Bohr Model of Hydrogen and Atomic Spectra: Study Notes

Bohr Model of Hydrogen

Energy Transitions in the Hydrogen Atom

The Bohr model describes the hydrogen atom as electrons orbiting the nucleus in discrete energy levels. When an electron transitions between these levels, energy is absorbed or emitted as electromagnetic radiation.

• Energy Change Formula: The change in energy ($riangle E$) for an electron moving from energy level $n_i$ to $n_f$ is given by: $riangle E = -2.18 imes 10^{-18} ext{ J} imes igg( rac{1}{n_f^2} - rac{1}{n_i^2} igg)$

• Wavelength of Emitted Light: The wavelength ($ext{λ}$) of the emitted photon is related to the energy change by: $riangle E = rac{hc}{ ext{λ}}$ where $h$ is Planck's constant ($6.626 imes 10^{-34}$ J·s) and $c$ is the speed of light ($3.00 imes 10^8$ m/s).

Example Transitions

1. n = 3 → n = 2: - Calculate $riangle E$ and $ext{λ}$ using the formulas above. - This transition is part of the Balmer series, which can produce visible light.

2. n = 4 → n = 2: - Calculate $riangle E$ and $ext{λ}$. - Also part of the Balmer series; may produce visible light depending on the wavelength.

3. n = 2 → n = 1: - Calculate $riangle E$ and $ext{λ}$. - This transition is part of the Lyman series, which produces ultraviolet light (not visible).

Visible Light Range: The wavelengths of visible light range from 400 to 700 nm. Only transitions that result in wavelengths within this range will be visible to the human eye.

Application: By calculating the wavelength for each transition, you can determine which transitions produce visible light. Typically, transitions to n=2 from higher levels (Balmer series) fall within the visible spectrum.

Bright-Line Spectra and Element Identification

Atomic Emission Spectra

When atoms are excited, electrons move to higher energy levels. As they return to lower levels, they emit photons at specific wavelengths, producing a unique bright-line (emission) spectrum for each element.

• Bright-Line Spectra: Each element has a characteristic set of spectral lines, which can be used to identify the element in a sample.

• Unknown Samples: By comparing the spectral lines of unknown samples to those of known elements, you can determine the composition of the unknowns.

Element Identification Questions

1. List all the elements present in Unknown W.

2. List all the elements present in Unknown X.

3. List all the elements present in Unknown Y.

4. List all the elements present in Unknown Z.

Relation to Electron Transitions: The observed spectral lines correspond to specific electron transitions between energy levels in atoms. Each line represents a photon emitted as an electron drops from a higher to a lower energy level.

Summary Table: Hydrogen Atom Transitions and Spectral Series

Transition	Spectral Series	Typical Wavelength Range	Visible?
n ≥ 3 → n = 2	Balmer	400–700 nm	Yes
n ≥ 2 → n = 1	Lyman	Ultraviolet	No
n ≥ 4 → n = 3	Paschen	Infrared	No

Additional info: The Balmer series is the only hydrogen spectral series that falls within the visible range. The Lyman and Paschen series are in the ultraviolet and infrared regions, respectively.