Animated Solution for Chemistry - Atomic Structure: For emission line of atomic hydrogen from $n_i=8$ to $n_f=n$, the plot of wave number ($\bar{\nu }$) against $\left(\frac{1}{n^2}\right)$ will be (The Rydberg constant, $R_H$ is in wave number unit)

When an electron in a hydrogen atom jumps from a higher energy level to a lower one, it emits a photon. The energy, wavelength, and wave number of this photon can be precisely calculated using the Rydberg formula. This formula is a cornerstone of atomic physics, elegantly connecting the discrete energy levels of an atom to the observable spectral lines.

The Rydberg formula for the wave number $\bar{ u}$ is given by:

Here, $R_H$ is the Rydberg constant, $Z$ is the atomic number, $n_i$ is the initial energy level, and $n_f$ is the final energy level.

For our specific problem, we are dealing with atomic hydrogen, so $Z=1$. The electron is falling from the 8th orbit, meaning $n_i=8$, and it lands in some final orbit $n_f=n$. Substituting these values into our formula, we get:

Let's expand this expression to see its mathematical structure more clearly. By distributing the Rydberg constant $R_H$ inside the bracket, we separate the variable part from the constant part:

Now, take a close look at this equation. Does it remind you of something from coordinate geometry? It perfectly matches the standard equation of a straight line:

In our case, the y-axis represents the wave number $\bar{ u}$, and the x-axis represents the variable $\frac{1}{n^2}$.

By comparing the two equations, we can easily identify the slope and the y-intercept of our plot. The slope $m$ is the coefficient of our x-variable, which is simply the Rydberg constant $R_H$. The y-intercept $c$ is the constant term, which is $-\frac{R_H}{64}$.

Therefore, if we plot the wave number $\bar{ u}$ against $\frac{1}{n^2}$, we will obtain a straight line with a positive slope equal to $R_H$. This elegant linear relationship is a direct consequence of the quantized nature of atomic energy levels!