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Screengrab of an interactive example for the transition of the electrons using the Bohr model (click to enlarge)

The discovery of atomic structure was the key that unlocked the door to modern understanding of structure and reactivity. Let’s take a moment to review a bit of history before Bohr’s monumental discovery.

J.J. Thomson (1856-1940) first discovered the existence of electrons. He knew that they contributed only a very small fraction of the mass and postulated that, therefore they must only contribute a very small size as well.  He proposed that they were embedded in a positive uniform sphere, which is known as the “plum pudding” model.
As mentioned in the Physical chemistry text (section 11.2), Ernest Rutherford (1871-1937) disproved this model by passing alpha particles through a sheet of gold foil and discovering the positive center termed the nucleus and large areas of empty space that contained the electrons, which orbited the nucleus.
Unfortunately, classical physics decreed that the pull of the positive nucleus would force the orbiting electrons to lose their energy and eventually crash into the nucleus.  This is much like dropping a coin into one of those spiral wishing well funnels.  The coin will circle for a long time, but eventually it loses the battle and falls into the center hole.
Obviously all the atoms in the universe did not follow the coins fate.  The atomic line spectrum of the hydrogen atom provided the clues that provided proof for Bohr’s monumental discovery.
When energy was directed at the hydrogen atom, only some of that energy was absorbed with the rest passing through it!  When the energy source was removed, only those energies that were absorbed were reemitted.  In fact, only four different colored lines, or energies, were seen in the visible spectrum which is now called the Balmer series.
How in the world does the one electron in hydrogen produce four different lines?!  Why didn’t it absorb all the energy? How does it release energy back out to make the four different colored lines?
Niels Bohr (1855-1962) proposed a solution to these questions without even knowing it.  He postulated that the electrons must reside in specific, stable areas or shells around the nucleus and that transitioning between these shells either absorb energy (jumping from a lower shell to a higher one) or release energy (falling from a higher shell to a lower one).
The actual shells can be visualized as stairs.  You can travel from stair to stair, but may not actually stop a quarter or even halfway between them.  Likewise, the electrons can travel from shell to shell but may not stop a quarter or even halfway between.
The four different colored lines then must be the transition between four different shells and only the energy that exactly matched those transitions could be absorbed, the others would be of no use!  If only those four energies could be absorbed, then only those four energies could be released when the external energy source was removed.
The physical chemistry textbook provides two wonderful interactive examples (see screen grab above) for the transition of the electrons using the Bohr model.
By now I hope you are pondering an even more important question.

How come there were only four transitions that hydrogen’s electron could undergo? Given a strong energy source, what is stopping it from more transitions or even from shooting the electron so far away from the nucleus that it can’t get back?!

Answer: it does have more transitions and the electron can go so far from the nucleus that it is lost, it is called ionization.
Remember, I told you that only four lines were seen in the visible spectrum, all the other transitions can be “seen” in other areas of the electromagnetic spectrum.  These series are usually given the name of the scientist that discovered them.  Two other early discoveries are the Lyman series in the ultraviolet region and the Braschen series is in the infrared.  The text (in the physical chemistry book) also lists other series that have been discovered more recently.
Niels Bohr provided the answers to all the questions and he hadn’t even been working with spectrum!!!  Section 11.2 of the physical chemistry textbook gives a comprehensive look at Bohr’s theory using the actual language of science, MATH.
Hopefully this written description will help you understand what the equations are trying to tell you.  Good Luck![/vc_column_text][/vc_column][/vc_row]