Animated Solution for Physics - Atoms and Nuclei: If $13.6\text{ eV}$ energy is required to ionise the hydrogen atom, then the energy required to remove an electron from $n=2$ is

Imagine a tiny solar system. At the center lies a heavy, positively charged proton, and orbiting around it is a lightweight, negatively charged electron. This is the hydrogen atom. According to Niels Bohr's brilliant model, the electron cannot just orbit anywhere; it is restricted to specific, quantized paths called energy levels or shells.

The energy of an electron in any given shell $n$ is not arbitrary. It is governed by a beautifully simple yet profound equation:

Notice the negative sign? That is crucial. It signifies that the electron is bound to the nucleus. It is trapped in a potential energy well. To free the electron—to push it out of the well and into the vastness of infinity where its energy would be zero—we must supply energy. This required energy is known as the ionization energy.

The problem states that $13.6\text{ eV}$ of energy is required to ionize the hydrogen atom. This perfectly aligns with our formula. In its natural, unexcited state (the ground state), the electron resides in the lowest energy level, $n=1$.

Plugging $n=1$ into our formula gives an energy of $-13.6\text{ eV}$. To bring this energy up to $0\text{ eV}$ (the threshold of freedom), we must add exactly $+13.6\text{ eV}$.

But what if the electron is already excited? What if it has absorbed some energy and jumped to the second orbit, $n=2$? The question asks us to find the energy required to remove the electron from this specific state.

First, let's find out how much energy the electron has while sitting in the $n=2$ orbit. We use our trusty formula:

So, in the second orbit, the electron is bound with an energy of $-3.4\text{ eV}$. It is closer to the edge of the well than it was in the ground state.

To completely remove the electron from this state to infinity ($E_{\infty }=0\text{ eV}$), we need to supply enough energy to overcome this binding energy.

Therefore, the energy required to remove an electron from the $n=2$ state is exactly $3.4\text{ eV}$. It takes significantly less energy to free an electron that is already in an excited state compared to one in the ground state.