Solution Overview

Uranium (atomic number 92) has the ground-state electron configuration [Rn] 5f³ 6d¹ 7s². For the U(III) ion, three electrons are removed from the outermost orbitals (7s² and 6d¹), yielding [Rn] 5f³, where the valence orbital is 5f.

Radial nodes in an orbital are calculated using the formula: number of radial nodes = n – l – 1, where n is the principal quantum number and l is the azimuthal quantum number.

For the 5f valence orbital, n = 5 and l = 3 (f orbitals), so radial nodes = 5 – 3 – 1 = 1.

Detailed Solution

Neutral U loses electrons sequentially from 7s (highest energy in actinides) then 6d, leaving 5f electrons as valence for U³⁺. The 5f orbital defines chemical reactivity in this ion, consistent with f-block actinide behavior.

Radial Node Formula Application

• Principal quantum number (n) = 5
• Azimuthal quantum number (l) = 3
• Radial nodes = 5 – 3 – 1 = 1

This matches GATE Life Sciences 2023 data, confirming 1 radial node.

Step-by-Step Calculation

1. Neutral U: [Rn] 5f³ 6d¹ 7s²
2. U(III): Remove 7s² + 6d¹ → [Rn] 5f³
3. Valence orbital: 5f (n=5, l=3)
4. Nodes calculation:
   • Radial nodes = 1
   • Angular nodes = l = 3
   • Total nodes = n-1 = 4

Radial Nodes Formula Explained

Radial nodes represent spherical regions of zero electron probability. Use:

Radial nodes = n – l – 1

Node Types Summary