Bohr Atom Energy Change Example Problem

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)

E_n=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

E_n=1

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

ΔE = E_n=3 - E_n=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.