Animated Solution for Physics - Atoms and Nuclei: If the binding energy of the electron in a hydrogen atom is $13.6\text{ eV}$, the energy required to remove the electron from the first excited state of ${\text{Li}}^{2+}$ is

When we dive into the quantum world of atoms, Bohr's model provides a beautifully simple yet powerful framework for understanding single-electron species. These are known as hydrogen-like ions, such as ${\text{He}}^{+}$ or ${\text{Li}}^{2+}$. The problem asks us to find the energy required to completely remove an electron from the first excited state of a ${\text{Li}}^{2+}$ ion.

The energy of an electron in the $n^{\text{th}}$ orbit of any hydrogen-like ion is given by the generalized Bohr formula:

Here, $Z$ represents the atomic number of the nucleus, and $n$ is the principal quantum number (the orbit number). The negative sign is crucial—it indicates that the electron is bound to the nucleus. To free the electron, we must supply positive energy to overcome this binding.

Before we plug in the numbers, we must carefully decode the terminology. The question specifies the first excited state of ${\text{Li}}^{2+}$.

For Lithium, the atomic number is $Z=3$.

The ground state is always $n=1$. Therefore, the first state above the ground state—the first excited state—corresponds to $n=2$. This is a classic trap where many students mistakenly use $n=1$.

Now, let's substitute our values into the master equation to find the energy of the electron while it is sitting in this $n=2$ orbit:

Dividing $-13.6$ by $4$ gives $-3.4$. Multiplying this by $9$ yields:

This means the electron is bound to the nucleus with an energy of $-30.6\text{ eV}$.

To "remove the electron" means to ionize the atom. In quantum terms, this requires taking the electron from its current state ($n=2$) all the way to infinity ($n=\infty$), where it is completely free from the nucleus's electrostatic pull. At infinity, the potential energy is zero, so $E_{\infty }=0$.

The energy required for this transition is simply the difference between the final and initial states:

Thus, we must supply exactly $30.6\text{ eV}$ of energy to liberate the electron. This elegant calculation highlights the predictive power of Bohr's model for hydrogen-like species.