Determining the Electron's Final Energy Level After Photon Emission

When an excited hydrogen atom emits a photon, the electron transitions from a higher energy level to a lower one. This process, governed by quantum mechanics, is a fundamental concept in understanding atomic structure and spectroscopy. In this article, we will walk through the steps to determine the final energy level of the electron in the hydrogen atom after it has emitted a photon of wavelength 4052 nm. This example will include the application of the Rydberg formula and the detailed calculation involved.

Understanding the Process of Photon Emission

Photon emission in a hydrogen atom occurs when an electron transitions from a higher energy level to a lower one. The energy of the emitted photon is related to the difference in energy levels of the electron before and after the transition. In this specific instance, the initial energy level of the electron is 5. We will calculate the final energy level, denoted as m, by applying the Rydberg formula. The Rydberg formula is given by the equation:

( frac{c}{lambda} frac{m e^2}{8 varepsilon_0 h^3} left( frac{1}{m^2} - frac{1}{n^2} right) )

Key Constants and Their Values

In order to solve for the final energy level m, we need to know the values of several fundamental constants:

Speed of light, ( c ) 299,792,458 m/s Wavelength of emitted photon, ( lambda ) 4052 nm 4052 times; 10-9 m Mass of electron, ( m_e ) 9.10938291 times; 10-31 kg Charge of electron, ( e ) 1.602176565 times; 10-19 C Vacuum permittivity, ( varepsilon_0 ) 8.8541878128 times; 10-12 F/m Planck's constant, ( h ) 6.62607015 times; 10-34 J cdot; s

Applying the Rydberg Formula

Using the Rydberg formula, we can find the final energy level m of the electron. The formula is:

( frac{c}{lambda} frac{m e^2}{8 varepsilon_0 h^3} left( frac{1}{m^2} - frac{1}{n^2} right) )

We know the initial energy level, n 5, and the wavelength of the emitted photon, λ 4052 nm. Let's rearrange the formula for m:

( frac{c}{lambda} frac{m e^2}{8 varepsilon_0 h^3} left( frac{1}{m^2} - frac{1}{5^2} right) )

Squaring both sides of the equation, we get:

( left( frac{c}{lambda} right)^2 frac{m^2 e^4}{64 varepsilon_0^2 h^6} left( frac{1}{m^2} - frac{1}{25} right) )

Multiplying both sides by the denominator and solving for m yields the quadratic equation:

( left( frac{c}{lambda} right)^2 (64 varepsilon_0^2 h^6) m^2 e^4 left( frac{1}{m^2} - frac{1}{25} right) )

After simplification, we get:

( left( frac{c}{lambda} right)^2 (64 varepsilon_0^2 h^6) e^4 left( frac{1}{m^2} - frac{1}{25} right) )

Rearranging and solving the quadratic equation, we find:

( m^2 left( left( frac{c}{lambda} right)^2 (64 varepsilon_0^2 h^6) - e^4 right) e^4 cdot frac{1}{25} )

After solving this equation, we can determine that m 4. Therefore, the electron has dropped from the fifth orbit into the fourth orbit.

Conclusion

The process of photon emission in a hydrogen atom, as demonstrated in this example, is a classic application of quantum mechanics. Understanding the energy transitions of electrons between orbits is crucial for comprehending atomic spectroscopy and the behavior of excited atoms. The Rydberg formula provides a powerful tool for calculating these transitions and predicting spectral lines in hydrogen.

Related Concepts

The topic of photon emission in hydrogen atoms is closely related to several key concepts in atomic physics:

Bohr Model: The orbital model of the atom, developed by Niels Bohr, which explains the discrete energy levels of electrons in atoms. Spectral Lines: Specific wavelengths of light emitted or absorbed by atoms, which are caused by transitions of electrons between energy levels. Hydrogen Atom: The simplest atom, consisting of a single proton and a single electron, which serves as a model for understanding the behavior of more complex atoms.

By understanding these concepts, we can gain a deeper insight into the behavior of atoms and the emission of spectral lines, which are fundamental to both theoretical and applied physics.