Question

The number of radial nodes in 3s and 2p orbitals, respectively are

• A. \( 2 : 2 \)
• B. \( 2 : 0 \)
• C. \( 0 : 0 \)
• D. \( 3 : 2 \)

Hint:

Radial nodes formula: \( n - l - 1 \). Quick reference table: \[ \begin{array}{|c|c|c|c|} \hline \text{Orbital} & n & l & \text{Radial nodes} \hline 1s & 1 & 0 & 0 2s & 2 & 0 & 1 2p & 2 & 1 & 0 3s & 3 & 0 & 2 3p & 3 & 1 & 1 3d & 3 & 2 & 0 \hline \end{array} \]

Answer 1

The Correct Option is B

Concept: In quantum mechanics, nodes are regions where the probability of finding an electron is zero.
• Total nodes = \( n - 1 \)
• Angular nodes (also called nodal planes) = \( l \)
• Radial nodes (spherical surfaces) = \( n - l - 1 \) Here, \( n \) = principal quantum number, \( l \) = azimuthal (angular momentum) quantum number.

Step 1: Identify quantum numbers for 3s orbital.
• For s orbital: \( l = 0 \)
• Given orbital: 3s \( \Rightarrow \) \( n = 3 \), \( l = 0 \) Radial nodes = \( n - l - 1 = 3 - 0 - 1 = 2 \). Verification:
• Total nodes = \( n - 1 = 2 \)
• Angular nodes = \( l = 0 \)
• Radial nodes = Total nodes - Angular nodes = \( 2 - 0 = 2 \)\checkmark

Step 2: Identify quantum numbers for 2p orbital.
• For p orbital: \( l = 1 \)
• Given orbital: 2p \( \Rightarrow \) \( n = 2 \), \( l = 1 \) Radial nodes = \( n - l - 1 = 2 - 1 - 1 = 0 \). Verification:
• Total nodes = \( n - 1 = 1 \)
• Angular nodes = \( l = 1 \)
• Radial nodes = Total nodes - Angular nodes = \( 1 - 1 = 0 \)\checkmark

Step 3: Write the required ratio. \[ \text{3s radial nodes : 2p radial nodes} = 2 : 0 \]

Step 4: Physical interpretation.
• 3s orbital has 2 spherical radial nodes where the radial wavefunction changes sign.
• 2p orbital has no radial node; its only node is the angular node (nodal plane through the nucleus).