Balmer Formula Calculations

The Balmer Formula is:

with n = 2 and m = 2, 3, 4, . . . . Balmer used 3645.6 x 10¯7 mm. as the value of the constant.

The four calculations shown below generate the wavelengths of the four visible lines of the hydrogen spectrum. Please feel free to carry out each calculation to verify the answers.

Line No. 1: Modern Symbol = Ha

Line No. 2: Modern Symbol = Hb

Line No. 3: Modern Symbol = Hg

Line No. 4: Modern Symbol = Hd

This ends the visible lines of the hydrogen spectrum. Keep in mind that Balmer discused two points related to the spectrum:

"If the formula for n = 2 is correct for all the main lines of the hydrogen spectrum, then it implies that towards the utraviolet end these spectral lines approach the wavelength 3645.6 in closer and closer sequence, but cannot cross this limit; while at the red end [of the spectrum] the C-line [today called Ha] represents the line of longest possible [wavelength]. Only if in addition lines of higher order existed, would further lines arise in the infrared region."

By higher order, he means allow n to take on higher values, such as 3, 4, 5, and so on.

With regard to his first point, the next line with n = 2 would take this form:

This line, named He, is in the ultraviolet region of the spectrum.

With regard to his second point no other series of lines, other than the above, was known to exist. However, with the Balmer formula, production of wavelengths was quite easy and, as techniques improved, each other series was discovered.

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