Animated Solution for Physics - Atoms and Nuclei: The diagram shows the energy levels for an electron in a certain atom. Which transition shown represents the emission of a photon with the most energy?

The problem presents us with a classic energy level diagram of an atom, showing four distinct energy states ($n=1$ to $n=4$) and four possible electron transitions labeled I, II, III, and IV. Our mission is to identify which of these transitions results in the emission of a photon with the maximum possible energy.

Before we dive into any calculations, we must carefully read the question. It specifically asks for the emission of a photon.

In the quantum world, an electron emits a photon when it loses energy, which means it must drop from a higher energy level to a lower one. Visually, this is represented by a downward-pointing arrow.

If we look at transition I, the arrow points upwards from $n=1$ to $n=2$. This indicates that the electron is absorbing energy from an incoming photon to jump to a higher state. Since it's not an emission process, transition I is immediately eliminated from our list of candidates.

For the remaining downward transitions (II, III, and IV), we need a way to compare the energy they release. The energy $E$ of an emitted photon is given by the Rydberg formula:

Since we are dealing with the same atom (so $Z$ is constant) and we only need to find the relative maximum, we can strip away the constants and focus purely on the proportional relationship:

Let's evaluate this bracket for each contender.

Transition IV ($n=4\to n=3$): The electron drops from the fourth orbit to the third. Finding a common denominator (144), we get:

Transition II ($n=4\to n=2$): Here, the electron falls from the fourth orbit down to the second. This simplifies nicely: Notice that this is already significantly larger than transition IV.

Transition III ($n=3\to n=1$): Finally, the electron makes a massive leap from the third orbit straight down to the ground state.

Comparing our calculated proportional values: - $E_{IV}\approx 0.048$ - $E_{II}=0.1875$ - $E_{III}\approx 0.888$

It is abundantly clear that Transition III releases the most energy by a massive margin. Therefore, option (a) is the correct answer.

While the math proves our answer rigorously, there is a powerful conceptual shortcut that can save you precious time in competitive exams like JEE.

You must remember that energy levels in an atom are not evenly spaced. As the principal quantum number $n$ increases, the energy levels get progressively closer together. The energy gap between the ground state ($n=1$) and the first excited state ($n=2$) is the largest gap in the entire atom. For hydrogen, this single gap is $10.2\text{ eV}$, while the entire remaining energy from $n=2$ to $n=\infty$ is only $3.4\text{ eV}$!

Because transition III is the only emission that drops all the way down to $n=1$, it crosses this massive initial energy gap. This physical reality guarantees that it will have the highest energy among the given choices, allowing you to solve the problem in seconds without writing down a single fraction.