Bohr Equation: Videos & Practice Problems

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 Equations

The Bohr model provides two essential equations for calculating the energy transitions of electrons between different orbital levels, which are crucial for understanding atomic structure and spectral lines. The first equation focuses on energy changes when an electron transitions between two shells, represented by the principal quantum numbers \( n \). The formula is expressed as:

\[\Delta E = -R_e \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right)\]

In this equation, \( \Delta E \) denotes the change in energy of the electron measured in joules, while \( R_e \) is the Rydberg constant for energy, valued at \( 2.178 \times 10^{-18} \) joules. The variables \( n_{\text{final}} \) and \( n_{\text{initial}} \) represent the final and initial orbital levels, respectively.

The second equation is utilized when calculating the wavelength of light emitted or absorbed during the transition between two orbital levels. This equation is given by:

\[\frac{1}{\lambda} = -R_\lambda \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right)\]

Here, \( \lambda \) represents the wavelength, and \( R_\lambda \) is the Rydberg constant for wavelength, which is \( 1.0974 \times 10^7 \) meters^-1. The structure of this equation mirrors that of the energy equation, with the primary distinction being the focus on wavelength rather than energy. This highlights the relationship between energy transitions and the electromagnetic spectrum.

In summary, both equations are pivotal for calculating the energy and wavelength associated with electron transitions between quantized energy levels in an atom. The choice of equation depends on whether the focus is on energy (in joules) or wavelength (in meters), with the Rydberg constant adapting accordingly to reflect these different units.

Bohr Equation Example

To determine the energy of a photon released during a transition from the n = 4 state to the n = 1 state in a hydrogen atom, we can use the formula for energy change:

$$\Delta E = -R_e \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right)$$

In this equation, \(R_e\) is the Rydberg constant, which for energy in joules is:

$$R_e = -2.178 \times 10^{-18} \text{ joules}$$

For this transition, the final state \(n_{\text{final}} = 1\) and the initial state \(n_{\text{initial}} = 4\). Plugging these values into the equation gives:

$$\Delta E = -2.178 \times 10^{-18} \left( \frac{1}{1^2} - \frac{1}{4^2} \right)$$

Calculating the fractions:

$$\frac{1}{1^2} = 1$$

$$\frac{1}{4^2} = \frac{1}{16} = 0.0625$$

Thus, the difference is:

$$1 - 0.0625 = 0.9375$$

Now substituting this back into the energy equation:

$$\Delta E = -2.178 \times 10^{-18} \times 0.9375$$

This results in:

$$\Delta E \approx -2.0419 \times 10^{-18} \text{ joules}$$

Rounding to three significant figures, the energy released as the electron transitions from the fourth orbital level to the first orbital level is:

$$\Delta E \approx -2.04 \times 10^{-18} \text{ joules}$$

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

Answer: (D)

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

Answer: (B)

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

Answer: (B)

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 = - R H ⋅ 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 ∞ 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.