Animated Solution for Physics - Atoms and Nuclei: Hydrogen atom is excited from ground state to another state with principal quantum number equal to 4. Then, the number of spectral lines in the emission spectra will be

Imagine you are standing on the fourth floor of a building and you need to get to the ground floor. You could jump straight down (not recommended!), or you could take the stairs, stopping at the third floor, then the second, and finally the ground floor. Or maybe you skip a floor and jump from the third to the first.

This is exactly what happens to an electron in a hydrogen atom when it is excited to a higher energy level. In our problem, the electron has been pumped up to the state with principal quantum number $n=4$.

Excited states are highly unstable. The electron desperately wants to return to the cozy, low-energy ground state ($n=1$). As it drops to lower energy levels, it sheds its excess energy by emitting photons of light. Each unique jump between two energy levels produces a photon of a specific wavelength, which appears as a distinct "spectral line" in the emission spectrum.

Let's trace all the possible paths our electron can take: 1. Direct Jumps from $n=4$: The electron can jump directly to $n=3$, $n=2$, or $n=1$. That gives us 3 distinct spectral lines. 2. Jumps from $n=3$: If the electron stopped at $n=3$, it can then jump to $n=2$ or $n=1$. That adds 2 more lines. 3. Jumps from $n=2$: Finally, if it finds itself at $n=2$, it has only 1 path left: down to $n=1$.

Adding these up, we get $3+2+1=6$ total possible transitions.

While counting paths is easy for $n=4$, imagine doing this for $n=10$! Thankfully, mathematics provides an elegant shortcut. The number of spectral lines is simply the number of ways you can choose any 2 energy levels out of the $n$ available levels. In combinatorics, this is represented as ${}^n{C}_2$.

The formula for the total number of emitted spectral lines $N$ from the $n^{\text{th}}$ state is:

Let's substitute our given value of $n=4$ into this master equation:

Both our logical counting and our mathematical formula lead us to the same beautiful result. When a sample of hydrogen atoms is excited to the $n=4$ state, the collective de-excitation of all those atoms will produce exactly 6 distinct spectral lines in the emission spectrum.