The z component of the angular momentum of an atom, Lz, is...

The $z$ component of the angular momentum of an atom, $L_{z}$, is measured as a function of the angle that the angular momentum vector $\vec{L}$ makes with the positive $z$ axis, $\theta .$ The resulting data are given in the table, where the angular momentum is measured in units of $\hbar$. $$ \begin{array}{cc} \hline \boldsymbol{\theta}\left({ }^{\circ}\right) & \boldsymbol{L}_{z}(\hbar) \\ \hline 63.7 & 1.97 \\ 79.2 & 1.05 \\ 90.4 & -0.03 \\ 102 & -0.95 \\ 118 & -2.02 \\ \hline \end{array} $$ Make a plot of $L_{z}$ as a function of $\cos \theta$. Using a linear "best fit" to the data, determine (a) the magnitude of the angular momentum vector in units of $\hbar$ and (b) the corresponding angular momentum quantum number.

The $z$ component of the angular momentum of an atom, $L_{z}$, is measured as a function of the angle that the angular momentum vector $\vec{L}$ makes with the positive $z$ axis, $\theta .$ The resulting data are given in the table, where the angular momentum is measured in units of $\hbar$.
$$
\begin{array}{cc}
\hline \boldsymbol{\theta}\left({ }^{\circ}\right) & \boldsymbol{L}_{z}(\hbar) \\
\hline 63.7 & 1.97 \\
79.2 & 1.05 \\
90.4 & -0.03 \\
102 & -0.95 \\
118 & -2.02 \\
\hline
\end{array}
$$
Make a plot of $L_{z}$ as a function of $\cos \theta$. Using a linear "best fit" to the data, determine (a) the magnitude of the angular momentum vector in units of $\hbar$ and (b) the corresponding angular momentum quantum number.

Key Concepts

Effective Nuclear Charge and Shielding

Effective nuclear charge is the net positive charge experienced by an electron in an atom, taking into account the full nuclear charge and the shielding effect of the other electrons. As electrons are removed, remaining electrons feel a stronger attraction to the nucleus because there is less shielding. This increase in effective nuclear charge contributes to the greater energy required to remove additional electrons, particularly when transitioning from valence to core electrons.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom or ion in the gaseous state. It is a fundamental concept that reflects how strongly an electron is bound to the nucleus. Successive ionization energies increase because with each electron removed, the remaining electrons experience a stronger effective nuclear charge and are held more tightly, thus requiring more energy for removal.

Electron Configuration

Electron configuration describes the distribution of electrons in an atom's orbitals. It explains why removing electrons from outer (valence) shells generally requires less energy than removing electrons from inner (core) shells. After the valence electrons have been removed, the next electrons come from a more stable, closed-shell configuration which is harder to remove, resulting in significantly higher ionization energies.

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