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Wize University Chemistry Textbook > Early Atomic Theory to Quantum Theory

Bohr Model Equations

Understanding the Bohr ModelPrevious Section
How Atomic Spectra Relates to Electronic Transitions + Other Applications of the Bohr Model Next Section
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Bohr Model Equations

There are a few important equations we should be familiar with that are related to the Bohr model. These equations can only be used for one electron systems! Recall that Bohr's model was only able to explain species with 1 electron.

Calculating the Energy of a Specific n Level

E= (−2.178×10−18J)(Z2)n2\boxed{E=\frac{\ \left(-2.178\times10^{-18}\text{J}\right)\left(Z^2\right)}{n^2}}

where n is the shell # and it can be 1, 2, 3...etc
E represents the energy of a specific n level
Z=atomic # (# of protons) in an atom (ex. Hydrogen has 1 proton, Z=1, He+ has 2 protons, Z=2)


Calculating the Energy Difference Between n Levels

ΔE=−2.178 ×10−18J(Z2) (1nf2−1ni2)\boxed{\Delta E=-2.178\ \times10^{-18}J\left(Z^2\right)\ \left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)}

where ΔE represents the difference in energy between 2 energy levels
Z=atomic # (# of protons) in an atom
nf is referring to the final energy level
ni is referring to the initial energy level

  • If ΔE is positive, energy was (absorbed/released)
    absorbed
    and the electron should go to a (higher/lower)
    higher
    energy level
  • If ΔE is negative, energy was (absorbed/released)
    released
    and the electron should go to a (higher/lower)
    lower
    energy level

Watch Out!
Be very careful when entering nf and ni values into this equation!
Example: If we are told energy is absorbed and an electron goes from n=1 to n=2
  • n=1 is the initial energy level, n=2 is the final energy level.

Example: If we are told energy is released as an electron goes from n=3 to n=2
  • n=3 is the initial energy level, n=2 is the final energy level.


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Calculating the Wavelength of the Photon Absorbed or Emitted

Method 1:
  • Use the above equation to solve for ΔE.
  • Then use E=hcλE=\frac{hc}{\lambda} to solve for the wavelength of the photon
  • Even if ΔE is negative, use the positive value to solve for the wavelength so you get a positive answer for wavelength (negative wavelength doesn't make sense)
Method 2:
  • Use the following equation and then take the inverse to solve for λ
1λ=RH(1ni2−1nf2)\boxed{\frac{1}{\lambda}=R_H\left(\frac{1}{n_i^2}-\frac{1}{n_f^2}\right)}

λ=wavelength of the photon (m) -always enter answer as a positive value!
RH=Rydberg constant=1.097x107m−1
ni=initial energy level, nf=final energy level


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Example: Releasing or Absorbing Energy and How That Differs For Different Atoms

An electron in a one-electron species transitions from the n = 4 state to the n = 2 state.

a) Is energy released or absorbed?
Energy (light) is released since the electron came to a lower energy level.

b) Would more energy be absorbed/released from a sample of Hydrogen atoms or Li2+ ions?
Consider the equation for the energy difference between energy levels:

ΔE=−2.178 ×10−18J(Z2) (1nf2−1ni2)\Delta E=-2.178\ \times10^{-18}J\left(Z^2\right)\ \left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)
If we consider the equation for a hydrogen atom vs a Li2+ ion, the only thing that would be different is Z (atomic #).
Z for H: 1, Z2=1
Z for Li2+: 3, Z2=9

In the equation, some number would be multiplied by 1 for H and would be multiplied by 9 for Li2+.
The energy released will be 9 times greater for Li2+ than H based on the equation shown above.
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Example: Excitation and Relaxation in Atoms


Calculate the wavelength, in nm, of a photon released when an electron in a hydrogen atom relaxes from n = 3 to n = 1. What is the energy of the photon if the electron is re-absorbed from n = 1 to n = 3?


