4.1: Rudiments of Atomic Spectroscopy Using Mathcad
- Page ID
- 150633
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\[ \begin{matrix} \text{Planck's constant:} & h = 6.62608 10^{-34} & \text{Speed of light:} & c = 2.9979 10^{8} \frac{m}{sec} \end{matrix} \nonumber \]
\[ \begin{matrix} \text{Conversion factors:} & nm = 10^{-9} m & pm = 10^{-12}m & aJ = 10^{-18} \text{Joule} \end{matrix} \nonumber \]
\[ \begin{matrix} \text{Energy of a photon:} & E_{photon} = h \nu = \frac{hc}{ \lambda} \end{matrix} \nonumber \]
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.
\[ \begin{matrix} \nu = \nu & n_i = 1 & n_f = 3 & \begin{array}{c|c} \nu = \frac{-2.178 aJ}{n_i^2} + h \nu = \frac{-2.178 aJ}{n_f^2} ~ & _{float,~4} ^{solve,~ \nu} \rightarrow \frac{0.2922e16}{sec} \end{array} \end{matrix} \nonumber \]
\[ \begin{matrix} \text{Calculate the wavelength of the photon:} & \lambda = \frac{c}{ \nu} & \lambda = 102.598 nm \end{matrix} \nonumber \]
\[ \begin{matrix} \text{Calculate the energy of the photon:} & h \nu = 1.936 aJ \end{matrix} \nonumber \]