( a ) Compute the energy of an electron in the n=1 ( K...

( $a$ ) Compute the energy of an electron in the $n=1$ ( $K$ shell) of tungsten, using $Z-1$ for the effective nuclear charge. ( $b$ ) The experimental result for this energy is $69.5 \mathrm{keV}$. Assume that the effective nuclear charge is $Z-\sigma,$ where $\sigma$ is called the screening constant, and calculate $\sigma$ from the experimental result.

Key Concepts

Screening Constant

The screening constant is a numerical value used to quantify the extent to which inner electrons offset the full nuclear charge, thereby reducing the effective nuclear charge felt by an outer electron. It is an important parameter in calculating energy levels and explains the deviation of actual energies from those predicted by simple models that assume no electron-electron repulsion.

Energy Levels in Hydrogen-like Atoms

This concept involves the quantized energy states of an electron in an atom that can be approximated by the hydrogen atom model. The energy levels are determined primarily by the principal quantum number (n) and the effective nuclear charge, and they follow a formula in which the energy is inversely proportional to the square of n. This idea is key to understanding electron binding energies in atoms where a single electron approximation is valid.

Effective Nuclear Charge

Effective nuclear charge is the net positive charge experienced by an electron in an atom. It is reduced from the full nuclear charge by the repulsion caused by other electrons, especially those in inner shells. This concept is essential for explaining and calculating how strongly an electron is attracted to the nucleus, and it plays a critical role in predicting trends in atomic properties such as ionization energies and electron affinities.

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