Animated Solution for Chemistry - Atomic Structure: The shortest wavelength of H atom in the Lyman series is ${\lambda }_1$. The longest wavelength in the Balmer series of ${\text{He}}^{+}$ is

When an electron in an atom or a hydrogen-like ion transitions from a higher energy state to a lower one, it sheds its excess energy in the form of a photon. The energy of this emitted photon is given by the famous Planck-Einstein relation:

This equation reveals a beautiful, inverse relationship: the greater the energy gap ($\Delta E$) between the two shells, the shorter the wavelength ($\lambda$) of the emitted light. Conversely, a tiny energy hop produces a long, stretched-out wavelength. To calculate the exact wavelength for any transition, we rely on the Rydberg formula:

Here, $R_H$ is the Rydberg constant, $Z$ is the atomic number, $n_1$ is the lower energy level, and $n_2$ is the higher energy level.

Chasing the Shortest Wavelength

The problem first asks us to look at the shortest wavelength in the Lyman series for a Hydrogen atom. The Lyman series is defined by transitions that end at the ground state, so $n_1=1$.

To get the shortest possible wavelength, we need the maximum possible energy jump. Imagine an electron falling from the very edge of the universe into the nucleus! That means our starting point is $n_2=\infty$. Since we are dealing with Hydrogen, $Z=1$. Let's plug these into our master equation:

Since $1/{\infty }^2$ is zero, the equation simplifies beautifully to:

The Longest Wavelength for Helium Ion

Next, we shift our focus to the ${\text{He}}^{+}$ ion and its Balmer series. The Balmer series is famous for its visible light transitions, all of which end at the second shell, so $n_1=2$.

This time, we are hunting for the longest wavelength, which corresponds to the smallest possible energy gap. The smallest step an electron can take to reach $n=2$ is from the immediately adjacent shell, $n_2=3$.

Caution: We are now dealing with a Helium ion, not Hydrogen! Helium has two protons, so its atomic number $Z=2$. This is a classic trap where many students lose marks. Let's set up the equation:

Connecting the Dots

Now, it's just a matter of careful fraction arithmetic. Let's solve the terms inside the bracket:

Flipping both sides gives us the expression for ${\lambda }_2$:

Finally, we remember our result from the first part: $\frac{1}{R_H}$ is exactly equal to ${\lambda }_1$. Substituting this back in, we get our final, elegant relationship:

And there we have it! By carefully mapping the physical concepts of 'shortest' and 'longest' to their mathematical limits, we've successfully navigated through the quantum jumps.