Question

The figures below show: Which of the following points in Figure 2 most accurately represents the nodal surface shown in Figure 1?

• A. C
• B. D
• C. B
• D. A

Hint:

Nodes are identified where the wave function becomes exactly zero, not where probability is merely low.

Answer 1

The Correct Option is C

Concept: Nodes in atomic orbitals A {node} is a region in space where the probability of finding an electron is zero. Mathematically, this occurs where the {wave function} \(\psi = 0\). For hydrogen-like orbitals:
Radial nodes occur where the radial part of the wave function becomes zero.
The number of radial nodes is given by: \[ n - l - 1 \]
Step 1: Identify nodes in the \(2s\) orbital For the \(2s\) orbital: \[ n = 2,\quad l = 0 \] \[ \text{Number of radial nodes} = 2 - 0 - 1 = 1 \] Thus, the \(2s\) orbital has one spherical nodal surface, as shown in Figure 1.
Step 2: Relation between probability density and wave function
Electron probability density is proportional to \(|\psi|^2\).
A nodal surface corresponds to \(\psi = 0\), not merely a minimum. Hence, we must locate the point in Figure 2 where the wave function crosses the \(x\)-axis.
Step 3: Analyse Figure 2 From the graph of \(\psi_{2s}(x)\):
Point \(A\): \(\psi\) is positive and maximum.
Point \(B\): \(\psi = 0\) (crosses the axis).
Point \(C\): \(\psi\) is negative and minimum.
Point \(D\): \(\psi\) is negative but non-zero. Only point \(B\) satisfies the condition: \[ \psi_{2s} = 0 \]
Step 4: Connect Figure 1 and Figure 2 The spherical nodal surface in Figure 1 corresponds to a specific radius where: \[ \psi_{2s}(r) = 0 \] When represented along a single axis (Figure 2), this node appears as a zero-crossing of the wave function. Final Answer: \[ \boxed{\text{Point B}} \]