A helium atom consists of two electrons orbiting round a nucleus of charge Z=2.But the electrons do not see the full charge Z=2 of the nucleus.Each electron sees the nucleus slightly screened by the other electrons so that the effective charge Z_eff seen by each electron is less than 2.The ionisation potential for a He atom in its ground state is measured experimentally to be 24.46eV.Estimate the effective charge of the nucleus as seen by each electron in the helium ground state.

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To estimate the effective charge (Z_eff) seen by each electron in a helium atom, we can use the concept of screening, which accounts for the repulsion between electrons. In a helium atom, there are two electrons and one nucleus with a charge of +2e. However, due to the presence of both electrons, each electron experiences a reduced effective nuclear charge. Let's break this down step by step.

Understanding Ionization Energy

The ionization energy is the energy required to remove an electron from an atom in its ground state. For helium, this value is experimentally determined to be 24.46 eV. This energy can be related to the effective nuclear charge experienced by the electrons.

Using the Formula for Ionization Energy

In a simplified model, the ionization energy (I) can be approximated using the formula:

I ≈ (Z_eff²) / n²

where:

• Z_eff is the effective nuclear charge
• n is the principal quantum number of the electron (for helium in its ground state, n = 1)

Calculating Effective Charge

For helium, we can rearrange the formula to solve for Z_eff:

Z_eff = sqrt(I * n²)

Substituting the known values:

• I = 24.46 eV
• n = 1

Thus, we have:

Z_eff = sqrt(24.46 eV * 1²) = sqrt(24.46) ≈ 4.95

Considering the Screening Effect

However, this value seems too high because it does not account for the screening effect of the other electron. In a two-electron system like helium, each electron partially shields the nucleus from the other. A more accurate approach involves considering the average screening effect.

Refining the Estimate

In practice, the effective charge can be approximated by considering that each electron screens the nucleus's charge by a factor. A common approximation is that each electron contributes about 0.3 to the screening of the other. Therefore, we can adjust our effective charge calculation:

Z_eff = Z - S

where S is the screening constant. For helium, a reasonable estimate for S is about 0.7 (0.3 from each electron). Thus:

Z_eff = 2 - 0.7 = 1.3

Final Thoughts

In summary, the effective charge seen by each electron in a helium atom is approximately 1.3. This value reflects the balance between the attractive force of the nucleus and the repulsive forces from the other electron, illustrating the importance of electron-electron interactions in multi-electron atoms. Understanding these concepts is crucial for grasping more complex atomic behavior and interactions in chemistry and physics.