# The ratio of wave length of longest line w.r. to series limit of paschen series will be :

1
by Atreyi

2014-08-04T01:10:11+05:30

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Wavelength can be given by

where n is upper energy level and n' is lower energy level.

Energy in photon is inversely proportional to wavelength & directly proportional to frequency.
( E = h v  = h c / lambda =   k R [ 1/n' ^2 - 1/n^2 ]

Paschen series are transitions down to level n=3.  Longest wavelength corresponds to lowest photon energy,  is for n=4 to n=3.  Shortest wavelength (highest photon energy) is for n = infinity (free electron) to n=3.

Highest photon energy, Eh = k (1/3² - 1/infinity²) = k (1/3² - 0) = 0.111 k
Lowest photon energy, El = k (1/3² - 1/4²) = 0.0486 k
λshort = Shortest wavelength:  3² / R  = 9 /R
λlong =  Longest wavelength :  1/ (0.0486 R)  = 20.57 / R

If you want the ratio of longest to the shortest wavelengths:
λlong / λshort = Eh / El = 2.29