6: Optical Spectroscopy of Atoms (Graph)
- Page ID
- 496122
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Key Points
- Three graphs will be produced for various \( n_{i} \) values.
- The quantum number of the final state, \(n_{f}\) can be found using the slope and intercept of the most accurate graph.
Summarized Procedure
- Obtain the wavelengths of each emitted line in nm.
- Take the inverse of each wavelength and convert to the units of cm.
- Three graphs will be made plotting \( \frac{1}{\lambda} \) against \( \frac{1}{n_i^2} \) for the following three trial sets of values: \( n_i \) = 2, 3, 4, 5; \( n_i \) = 3, 4, 5, 6; and \( n_i \) = 4, 5, 6, 7. The wavelengths are associated with the \( n_i \)'s in inverse order for each set of values (red line with the smallest \( n_i \) and violet line with the largest \( n_i \)).
- The sequence of integers that gives the best straight line gives the correct assignment. The slope of the line is determined by least-squares analysis. In the provided spreadsheets, the slope and y-intercepts are calculated for you as well as the standard deviations in both parameters.
- The quantum number of the final state, \(n_{f}\). can be found in from the slope and intercept. Refer to the calculations for further information.
Calculations
Two main formulas that will be used to obtain \(n_{f}\) are the slope of the line equaling \( -R_{H} \) and the intercept equaling \( \frac {R_{H}}{n^{2}_{f}} \).
Plugging in the values obtained from the selected graph, we can then solve for the quantum number of the final state.
Making the Graphs:
| Example Graph | https://docs.google.com/spreadsheets...f=true&sd=true |
|
Student Graph (MKAE A COPY OF THIS, FILL IT OUT WITH YOUR DATA TO MAKE YOUR GRAPHS) |
https://docs.google.com/spreadsheets...f=true&sd=true |
Procedure for Graphs:
- Click "Student Graph" link above.
- Click "File" then "Make a Copy."
- Following the instructions on YOUR BLANK COPY, add your experimental data obtained from the lab into the YELLOW HIGHLIGHTED BOXES. The rest of the sheet is set to run the calculations for you. Do NOT adjust the other parts. If you do, you can go back to the original and obtain the necessary functions to fix it.
- The output part below your data input has data that is used in the calculations. Do not edit this part of the sheet.
- Your three graphs will automatically show up in this part of the sheet. The colors of the lines and dots on the graph are coordinated with the results on the left side. On the left, you will find the values plotted with respect to their \(n_{i}\) value. There will also be each graph's respective slope value, y-intercept, and standard deviations. Again, do not edit this part of the sheet.
- Here are your results for the lab! The slope and its standard deviation will be for the graph that produced the most linear line. You will now use this data accordingly to solve for the quantum number of the final state. The data is simply consolidated to this table for your ease. Again, do not edit this part of the sheet.