Atomic Structure

To correctly match the items from List I with List II, we need to understand the energy values associated with specific states and properties of hydrogen (H) and helium (He) atoms.

1. Understanding the Given Quantities:
   
   • Energy of ground state of He (A): This is the total energy of a helium atom in its lowest energy state.
   • Potential energy of 1s orbit of H-atom (B): This refers to the potential energy of the electron in the first energy level (1s) of a hydrogen atom.
   • Kinetic Energy of II excited state of He (C): This is the kinetic energy of an electron in the second excited state of a helium atom.
   • Ionization potential of He (D): The energy required to remove an electron from a helium atom.
2. Known Energy Values:
   
   • Hydrogen Atom:
     
     • Ground state energy: .
     • Potential energy ( ) is twice the total energy: .
     • Kinetic energy ( ) is equal in magnitude but opposite in sign to potential energy: .
   • Helium Atom:
     
     • Ground state energy: .
     • Ionization potential: (energy required to remove one electron).
     • For kinetic energy of an excited state, specific values depend on the energy level, but given the options, it aligns with .
3. Matching the Items:
   
   • A. Energy of ground state of He:
     Ground state energy of He = (IV)
   • B. Potential energy of 1s orbit of H-atom:
     (II)
   • C. Kinetic Energy of II excited state of He:
     (I)
   • D. Ionization potential of He:
     Ionization potential = (III)

Therefore, the correct matching is:
(A) - (IV), (B) - (II), (C) - (I), (D) - (III).

Step 1: Understanding Effective Nuclear Charge.
The effective nuclear charge ( ) is the net charge experienced by an electron in a multi-electron atom. It is calculated using the formula: where is the atomic number and is the shielding constant, which represents the effect of other electrons in the atom in shielding the nucleus.

Step 2: Calculation for Copper (Cu).
For copper ( ), the 4s electron is shielded by the inner electrons, especially the 1s, 2s, and 2p electrons. The shielding constant for the 4s electron can be estimated using Slater's rules or experimental values. For copper, the shielding constant is approximately 1.30. Therefore, the effective nuclear charge is:

Step 3: Conclusion.
Thus, the effective nuclear charge experienced by the 4s electron of Cu is , which corresponds to option (1).

Step 1: Number of Nodes in Orbitals.
The number of radial nodes in an orbital is given by the formula: where: - is the principal quantum number - is the azimuthal quantum number For the 4f orbital: - (because it is the 4th shell) - (for f-orbitals, ) Thus, the number of radial nodes is:

Step 2: Conclusion.
Hence, the correct number of radial nodes in the 4f orbital is , which corresponds to option (3).

Radial nodes are calculated as n - l - 1. For a 4f orbital, where n = 4 and l = 3, the number of radial nodes is:

4 - 3 - 1 = 1