A sodium atom (Z=11) contains 11 protons in its nucleus....

A sodium atom $(Z=11)$ contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply, because the neutral atom contains 11 electrons instead of a single electron. However, we can apply the model to the outermost electron as an approximation, provided that we use an effective value $Z_{\text {effective }}$ rather than 11 for the number of protons in the nucleus. Since electrons are sometimes between the nucleus and the outermost electron, $Z_{\text {effective }}$ is less than $11 .$ The repulsion from these “ inner" electrons offsets some of the attractive force that the positive nuclear charge exerts on the outermost electron; this is called "shielding." If the 10 inner electrons were permanently within the nucleus, we would have $Z_{\text {effective }}=1 .$ The inner electrons are in orbit, however. (a) Since the orbiting inner electrons spend part of the time on the side of the nucleus opposite to the outermost electron, they are not always between the nucleus and the outermost electron. Considering this fact, is $Z_{\text {effective }}$ less than or greater than $1 ?$ (b) Do you expect the radius of the outermost Bohr orbit to be greater with or without the effect of shielding to reduce $Z$ to a value smaller than $11 ?$ Explain your answers. (a) The ionization energy for the outermost electron in a sodium atom is 5.1 $\mathrm{eV}$. Use the Bohr model with $Z=Z_{\text {effective }}$ to calculate a value for $Z_{\text {effective }}$. (b) Using $Z=11$ and $Z=Z_{\text {effective }},$ determine the corresponding two values for the radius of the outermost Bohr orbit. Verify that your answers are consistent with your answers to the Concept Questions.

A sodium atom $(Z=11)$ contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply, because the neutral atom contains 11 electrons instead of a single electron. However, we can apply the model to the outermost electron as an approximation, provided that we use an effective value $Z_{\text {effective }}$ rather than 11 for the number of protons in the nucleus. Since electrons are sometimes between the nucleus and the outermost electron, $Z_{\text {effective }}$ is less than $11 .$ The repulsion from these “ inner" electrons offsets some of the attractive force that the positive nuclear charge exerts on the outermost electron; this is called "shielding." If the 10 inner electrons were permanently within the nucleus, we would have $Z_{\text {effective }}=1 .$ The inner electrons are in orbit, however. (a) Since the orbiting inner electrons spend part of the time on the side of the nucleus opposite to the outermost electron, they are not always between the nucleus and the outermost electron. Considering this fact, is $Z_{\text {effective }}$ less than or greater than $1 ?$ (b) Do you expect the radius of the outermost Bohr orbit to be greater with or without the effect of shielding to reduce $Z$ to a value smaller than $11 ?$ Explain your answers.
(a) The ionization energy for the outermost electron in a sodium atom is 5.1 $\mathrm{eV}$. Use the Bohr model with $Z=Z_{\text {effective }}$ to calculate a value for $Z_{\text {effective }}$. (b) Using $Z=11$ and $Z=Z_{\text {effective }},$ determine the corresponding two values for the radius of the
outermost Bohr orbit. Verify that your answers are consistent with your answers to the Concept Questions.

Key Concepts

Relationship Between Orbital Radius and Nuclear Charge

Within the Bohr model, the radius of an electron's orbit is inversely proportional to the effective nuclear charge. A higher effective nuclear charge pulls the electron closer to the nucleus, resulting in a smaller orbit, while a reduced effective nuclear charge due to shielding leads to a larger orbital radius. This relationship is fundamental to understanding atomic structure and the behavior of electrons in different energy levels.

Ionization Energy

Ionization energy is the amount of energy required to remove an electron from an atom or ion in its gaseous state. It provides insight into the strength of the attractive forces within an atom. In the context of the Bohr model and effective nuclear charge, changes in ionization energy reflect how modifications in the net nuclear attraction (due to shielding) influence the energy levels of electrons.

Electron Shielding

Electron shielding, or screening, refers to the phenomenon where inner electrons partially block the attractive force of the nucleus from reaching outer electrons. This reduction in nuclear attraction causes the outermost electrons to experience a lower effective nuclear charge than the actual nuclear charge. Shielding is an important factor in explaining variations in atomic properties, such as atomic size and ionization energy.

Effective Nuclear Charge

Effective nuclear charge (Z_effective) is the net positive charge experienced by an electron in an atom. It is lower than the actual nuclear charge because of the shielding effect exerted by inner electrons. This concept is crucial for understanding how electrons, especially those farther from the nucleus, experience a reduced attractive force due to the counteracting repulsion from intervening electrons.

Bohr Model

The Bohr model is an early quantum theory that describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits. These orbits have quantized energy levels, meaning an electron can only occupy certain allowed orbit radii and energy states. This model, although simplified, provides foundational insights into atomic structure, energy quantization, and spectral lines.

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