Animated Solution for Chemistry - Atomic Structure: Which of the following is the energy of a possible excited state of hydrogen?

The journey into the quantum world of the hydrogen atom is fascinating. When we talk about the energy levels of an atom, we are essentially looking at the allowed "orbits" or states where an electron can exist.

The Electron's Home

Imagine a hydrogen atom at room temperature. Its single electron is most likely found in the lowest possible energy state, which we call the ground state. This state corresponds to the principal quantum number $n=1$. Because the population of electrons is maximum at this ground state, any excitation—meaning the electron absorbing energy and jumping to a higher level—will most likely start from here.

Excitation

When the electron absorbs just the right amount of energy, it jumps to a higher energy level. The very next available level is $n=2$. This is the first possible excited state. It's like stepping onto the first rung of a ladder from the ground.

To find the exact energy of this state, we rely on the Bohr model's energy formula for a hydrogen-like species:

Here, $Z$ is the atomic number, and $n$ is the principal quantum number of the orbit. The negative sign is crucial—it indicates that the electron is bound to the nucleus. You would need to supply positive energy to free it completely (which would bring its energy to zero at $n=\infty$).

For our hydrogen atom, the atomic number $Z=1$. We are looking for the energy of the first excited state, so we plug in $n=2$:

And there we have it! The energy of the electron when it is in the first possible excited state is $-3.4\text{ eV}$. This elegant calculation not only gives us the answer but also beautifully demonstrates the quantized nature of atomic energy levels.