Animated Solution for Physics - Atoms and Nuclei: Hydrogen (${}_1{H}^1$), deuterium (${}_1{H}^2$), singly ionised helium (${}_{2}H{e}^4{)}^{+}$ and doubly ionised lithium (${}_{3}L{i}^8{)}^{++}$ all have one electron around the nucleus. Consider an electron transition form $n=2$ to $n=1$. If the wavelengths of emitted radiation are ${\lambda }_1,{\lambda }_2,{\lambda }_3$ and ${\lambda }_4$ respectively for four elements, then approximately which one of the following is correct?

Imagine a universe where every element behaves completely differently, with no underlying mathematical harmony. Thankfully, our universe isn't like that! Niels Bohr showed us that as long as an atom or ion has exactly one electron, it follows a beautifully predictable set of rules, regardless of how heavy or highly charged its nucleus is.

In this problem, we are introduced to a fascinating lineup: Hydrogen (${}_1{H}^1$), Deuterium (${}_1{H}^2$), singly ionized Helium (${}_{2}H{e}^4{)}^{+}$, and doubly ionized Lithium (${}_{3}L{i}^8{)}^{++}$. At first glance, they seem like a diverse bunch. However, if you count their electrons, you'll realize they all have exactly one electron orbiting their respective nuclei. This makes them all hydrogen-like species, allowing us to apply Bohr's model and the Rydberg formula universally across them.

When an electron transitions from a higher energy state to a lower one, it sheds its excess energy by emitting a photon. The wavelength $\lambda$ of this photon is governed by the Rydberg formula:

Here, $R$ is the Rydberg constant, $Z$ is the atomic number (the number of protons in the nucleus), and $n_1$ and $n_2$ are the principal quantum numbers of the lower and higher energy levels, respectively.

The problem gives us a massive shortcut: for all four species, the electron transitions from the first excited state ($n_2=2$) to the ground state ($n_1=1$). Because this transition is identical across the board, the entire bracketed term in our formula becomes a constant:

Since $R$ and the bracketed term are constants, we can strip away the clutter and establish a direct proportionality. The inverse of the wavelength is directly proportional to the square of the atomic number:

This elegant relation is all we need to solve the problem!

Let's apply our proportionality to each species one by one.

1. Hydrogen and Deuterium: Both Hydrogen and its heavier isotope, Deuterium, have an atomic number $Z=1$.

Since both are proportional to $1$, we can conclude that ${\lambda }_1={\lambda }_2$.

2. Singly Ionized Helium ($H{e}^{+}$): Helium has an atomic number $Z=2$.

To make comparisons easier, let's clear the fraction by multiplying both sides by 4:

3. Doubly Ionized Lithium ($L{i}^{++}$): Lithium has an atomic number $Z=3$.

Again, clearing the fraction gives us:

Look at what we've achieved! We have expressed all our wavelengths in terms of the same proportionality constant (which is $1$ in our relative scale).

Since ${\lambda }_1$, ${\lambda }_2$, $4{\lambda }_3$, and $9{\lambda }_4$ are all proportional to the exact same value, we can confidently equate them all together:

This perfectly matches option (c).

The Isotope Effect

While our calculation shows ${\lambda }_1={\lambda }_2$, a curious physics student might ask: Are they exactly identical in the real world?

The answer is a resounding no. In our derivation, we assumed the nucleus is infinitely massive compared to the electron, meaning it remains perfectly stationary. In reality, both the electron and the nucleus orbit their common center of mass. To account for this, advanced physics replaces the electron mass with the reduced mass ($\mu$) of the system:

Because the Deuterium nucleus is roughly twice as heavy as the Hydrogen nucleus, its reduced mass is slightly larger. This makes the effective Rydberg constant for Deuterium slightly larger than that of Hydrogen, resulting in a slightly shorter wavelength (${\lambda }_2<{\lambda }_1$). This subtle shift is known as the isotope effect, a beautiful reminder that physics always has deeper layers waiting to be explored!