For atomic hydrogen, the paschen series of lines occurs when nf = 3, whereas the brackett series occurs when nf = 4 in the equation

1/? = 2p2 mk2 e4/ h3c (z2) (1/n2f - 1/n2i)

using this equation, show that the ranges of wavelengths in these two series overlap.
shortest wavelength (m) longest wavelength (m)

paschen series

the highest value of the Paschen series is 18805 μm which is greater than the shortest value of the Brackett series (14617 μm).

Explanation:

Hydrogen atom transitions are described by the Bohr model

        = -13.606 / n²

Where n is an integer

Transitions occur between two states with different quantum numbers n

         ΔE = -13.606 (1 / ² - 1 / ²)

Where >

For the wavelength we use the Planck equation

         E = h f

         c = λ f

          E = h c / λ      

         λ = h c / E

         λ = 6.63 10⁻³⁴ 3 10⁸ / E

         λ = 19.89 10⁻²⁶ / E

Let's reduce to eV

         λ = 19.89 10⁻²⁶ / E[J]  (Ev / 1.6 10⁻¹⁹ J)

         Lam = 12430 / E [eV]

Let's calculate the energy for transitions

Paschen Series

        ΔE (eV)       λ  (um)

4       3       0.661          18805

5        3       0.9657        12871

10       3       1.3757          9035

100     3      1.5104           8229

∞       3      1.5117            8222

Brackett Series

           ΔE (eV)        λ  (um)

5      4         0.3061        40607

6      4         0.4724        26312

10    4          0.7143         17401

100  4         0.8490        14640

∞    4         0.8504        14617

When we analyze the wavelength values we see that the highest value of the Paschen series is 18805 μm which is greater than the shortest value of the Brackett series (14617 μm).

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