Calculate the longest wavelength of radiation for the Paschen Series. (Answer: 1875 nm) 



Data Given:


Paschen Series:

∴ Rydberg constant = R = 1.097 ｘ 10⁷ m⁻¹


To Find:

Longest Wavelength = `\λ_{max}` =  ?



Solution:

Rydberg formula for Paschen Series:

`\frac {1}{λ}` = R`(\ frac {1}{3²} - frac {1}{n²})` --------(1)

where n = 4,5,6,7,..............


For the longest Wavelength, the number of orbital n = 4 is used (ie. the electron jumps from orbital 4 to orbital no. 3)


Putting values in eqn (1)


`\frac {1}{λ_{max}}` = 1.097 ｘ 10⁻⁷ m⁻¹ `(\ frac {1}{3²} - frac {1}{4²})`


`\frac {1}{λ_{max}}` = 1.097 ｘ 10⁻⁷ m⁻¹ `(\ frac {1}{9} - frac {1}{16})`


`\frac {1}{λ_{max}}`= 1.097 ｘ 10⁻⁷ m⁻¹ `(\ frac {16 - 9}{144})`


`\frac {1}{λ_{max}}` = 1.097 ｘ 10⁻⁷ m⁻¹ `(\ frac {7}{144})`


`\frac {1}{λ_{max}}` = `(\ frac {7.679 ｘ 10⁻⁷ m⁻¹}{144})`


by flipping the fraction on both sides of the equation we get


`\λ_{max}` = `\ frac {144}{7.679 ｘ 10⁻⁷ m⁻¹}`


or


`\λ_{max}`  = 18.752 ｘ 10⁷ m


or


`\λ_{max}` = 1875.2 ｘ 10⁹ m


`\λ_{max}`  = 1875 nm ---------------Ans.


If expressing in Angstrom)
`\λ_{min}` = 18752 ｘ 10¹⁰ m
`\λ_{min}` = 18752 A°


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