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.