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Bohr Atom Energy Change Example Problem

Finding Energy Change of an Electron in a Bohr Atom

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The Bohr Model of the atom is a planetary model in which the electrons orbit around the nucleus.

The Bohr Model of the atom is a planetary model in which the electrons orbit around the atomic nucleus.

JabberWok, Wikipedia Commons
This example problem demonstrates how to find the energy change that corresponds to a change between energy levels of a Bohr atom.

Problem:

What is the energy change when an electron drops from the n=3 energy state to the 𝑛=1 energy state in a hydrogen atom?

Solution:

E = hν = hc/λ

According to the Rydberg formula:

1/λ = R(Z2/n2) where

R = 1.097 x 107 m-1
Z = Atomic number of the atom (Z=1 for hydrogen)

Combine these formulas:

E = hcR(Z2/n2)

h = 6.626 x 10-34 J·s
c = 3 x 108 m/sec
R = 1.097 x 107 m-1

hcR = 6.626 x 10-34 J·s x 3 x 108 m/sec x 1.097 x 107 m-1
hcR = 2.18 x 10-18 J

E = 2.18 x 10-18 J(Z2/n2)

En=3

E = 2.18 x 10-18 J(12/32)
E = 2.18 x 10-18 J(1/9)
E = 2.42 x 10-19 J

En=1

E = 2.18 x 10-18 J(12/12)
E = 2.18 x 10-18 J

ΔE = En=3 - En=1
ΔE = 2.42 x 10-19 J - 2.18 x 10-18 J
ΔE = -1.938 x 10-18 J

Answer:

The energy change when an electron in the n=3 energy state to the n=1 energy state of a hydrogen atom is -1.938 x 10-18 J. The negative value means the atom loses this energy during the transition. This energy is carried away by a photon emitted by the atom.

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