Both Bohr’s equation and Balmer’s equation try their best to describe and characterize changes in either energy or wavelength when a photon or an electron goes from one shell to another shell.
In the Bohr equation, we’re going to say this is an equation that relates energy to a photon electronic transitions. We’re going to say here we’re talking about an electron or photon going from one shell to another shell and how that affects the energy (either absorbed or released).
The equation is delta E = RH (1/n2final - 1/n2 initial).
Here, our RH value is 2.18 x 10-18 Joules. We’d use this version of the Bohr equation when they’re giving us or discussing two shells like you’re going from shell 3 to shell 1 and they reference energy. Again, if they’re talking about two shells, and they’re referencing energy, we use Bohr’s equation.
In Balmer’s equation, this equation relates wavelength to a photon’s electronic transitions. In the above example, it tries to relate changes in your shell number to the wavelength. Here, R now becomes 1.0974 x 107 m-1. Realize here that when we use the term Balmer’s equation, Balmer in terms of emissions which means we’re going from a higher shell number to a lower shell number, it usually means we’re starting off at a higher shell number and we’re going down to the second shell. That represents a Balmer emission. That deals with the origin of this equation.
When they were talking about Balmer’s equation, the final n final would’ve been 2. It doesn't necessarily always have to equal 2 now because they could give you a different value for the final shell number. Just realize that in the origins of this equation are
n final = 2. This is because a Balmer emission is going from a higher shell number down to shell number 2, making shell number 2 our final shell.
Remember that with Bohr’s equation and Balmer’s equation, they try to understand what’s happening to a photon as it goes from one shell to another. With Bohr’s equation, we’re relating shell numbers to energy. With the Balmer equation, we’re relating shell numbers to wavelength.