One of the early pieces of experimental evidence that help convince scientists t

Question

One of the early pieces of experimental evidence that help convince scientists that electrons in atoms have quantized energies was the hydrogen line spectrum. Rydberg determined a formula that could describe these transition for hydrogen. He realized that this worked for any hydrogen-like atom (meaning any atom with only one electron). What is the energy of the transition (in eV - don't include this when you inputyour answer) from the n=5 shell to the n=2 shell for Be3+? (Note that for hydrogen, this transition is part of the Balmer series and shown up in the visible region. Consider the implications for the energy levels that having 4 protons vs. 1 in hydrogen has.)

Explanation / Answer

We know that 1/ = Rz2 [ 1/n1^2- 1/n2^2]

= wavelength

R = Rydberg constant = 1.097 x 107 m-1

   Z = atomic number

Atomic number of Be, Z = 4

Given that n1 = 2, n2 = 5

E = hc/

Hence,

E = hcRz2 [ 1/n1^2- 1/n2^2]

   = (6.626 x 10-34 J.s) ( 3 x 108 m/s) (1.097 x 107 m-1) (42) [ 1/ 2^2 - 1/ 5^2]

= (6.626 x 10-34 J.s) ( 3 x 108 m/s) (1.097 x 107 m-1) (42) [21/100]

= 73.3 x 10-19 J

E = 73.3 x 10-19 J

Energy in electron - volts (eV) :

1 eV = 1.602 x 10-19 J

Then, 1 J = 0.6242 x 1019 eV

Hence,

E = 73.3 x 10-19 J

= ( 73.3 x 10-19 J ) (0.6242 x 1019 eV)

= 45.75 eV

E = 45.75 eV

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