Animated Solution for Physics - Atoms and Nuclei: If the series limit frequency of the Lyman series is ${\nu }_L$, then the series limit frequency of the Pfund series is

The quantum world of the hydrogen atom is a fascinating landscape of discrete energy levels and precise mathematical rules. When we talk about the emission spectrum of hydrogen, we are essentially looking at the fingerprints of electrons jumping between these allowed energy states.

In this problem, we are exploring a very specific type of electron jump: the series limit. Let's dive deep into what this means and how we can elegantly connect the frequencies of different spectral series.

Imagine an electron that has been completely removed from the hydrogen atom. It is sitting at an infinite distance away, with zero potential energy. We say this electron is at the energy level $n=\infty$.

Now, suppose this electron is captured by the proton and falls all the way down to one of the bound energy states. As it falls, it loses energy, and this lost energy is emitted as a photon of light.

A spectral series is a collection of all such transitions that end at the same specific energy level. For example, any transition that ends at the ground state ($n=1$) belongs to the Lyman series. Any transition that ends at the fifth energy level ($n=5$) belongs to the Pfund series.

The series limit is the most energetic transition possible within a given series. It occurs when the electron falls from the highest possible starting point—infinity—down to the base level of that series. Because it's the largest energy jump, it corresponds to the highest frequency (and shortest wavelength) photon in that series.

Rydberg Formula

To calculate the energy of these emitted photons, we use the Bohr model's energy equation, often expressed in a form similar to the Rydberg formula. The energy of a photon emitted during a transition from an initial state $n_2$ to a final state $n_1$ is given by:

Here, $h$ is Planck's constant, $ u$ is the frequency of the photon, and $E_0$ is the ground state energy magnitude of the hydrogen atom (approximately $13.6\text{ eV}$).

For any series limit, the electron starts from infinity, so we always set $n_2=\infty$. Since $\frac{1}{\infty }=0$, the formula simplifies beautifully for series limits:

This simple, elegant equation is the key to solving our problem.

Let's apply our simplified formula to the Lyman series. The defining characteristic of the Lyman series is that all transitions end at the ground state, so $n_1=1$.

Substituting this into our equation, we get the energy of the Lyman series limit photon:

This tells us that the energy of the Lyman series limit photon is exactly equal to the ionization energy of the hydrogen atom, $E_0$. This makes perfect physical sense: the energy released when an electron falls from infinity to the ground state is exactly the same amount of energy required to kick it from the ground state out to infinity!

Now, let's turn our attention to the Pfund series. For the Pfund series, the base energy level is $n_1=5$.

Again, we are looking for the series limit, so the electron falls from $n_2=\infty$. Plugging these values into our master equation:

The energy of the Pfund series limit photon is exactly one twenty-fifth of the ground state energy magnitude.

We now have two crucial pieces of information: 1. $E_0 = h u_L$ 2. $h u_P = \frac{E_0}{25}$

Our goal is to find the relationship between the two frequencies, $ u_P$ and $ u_L$. This is where the magic of algebra comes in. We can simply substitute the expression for $E_0$ from the first equation into the second equation.

Replacing $E_0$ with $h u_L$ in the Pfund equation gives us:

Notice that Planck's constant, $h$, appears on both sides of the equation. We can divide both sides by $h$ to cancel it out, leaving us with a direct relationship between the frequencies:

And there we have it! The series limit frequency of the Pfund series is exactly one twenty-fifth of the series limit frequency of the Lyman series.

This result highlights a profound symmetry in the hydrogen atom. The series limit frequency for any spectral series ending at level $n$ will always be $\frac{ u_L}{n^2}$. It's a beautiful demonstration of how simple integer ratios govern the quantum mechanics of atoms.