Animated Solution for Physics - Atoms and Nuclei: A ${\text{He}}^{+}$ ion is in its first excited state. Its ionisation energy is

When we dive into the quantum world of atoms, the Bohr model serves as our trusty map. It beautifully explains how electrons orbit the nucleus in specific, quantized energy levels. But what happens when we want to kick an electron completely out of the atom? That's where the concept of ionisation energy comes into play. Let's break down this fascinating problem step by step.

The Bohr model isn't just for hydrogen; it works perfectly for any ion that has only one electron left, such as ${\text{He}}^{+}$, ${\text{Li}}^{2+}$, or ${\text{Be}}^{3+}$. These are called hydrogen-like ions. The energy of an electron in the $n^{\text{th}}$ orbit of such an ion is given by the elegant formula:

Here, $Z$ represents the atomic number (which tells us the positive charge of the nucleus), and $n$ is the principal quantum number (the orbit number). The negative sign is crucial—it signifies that the electron is bound to the nucleus. An energy of zero would mean the electron is completely free.

Our problem features a Helium ion (${\text{He}}^{+}$). Since Helium is the second element on the periodic table, its atomic number is $Z=2$.

The question states that the electron is in the first excited state. This is a classic trap where many students stumble! The lowest possible energy level, the ground state, corresponds to $n=1$. When the electron absorbs energy and jumps to the very next available level, it reaches the first excited state. Therefore, the first excited state corresponds to $n=2$.

Now, let's plug our values into the Bohr energy formula to find the energy of the electron in this specific state:

Notice the beautiful symmetry here. The $2^2$ in the numerator (from $Z$) and the $2^2$ in the denominator (from $n$) perfectly cancel each other out.

This means the electron is currently sitting in an energy well of $-13.6\text{ eV}$.

Ionisation Energy

Ionisation energy is defined as the minimum amount of energy required to completely remove an electron from its current state to infinity. At an infinite distance from the nucleus, the electron feels no attractive force, and its potential energy is zero ($E_{\infty }=0$).

To find the required energy, we simply calculate the difference between the final state and the initial state:

And there we have it! It takes exactly $13.6\text{ eV}$ of energy to liberate the electron from the first excited state of a ${\text{He}}^{+}$ ion. This matches option (b) perfectly. Always remember to carefully decode the terminology like "excited state" and keep track of your atomic numbers, and these problems will become a walk in the park!