Calculate the shortest and longest wavelength of radiation for the Paschen Series. (Answer: 1875 nm, 820 nm) 



Data Given:


Paschen Series:

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


To Find:

(a) Shortest Wavelength  = `\λ_{min}` =  ?

(b) Longest Wavelength = `\λ_{max}` =  ? 



Solution:

Rydberg formula for Paschen Series:

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

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


(a) Shortest Wavelength = `\λ_{min}` =  ?

For the shortest Wavelength the number of orbitals is used:
n = ∞ (ie. the electron jumps from infinite orbital to the orbital no. 3)

Using Rydberg formula:

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





`\frac {1}{λ_{min}}` = 1.097 ｘ 10⁷ m⁻¹ `(\ frac {1}{3²} - frac {1}{∞²})`


`\frac {1}{λ_{min}}` = 1.097 ｘ 10⁷ m⁻¹ `(\ frac {1}{9} - 0 )`


`\frac {1}{λ_{min}}` = `\ frac {1.097 ｘ 10⁷ m⁻¹}{9}`


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


`\λ_{min}` = `\ frac {9}{1.097 ｘ 10⁷ m⁻¹}`


or


`\λ_{min}` = 8.204 ｘ 10⁻⁷ m


or


`\λ_{min}` = 820.4 ｘ 10⁻⁹ m


`\λ_{min}` = 820.4 nm ------------Ans. 1



Or (expressing in Angstrom) 
`\λ_{min}` = 8204.0 ï½˜ 10¹⁰ m
`\λ_{min}` = 8204 A° 



(b) Longest Wavelength = `\λ_{max}` =  ?

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


Using the Rydberg formula for longest wavelength:

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


by putting values


`\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. 2


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



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