Calculate the wavelength in the first line in the Lyman series. The Rydberg constant 𝑅 sub 𝐻 equals 1.09737 times ten to the seventh inverse meters. State your answer to four significant figures.
We want to solve for a wavelength we’ll call 𝜆. We’re told in this statement the value of a constant called the Rydberg constant. Knowing this, we can begin our solution by recalling the mathematical relationship for the Lyman series. This series expresses the wavelength of photons that are created when electrons in a hydrogen atom transition from higher-energy levels down to the 𝑛 equals one energy level.
The wavelength of the emitted photon depends on the starting energy level only, since all the electrons end up down at the 𝑛 equals one level. In our statement, we’re told we wanna solve for the wavelength in the first line in the Lyman series. That means we’re transitioning from 𝑛 equals two to 𝑛 equals one.
The Lyman series’ mathematical relationship says that one over the wavelength 𝜆 equals the Rydberg constant 𝑅 sub 𝐻 times one divided by the final 𝑛 value squared, which for the Lyman series is one minus one divided by the initial 𝑛 value squared.
In our case, because we’re looking for the first line in the series, 𝑛 sub 𝘪 is two, and therefore our expression in parentheses simplifies to one minus one-fourth or three-fourths. We rearrange this equation then for 𝜆 and we find that 𝜆 equals four divided by three times 𝑅 sub 𝐻.
Plugging in for 𝑅 sub 𝐻, the Rydberg constant, when we enter these values on our calculator, we find that 𝜆, to four significant figures, is 121.5 nanometers. That is the wavelength of the first line in the Lyman series.