Match Atomic Orbitals with Radial and Nodal Planes

To determine the correct matching between the orbitals in List-I and their corresponding nodes in List-II, we use the standard formulas for the radial nodes and angular nodes (nodal planes) of an atomic orbital:

1. Number of radial nodes $= n - l - 1$
2. Number of angular nodes (nodal planes) $= l$

where $n$ is the principal quantum number and $l$ is the azimuthal quantum number ($l = 0$ for $s$-orbital, $l = 1$ for $p$-orbital, $l = 2$ for $d$-orbital).

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Step-by-step Evaluation:

1. Orbital A: $2s$

   • $n = 2, \, l = 0$
   • Radial nodes $= 2 - 0 - 1 = 1$
   • Nodal planes $= l = 0$ (No nodal plane)
   • Matching: A $\rightarrow$ IV ($1\text{ Radial node} + \text{No nodal plane}$)

2. Orbital B: $3s$

   • $n = 3, \, l = 0$
   • Radial nodes $= 3 - 0 - 1 = 2$
   • Nodal planes $= l = 0$ (No nodal plane)
   • Matching: B $\rightarrow$ III ($2\text{ Radial nodes} + \text{No nodal plane}$)

3. Orbital C: $3p$

   • $n = 3, \, l = 1$
   • Radial nodes $= 3 - 1 - 1 = 1$
   • Nodal planes $= l = 1$ (one nodal plane)
   • Matching: C $\rightarrow$ II ($1\text{ Radial node} + \text{one nodal plane}$)

4. Orbital D: $4d$

   • $n = 4, \, l = 2$
   • Radial nodes $= 4 - 2 - 1 = 1$
   • Nodal planes $= l = 2$ (two nodal planes)
   • Matching: D $\rightarrow$ I ($1\text{ Radial node} + \text{two nodal planes}$)

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Conclusion:

The correct matching combination is: $\text{A-IV, B-III, C-II, D-I}$

Hence, the correct option is D.