Question

compute the following:

(a) Calculate the wavelengths for the Paschen series for Hydrogen

(b) If the average speed of a Hydrogen electron is 4.0*10^6 m/s with a 0.95% uncertainty. What is the uncertainty of its position? ( Hint: the mass of an electron is 9.109*10^-31)

Answer 1

Paschen series for Hydrogen obtained when electron jumps from higher energy levels 4,5,6,..... to n = 3

From Rydburg's Equation , 1/ λ = R[(1/n_i²) – (1/n_f²)]
   Where R = Rydburg's constant = 10.96 x10⁶ m^-1
               λ = wavelength = ?
               n_i = 3
               n_f = 4
Plug the values we get λ = 1.876x10^-6 m

when   n_i = 3
               n_f = 5 ----> λ = 1.283x10^-6 m

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According to Heisenberg uncertainty principle,

m*Δx*Δv = h/4π

Where

m = mass of electron = 9.109x10^-31 kg

Δx = uncertainty in position = ?

Δv = uncertainty in velocity =0.95% = 0.95/100 = 9.5x10^-3 m/s

h = plank's constant = 6.625x 10^-34 Js

Plug the values we get

Δx = h / (4π*m*Δv )

    = (6.625x 10^-34 Js)/(4πx 9.109x10^-31x9.5x10^-3 )

   = 6.09x10^-3 m

Therefore the uncertainty in position is 6.09x10^-3 m