1a) The binding energy of a calcium 2p electron is -349.7 eV. What would be the effective nuclear charge experienced by a calcium 2p electron? (RH = 13.6 eV)

To determine the effective nuclear charge experienced by a calcium 2p electron, we can use the concept of binding energy and the Rydberg constant for hydrogen-like atoms. The binding energy of an electron in an atom is related to the effective nuclear charge (Z_eff) that the electron experiences. For a hydrogen-like atom, the binding energy can be expressed using the formula:

Understanding the Relationship Between Binding Energy and Effective Nuclear Charge

The binding energy (E) of an electron in a hydrogen-like atom is given by the equation:

E = -Z_eff² * RH / n²

Where:

• E is the binding energy of the electron (in eV).
• Z_eff is the effective nuclear charge experienced by the electron.
• RH is the Rydberg constant for hydrogen (approximately 13.6 eV).
• n is the principal quantum number of the electron's orbital (for 2p, n = 2).

Calculating the Effective Nuclear Charge

Given that the binding energy of the calcium 2p electron is -349.7 eV, we can rearrange the equation to solve for Z_eff:

Z_eff = sqrt(-E * n² / RH)

Substituting the values we have:

• E = -349.7 eV
• n = 2
• RH = 13.6 eV

Now, plug these values into the equation:

Z_eff = sqrt(-(-349.7 eV) * (2)² / 13.6 eV)

This simplifies to:

Z_eff = sqrt(349.7 * 4 / 13.6)

Calculating the values inside the square root:

Z_eff = sqrt(1398.8 / 13.6)

Z_eff = sqrt(102.0)

Z_eff ≈ 10.1

Interpreting the Result

The effective nuclear charge experienced by a calcium 2p electron is approximately 10.1. This value indicates that the 2p electron feels a significant attraction from the nucleus, but it is also influenced by the shielding effect of the other electrons in the atom. In calcium, which has 20 electrons, the inner electrons shield the outer electrons from the full nuclear charge, resulting in a lower effective nuclear charge than the actual nuclear charge of +20.

This calculation illustrates how binding energy and effective nuclear charge are interconnected, providing insight into the electron's behavior in an atom. Understanding these concepts is crucial for grasping the fundamentals of atomic structure and electron interactions.