Animated Solution for Physics - Atoms and Nuclei: A hydrogen atom and a ${\text{Li}}^{2+}$ ion are both in the second excited state. If $l_H$ and $l_{Li}$ are their respective electronic angular momenta, and $E_H$ and $E_{Li}$ their respective energies, then

Let's begin by visualizing the Hydrogen atom and the Lithium ion (${\text{Li}}^{2+}$) side by side. The problem states that both are in the second excited state.

It is crucial to remember the terminology of energy states. The ground state corresponds to $n=1$. The first state above it is the first excited state ($n=2$), which means the second excited state corresponds to a principal quantum number of $n=3$ for both atoms.

First, let's compare their electronic angular momenta. According to Bohr's quantization postulate, the angular momentum $l$ of an electron in the $n^{\text{th}}$ orbit is an integral multiple of $\frac{h}{2\pi }$. The formula is given by:

Notice how this formula depends only on the principal quantum number $n$. It does not depend on the atomic number $Z$ or the mass of the nucleus. Since both the Hydrogen atom and the Lithium ion have their electron in the $n=3$ orbit, we can directly conclude that their angular momenta must be exactly the same.

The heavier, more highly charged nucleus of Lithium doesn't change the angular momentum of the electron in a given orbit.

Now, let's evaluate their energies. The total energy of an electron in the $n^{\text{th}}$ orbit of a hydrogen-like species is given by the expression:

Here, the atomic number $Z$ plays a crucial role. For Hydrogen, the atomic number $Z_H$ is $1$. But for the Lithium ion, $Z_{Li}$ is $3$. Because the magnitude of energy $|E_n|$ is directly proportional to $Z^2$ (for a constant $n$), we can compare them easily.

This means the energy magnitude for Lithium will be three squared, which is nine times the energy magnitude of Hydrogen. Therefore, $|E_{Li}|=9|E_H|$, which implies $|E_H|<|E_{Li}|$.

Putting it all together, the angular momenta are strictly equal because they depend only on $n$, but the magnitude of energy for the Lithium ion is strictly greater than that of the Hydrogen atom because it scales with $Z^2$. This perfectly matches option (b).