Animated Solution for Physics - Atoms and Nuclei: The electron in a hydrogen atom makes a transition from an excited state to the ground state. Which of the following statement is true ?

Imagine an electron in a hydrogen atom, currently residing in an excited state. When it makes a transition down to the ground state, it moves closer to the nucleus. In quantum mechanical terms, this means its principal quantum number, $n$, decreases.

This simple geometric shift—moving from a larger orbit to a smaller one—has profound implications for the electron's energy distribution. To understand exactly what happens, we need to break down the electron's total energy into its kinetic and potential components.

Let's first look at the electron's kinetic energy. According to Bohr's model, the kinetic energy $K_n$ of an electron in the $n^{\text{th}}$ orbit is given by:

Notice that the kinetic energy is inversely proportional to the square of the principal quantum number $n$. Since $n$ is decreasing as the electron jumps down to the ground state, the fraction $\frac{1}{n^2}$ must increase.

Physically, this makes perfect sense. As the electron gets closer to the positively charged nucleus, the electrostatic force of attraction becomes stronger. To maintain a stable orbit and avoid spiraling into the nucleus, the electron must revolve faster. Therefore, its kinetic energy increases.

Now, what about the potential energy? The potential energy $U_n$ of the electron is given by:

Like kinetic energy, potential energy is also inversely proportional to $n^2$. However, there is a crucial catch here—it has a negative sign!

As $n$ decreases, the magnitude $\frac{1}{n^2}$ increases. But because of the negative sign, the overall value drops deeper into negative territory. For example, moving from $-1.51\text{ eV}$ to $-13.6\text{ eV}$ is a decrease in value. Therefore, the potential energy decreases. The electron is falling deeper into the electrostatic potential well created by the nucleus.

The Big Picture

Finally, let's examine the total energy. The total energy $E_n$ is simply the sum of the kinetic and potential energies, which yields:

The total energy is exactly the negative of the kinetic energy, and it follows the exact same mathematical trend as the potential energy. Because of the negative sign, as $n$ decreases, the total energy also decreases.

Where does this lost energy go? It is emitted into the universe in the form of an electromagnetic photon!

Putting it all together: as the electron drops to the ground state, it speeds up (kinetic energy increases), but it falls deeper into the potential well (potential and total energy decrease). This perfectly matches our first option, making it the correct answer.