4.1: Rudiments of Atomic Spectroscopy Using Mathcad

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\[ \begin{matrix} \text{Planck's constant:} & h = 6.62608 10^{-34} & \text{Speed of light:} & c = 2.9979 10^{8} \frac{m}{sec} \end{matrix} \nonumber \]

\[ \begin{matrix} \text{Conversion factors:} & nm = 10^{-9} m & pm = 10^{-12}m & aJ = 10^{-18} \text{Joule} \end{matrix} \nonumber \]

\[ \begin{matrix} \text{Energy of a photon:} & E_{photon} = h \nu = \frac{hc}{ \lambda} \end{matrix} \nonumber \]

Energy of the hydrogen atom: where n is a quantum number and can have integer values.

\[ E_{atom} = \frac{-2.178 aJ}{n^2} \nonumber \]

Emission Spectroscopy

In emission spectroscopy a photon is created as the electron undergoes a transition from a higher to a lower energy state. Energy conservation requires

\[ E_{atom} ~^{initial} = E_{atom} ~^{final} + E_{photon} \nonumber \]

Example: Calculate the frequency, wavelength, and energy of the photon emitted when an electron undergoes a transition from the n=2 to the n=1 state.

\[ \begin{matrix} n_i = 2 & n_f = 1 & \begin{array}{c|c} \nu = \frac{-2.178 aJ}{n_i^2} = \frac{-2.178 aJ}{n_f^2} + h \nu ~ & _{float,~4} ^{solve,~ \nu} \rightarrow \frac{0.2465e16}{sec} \end{array} \end{matrix} \nonumber \]

\[ \begin{matrix} \text{Calculate the wavelength of the photon:} & \lambda = \frac{c}{ \nu} & \lambda = 121.619 nm \end{matrix} \nonumber \]

\[ \begin{matrix} \text{Calculate the energy of the photon:} & h \nu = 1.633 aJ \end{matrix} \nonumber \]

Absorption Spectroscopy

In absorption spectroscopy a photon is absorbed and an electron is promoted to a higher energy level. Energy conservation requires

\[ E_{atom} ~^{initial} = E_{atom} ~^{final} + E_{photon} \nonumber \]

Example: Calculate the frequency, wavelength, and energy of the photon required to promote the electron from the n=1 to the n=3 level.

\[ \begin{matrix} \nu = \nu & n_i = 1 & n_f = 3 & \begin{array}{c|c} \nu = \frac{-2.178 aJ}{n_i^2} + h \nu = \frac{-2.178 aJ}{n_f^2} ~ & _{float,~4} ^{solve,~ \nu} \rightarrow \frac{0.2922e16}{sec} \end{array} \end{matrix} \nonumber \]

\[ \begin{matrix} \text{Calculate the wavelength of the photon:} & \lambda = \frac{c}{ \nu} & \lambda = 102.598 nm \end{matrix} \nonumber \]

\[ \begin{matrix} \text{Calculate the energy of the photon:} & h \nu = 1.936 aJ \end{matrix} \nonumber \]