Spectral Transitions

The width of spectral lines is a function of:

1. Collision Broadening: Atoms are in continual motion and frequent collisions perturb the energies of outer electrons, resulting in broadening of width of UV and visible spectral lines.
2. Doppler Broadening: Motion of particles causes Doppler shift; motion is random ==> shift in both high and low frequencies.
3. Heisenberg's Uncertainty Principle: absolutely unavoidable! If a state has lifetime $\delta t$ then its energy is uncertain to an extent $\delta E$ given by
(1)
\begin{align} \delta E \times \delta t \geq \hbar \end{align}

The intensity of spectral lines is a function of:

1. Transition Probability (gives rise to selection rules)

2. Population of States: Boltzmann distribution ==> at thermal equilibirum,

(2)
\begin{align} \frac{N_{2}}{N_{1}} = \frac{g_{2}}{g_{1}}\frac{e^{-E_{2}/kT}}{e^{-E_{1}/kT}} \end{align}

Here, $g_{i}$ is the degeneracy of state $E_{i}$.

3. Path Length of sample: The Beer Lambert law relates the intensities of incident and transmitted radiation by

(3)
\begin{align} \frac{I}{I_{0}} = e^{-\kappa c l} \end{align}

where $\kappa$ is a constant for the transition being studied. This is consistent with the exponential drop in intensity as successive samples of the substance are placed in the path of the radiation.

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