Understanding the Frequency, Wavelength, and Energy of the Lowest Frequency Spectral Line in the Lyman Series of Hydrogen

Introduction

The Lyman series is a set of hydrogen spectral lines formed by transitions of an electron to the first excited state (n1) from higher excited states (n ≥ 2). The spectral line with the lowest frequency in the Lyman series corresponds to the transition from n2 to n1. This article delves into the detailed calculation of the frequency, wavelength, and energy of this particular transition.

The Transitions and Calculation

The low-level transitions in the Lyman series are significant in understanding the fundamental properties of the hydrogen atom. Specifically, we can calculate key parameters like the wavelength, frequency, and energy for the transition from n2 to n1.

1. Wavelength Calculation

The Rydberg formula for hydrogen is a powerful tool in such calculations. The Rydberg formula is expressed as:

[frac{1}{lambda} R_H left(frac{1}{n_1^2} - frac{1}{n_2^2}right)]

where (R_H) is the Rydberg constant, approximately (1.097 times 10^7 ; text{m}^{-1}), (n_1 1), and (n_2 2). Substituting these values into the formula, we get:

[frac{1}{lambda} R_H left(1 - frac{1}{4}right) R_H left(frac{3}{4}right)]

This simplifies to:

[frac{1}{lambda} approx 1.097 times 10^7 times frac{3}{4} approx 8.2275 times 10^6 ; text{m}^{-1}]

To find (lambda), we take the reciprocal of this value:

[lambda approx frac{1}{8.2275 times 10^6} approx 1.216 times 10^{-7} ; text{m} 121.6 ; text{nm}]

2. Frequency Calculation

The frequency (f) can be calculated using the speed of light (c), which is approximately (3 times 10^8 ; text{m/s}), and the wavelength (lambda), which we have calculated as 121.6 nm (or (1.216 times 10^{-7} ; text{m})). Using the equation:

[f frac{c}{lambda}]

Substituting the values:

[f approx frac{3 times 10^8 ; text{m/s}}{1.216 times 10^{-7} ; text{m}} approx 2.46 times 10^{15} ; text{Hz}]

3. Energy Calculation

The energy (E) of the photon can be calculated using Planck's equation:

[E h f]

where (h) is Planck's constant, approximately (6.626 times 10^{-34} ; text{J s}). Therefore:

[E approx 6.626 times 10^{-34} ; text{J s} times 2.46 times 10^{15} ; text{Hz} approx 1.63 times 10^{-18} ; text{J}]

Summary

The values obtained through these calculations are:

Wavelength (lambda) 121.6 nm Frequency (f) (2.46 times 10^{15} ; text{Hz}) Energy (E) (1.63 times 10^{-18} ; text{J})

These values correspond to the spectral line of the lowest frequency in the Lyman series of the hydrogen atom.

The Bohr Model and Energy Calculations

Using the Bohr model, the energy of the (n)th shell is given by:

[E -frac{13.6 ; text{eV}}{n^2}]

For the Lyman series, where the electron transitions to the n1 shell:

[E -frac{13.6 ; text{eV}}{1^2} -13.6 ; text{eV}]

Additionally, the energy in terms of wavelength and frequency is given by the equation:

[E frac{hc}{lambda}]

Where (h) is Planck's constant, approximately (6.631 times 10^{-34} ; text{J s}), (c) is the speed of light in vacuum, approximately (3 times 10^8 ; text{m/s}), and (lambda) is the wavelength. This can be further simplified to:

[E hf]

Where (f) is the frequency. Using the frequency calculated above, you can verify the energy as:

[E approx 1.63 times 10^{-18} ; text{J}]

Conclusion

Understanding the frequency, wavelength, and energy of the spectral line transitions in the Lyman series is crucial in studying the properties of hydrogen atoms. These calculations provide insights into the quantum mechanical behavior of electrons in atoms, facilitating further research and applications in spectroscopy and physics.