Rydberg Formula
Rydberg formula is a perfect example of how science is done.
It all started in 1880's with numerous experimental results of spectral lines of hydrogen, emitted after its atoms are excited by some external energy, like electric field or heat.
These spectral lines had certain wave lengths observed through experiments.
Johann Jacob Balmer attempted to connect the wave lengths of observed spectral lines of hydrogen with some kind of empirical formula and found the one:
λ = B·n²/(n²−2²)
where
λ is a wave length of an observable spectral line,
B=3.6450682·10−7 m is a constant that Balmer has suggested,
n ≥ 3 is a sequence number of a spectral line.
Here are a few first wave lengths λ and colors of Balmer series for different sequence number n
| n | λ (nm) | Color |
| 3 | 656.1 | Red |
| 4 | 486.0 | Cyan |
| 5 | 433.9 | Violet |
| 6 | 410.1 | Violet |
| 7 | 396.9 | Violet |
| 8 | 388.8 | Violet |
| 9 | 383.4 | Violet |
Jumping forward, with introduction of Bohr's model with specific stationary electron orbits with fixed energy levels for each orbit of an atom, it was apparent that Balmer has described electron emitting radiation when jumping from some higher orbit to orbit #2.
When electron jumps from any higher orbit to orbit #1 (the closest to nucleus), the emitted radiation is in ultraviolet part of a spectrum and was not observed by Balmer.
A few years later Johannes Rydberg generalized the Balmer formula and described any jump of an electron in a hydrogen atom from orbit #n to orbit #m:
1/λ = R·(1/m² − 1/n²)
where
n ≥ 2 is an orbit number an electron jumps from,
1 ≤ m ≤ n−1 is an orbit an electron jumps to,
R is Rydberg constant.
For m=2 the Rydberg formula is
1/λ = R·(n² − 2²)/2²·n²)
or
λ = (4/R)·n²/(n² − 2²)
which corresponds to Balmer formula if B=4/R.
All the above formulas are empirical, obtained in the process of analyzing the results of experiments. The theoretical foundation of them would be known only after Bohr introduced his atom model in the beginning of the 20th century and his model would undergo certain improvements based on quantum physics.
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