Part 1: Solve for the energy difference between the energy levels, then solve for the wavelength of the photon
ΔE=−2.178 x 10−18J(Z2)(1nf2−1ni2) \Delta E_{ }=-2.178\ x\ 10^{-18}J\left(Z^{_{^2}}\right)\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)\ Z=1 for hydrogen

ΔE= −(2.178 x 10−18 J)(1(1)2−1(3)2) = −1.938 x 10−18 J\Delta E_{ }=\ -\left(2.178\ x\ 10^{-18}\ J\right)\left(\frac{1}{\left(1\right)^2}-\frac{1}{\left(3\right)^2}\right)\ =\ -1.938\ x\ 10^{-18}\ J

E =hcλ  →  λ = hc∣E∣ = (6.626 x 10−34 J⋅s)(3.0 x 108 ms)∣−1.938 x 10−18 J∣ = 1.03 x 10−7 m = 103 nmE\ =\frac{hc}{\lambda}\ \ \rightarrow\ \ \lambda\ =\ \frac{hc}{\left|E\right|}\ =\ \frac{\left(6.626\ x\ 10^{-34}\ J\cdot s\right)\left(3.0\ x\ 10^8\ \frac{m}{s}\right)}{\left|-1.938\ x\ 10^{-18}\ J\right|}\ =\ 1.03\ x\ 10^{-7}\ m\ =\ 103\ nm

Recall 1nm=1x10-9m if we have m and want to solve for nm:
m/1x10-9=nm

Part 2: Change the sign of the energy difference
If the photon is re-absorbed by the electron to move from the n = 1 state to the n = 3 state, this photon must have the same energy as the photon released to go from n = 3 to n = 1, except with OPPOSITE SIGN.

∴ ΔE = +1.938 x 10−18 J\therefore\ \Delta E\ =\ +1.938\ x\ 10^{-18}\ J

Practice: Using Bohr Model Equations to Determine an Energy Level

A photon of light was released by a hydrogen atom with an energy of 1.94 x 10-18 J. If this transition ended at n=1, from which level did this transition begin?
Extra Practice
Bohr model
Which of the following transitions would emit the shortest wavelength photon?
Bohr Model
A photon of light was released by a helium cation (He+) with an energy of 1.8312 x 10-18 J. If this transition ended at n = 2, from which level did this transition begin?
Bohr model: Energy
In a Hydrogen atom, for a transition to occur from n = 3 to n = 2, would energy need to be absorbed or emitted?
H atom
A hydrogen atom, initially in its lowest energy level, absorbs a photon with a wavelength of 93.1 nm. What is the final energy level?
Bohr Model: Absorption/emission spectra
An Emission spectrum for a one-electron species is shown below. The spectrum shows all lines resulting from transitions to the second excited state. The line B is 165nm.
List the transitions which correspond to A, B and C
Atomic Orbitals and Electronic Transition
An electron in H atom undergoes an electronic transition from I to II. What is the wavelength of absorbed/emitted photon?
Bohr Model
Below is the energy level diagram for the possible energy levels of a hydrogen atom (not to scale).
Answer following question about the hydrogen atom.
What is the energy change (Δ\DeltaE) corresponding to the transition labelled T?
Bohr model
Below is the energy level diagram for an electron in a hydrogen atom (not exactly to scale). Answer following question with reference to this diagram.
What is the energy change (in J) corresponding to the transition labelled D?
The Bohr Model: Energy levels
A photon of light was released by a hydrogen atom with an energy of 1.94 x 10-18 J. If this transition ended at n = 1, from which level did this transition begin?
Bohr Model: Energy levels
How much energy is required for an electron to move from n = 1 to n = 3 in Hydrogen?
The Bohr Model
A photon of light was released by a hydrogen atom with an energy of 1.94 x 10-18 J. If this transition ended at n = 1, from which level did this transition begin?
Bohr model
Which of the following transitions would emit the shortest wavelength photon?
Bohr Model: Energy
Which hydrogen emission line should you expect to have the greatest energy?
Bohr model: Energy
A sample of Hydrogen atoms is excites such that there exists a mixture of excited states from n = 1 to n = 6. the sample is then allowed to relax back to the ground state and the emission spectrum is recorded. What transition produces the lowest energy photon?
Bohr Model: Predicting Hydrogen Emission Wavelengths
Calculate the wavelength of light emitted when an electron in a hydrogen atom moves from n= 2 to n = 1.
Element: Calculating Ionization Energy of a Hydrogen-like Ion
Calculate the ionization energy of He+. Express your answer in Joules (J).