Bohr Equation - Online Tutor, Practice Problems & Exam Prep

The Bohr equation is pivotal in understanding the energy transitions of electrons between atomic shells. It quantifies the energy change (ΔE) in joules when an electron moves from one energy level to another. The equation:

$\mathrm{\Delta E}=-R_e(\frac{1}{n^2}-\frac{1}{n^2})$

involves the Rydberg constant (R_e = 2.178 x 10^-18 J), with 'n' representing the principal quantum number of the electron's initial and final orbits. Additionally, a modified version of the Bohr equation relates to the wavelength (λ) of emitted or absorbed light, given by:

$\frac{1}{\lambda }=-R_{\lambda }(\frac{1}{n^2}-\frac{1}{n^2})$

where the Rydberg constant for wavelength (R_λ) is 1.0974 x 10⁷ m^-1. Both forms of the equation are essential for predicting the spectral lines of hydrogen and other elements, reflecting the quantized nature of electron energy levels. Understanding these concepts is crucial for grasping atomic structure and electron behavior in quantum mechanics.

Bohr Equation

Now the Bohr equation is used to calculate the energy transition of an electron as it moves from one shell to another. So here we have two different Bohr equations. Here we're going to say in the first one, this formula is used when dealing with two orbital levels, so they'll give you two N values and they're discussing energy. Here we're going to say change in energy of our electron equals negative R_E because we're dealing with energy times 1n2 final - 1n2 initial.

Here delta E is the energy change for an electron in joules. Here we're going to say R_E is our Rydberg constant. Here we're using E to differentiate it from the other R we're going to see in the other equation. So here this is our Rydberg constant. Since it's in joules, it's 2.178×10-18 joules. Then we're going to have our N final, which represents our final orbital level. And then we're going to have N initial which represents our initial orbital or shell level.

Now the next Bohr equation, this formula is used when dealing with again two orbital levels. So an initial and final still and you're dealing with wavelength here. We're going to say one over wavelength equals negative R_λ, just to show that we're dealing with wavelength here. Again, usually on your formula sheet and in your book they just use the variable R. Here we're changing it up slightly just to show you that R_E is when we're dealing with energy and R_λ is when we're dealing with wavelength. And that's times 1n2 final - 1n2 initial.

Notice that these two formulas use the same portion here. What's changing is that we're dealing with energy here and wavelength here, and as a result of that it has a change in my Rydberg constant value. Now since we're dealing with wavelength, our Rydberg constant will have units of meters inverse. In this case R_λ equals 1.0974×107 meters inverse. So just remember the first Bohr equation is when we're dealing with different shell numbers, two shell numbers with energy. So remember the N values are your orbital levels or shell numbers and we're dealing with this second Bohr equation where we have two orbital levels and wavelength.

And remember, whether we're dealing with energy or wavelength, we're dealing with the Rydberg constant, but the values change based on if we're using joules versus meters inverse.

Bohr Equation Example 1

Here it states what is the energy of a photon in joules released during a transition from N = 4 to N = 1 state in the hydrogen atom. All right, so here what we're going to have is we're going to say

ΔE = - R E × 1 N 2 - 1 N 2

We're using this form of the equation because we're dealing with energy.

All we do now is we plug in the values that we know. First of all, the Rydberg constant, which is R. When we're dealing with Joules, it's

- 2.178 × 10 -18   Joules

Here we start off at N = 4, and we're going down to N = 1, so our final N value is 1, so that be

1 2

and our initial would be

1 4 2

So this comes out to be

- 2.178 × 10 -18   Joules

When we do

1 1 2 - 1 4 2

that comes out to .9375. When these two numbers multiply with each other, that's going to give us the energy involved with this electron transitioning from an N = 4 to an N = 1 state. So that comes out to be

- 2.0419 × 10 -18   joules

Here, let's just do it in terms of three sig figs, so

- 2.04 × 10 -18   joules

in this question. All we have are shell numbers, but there are only one sig figs. So here I feel more comfortable giving it three sig figs, right? So here

- 2.04 × 10 -18   joules

are released as the electron moves from the 4th orbital level to the first orbital level.

What is the wavelength of a photon (in nm) absorbed during a transition from the n = 2 to n = 5 state in the hydrogen atom?

• A. 369 nm
• B. 587 nm
• C. 293 nm
• D. 434 nm

Determine the end (final) value of n in a hydrogen atom transition, if the electron starts in n = 5 and the atom releases a photon of light with an energy of 4.5738 × 10^-19 J.

• A. 4
• B. 2
• C. 3
• D. 1

An electron releases energy as it moves from the 6^th shell to the 3^rd shell. If it releases 4.25 x 10⁹ kJ of energy at a wavelength of 915.7 nm, how many photons were released in the process?

• A. 8.96 x 10³¹ photons
• B. 1.96 x 10³¹ photons
• C. 1.82 x 10²⁹ photons
• D. 2.98 x 10³¹ photons

Here’s what students ask on this topic:

What is the Bohr equation?

The Bohr equation, named after the Danish physicist Niels Bohr, is a formula that relates the energy levels of electrons in an atom to the structure of the atom itself. Bohr proposed this model in 1913 to explain how electrons can have stable orbits around the nucleus. The equation is as follows:

$E_n=-\frac{R_H\cdot Z^2}{n^2}$

Here, En is the energy of the electron in the nth orbit, RH is the Rydberg constant (approximately 2.18 x 10^-18 Joules), Z is the atomic number (number of protons in the nucleus), and n is the principal quantum number, which can be any positive integer. The negative sign indicates that the energy is quantized and that electrons are bound to the nucleus; the energy is lower (more negative) for electrons closer to the nucleus. Bohr's model was a significant step forward in atomic physics, but it has since been superseded by the more comprehensive quantum mechanical models.

What is the Rydberg constant?

The Rydberg constant is a fundamental physical constant that is crucial to atomic physics, particularly in the study of spectral lines of hydrogen and hydrogen-like elements. It represents the highest wavenumber (the number of waves per unit distance) that can be emitted from an atom as an electron transitions from higher energy levels to the lowest possible energy level, known as the ground state.

The constant is named after the Swedish physicist Johannes Rydberg, who formulated the Rydberg formula in the late 19th century. This formula was used to predict the wavelengths of the spectral lines of hydrogen.

The value of the Rydberg constant $R_{\infty }$ is approximately 10,973,731.56816 m^-1. This value is obtained when the nucleus of the atom is assumed to have an infinite mass compared to the electron, allowing for a simplified model of the atom that does not take into account the motion of the nucleus. In reality, for hydrogen or any other one-electron system, the actual value slightly differs to account for the finite mass of the nucleus. The Rydberg constant is used to calculate the energy differences between levels in the Bohr model of the atom and to predict the wavelengths of emitted or absorbed light in spectroscopy.