psi_r = 0\) line (do not count the origin \(r=0\) or the far right end where it flattens out). - Graph C cuts the line twice \(\rightarrow\) 2 radial nodes.

Answer 1

The Correct Option is D

Concept: A radial node is a spherical region surrounding the atomic nucleus where the probability of finding an electron drops identically to zero. Mathematically, this corresponds to coordinates where the radial wavefunction changes sign, crossing the zero axis: \[ \psi_r = 0 \] On a schematic plot graphing \(\psi_r\) as a function of distance \(r\) from the nucleus:

• A radial node is graphically indicated by each instance where the wavefunction plot

crosses the horizontal zero line (\(r\)-axis), excluding the asymptotic approach to zero at infinite distance (\(r \rightarrow \infty\)).

Step 1: Analyzing Plot A
In Plot A, the wavefunction starts at a high positive value near the nucleus (\(r = 0\)) and decays smoothly and exponentially toward zero as distance increases. It never intersects or crosses the zero axis. This indicates

0 radial nodes (typical of a \(1s\) orbital).

Step 2: Analyzing Plot B
In Plot B, the curve begins at a high value at \(r = 0\), plunges downward to cross the zero baseline into negative values, reaches a local minimum, and then asymptotically rises back toward zero. It crosses the horizontal axis exactly once. This represents an orbital with

1 radial node (characteristic of a \(2s\) orbital).

Step 3: Analyzing Plot C
In Plot C, the curve begins at a high value at \(r = 0\), travels downward to cross the zero axis into negative territory, turns around to cross the zero axis a

second time back into positive values, and then levels out toward zero at extended distances. Since the line crosses the zero line at two separate finite distances, it represents an orbital possessing exactly

2 radial nodes (characteristic of a \(3s\) orbital).

Step 4: Analyzing Plot D
In Plot D, the wavefunction starts exactly at zero at the nucleus (\(r = 0\)), which is typical for non-s orbitals (\(p, d, f\), etc.) because the angular momentum quantum number \(l \gt 0\). The curve increases to a positive peak, drops down to cross the zero baseline once into a negative trough, and then approaches the axis asymptotically. It has exactly

1 radial node. Therefore, Figure

C is the correct plot containing exactly two radial nodes.