Quantum Mechanics Lab 2:
The Emission Spectrum of Atomic Hydrogen: A Simulation
Objective:
One of the greatest successes of quantum mechanics was its ability to interpret the line spectra of atoms. Such spectra, which clearly indicated that electronic transitions in atoms occurred between discrete electronic energy levels, could not be explained at all by classical Newtonian mechanics and electrostatics. It was only through the introduction of “new” concepts such as energy quantization, quantization of angular momentum and “space” or magnetic moment quantization that such spectra could be fully and satisfactorily understood. Indeed, the successful interpretation of the spectrum of atomic hydrogen was among the strongest evidence for the validity of the “new” theory of quantum mechanics in the early part of the 20th century.
In this exercise, you will use a simulation of a prism spectrograph to observe and measure the wavelength values for a portion of the visible line spectrum of atomic hydrogen. With this data, you will obtain an experimental value for the Rydberg constant, R_{H}, and calculate an experimental value for the ionization energy for the hydrogen atom. You can then compare your value to the well-known literatures values, for each of these two numbers.
Theory:
The emission spectrum of atomic hydrogen is highly regular and contains within it several obvious geometrical progressions (see Figure 1). In the years leading up to the application of quantum theory to the spectrum of hydrogen, scientists had laboured to find an empirical formula or relationship that would explain this pattern, and in 1885, an amateur Swiss scientist named Johann Balmer did just that. He demonstrated that a plot of the energy or frequency of the lines in the visible portion of the hydrogen atomic emission spectrum was linearly related to 1/n^{2}, where n was an arbitrary integer. In particular, he showed that the frequency of the emission lines
1) = 8.2202×10^{14 }_{}1− _{n}^{4}_{2 }^{}_{ }s^{-1} where n = 3,4,5,…
_{}
Later this integer, n, was shown to be a quantum number associated with the energy levels involved in the electronic transition of the atom, which gives rise to the observed emission. In essence, atomic H^{+} recombines with an electron to form an excited state H atom. When the excited H atom relaxes to its lowest energy state, light is released with a photon energy corresponding to the energy difference for the transition of the electron from the excited state energy level, to the ground state energy level. Each observed line in the spectrum corresponds to a different electronic transition, in turn.
Figure 1
A schematic representation of the various series in the hydrogen atomic emission spectrum. Reprinted from “Quantum chemistry” by Donald A. McQuarrie, University Science Books. 1983, p. 19.
The formula introduced by Balmer was later generalized by Johannes Rydberg, who was able to account for all the lines in the atomic hydrogen spectrum, including those in the UV (Lyman) and the near infrared (Paschen). The form of the Balmer equation he developed was in terms of the wavenumber of the emission lines,
1 1 -1
2) =109,680 2 − n22 cm
n1
where n_{1} and n_{2} are integers such that n_{2} > n_{1}. The constant at the beginning of this expression is the Rydberg constant, R_{H}, one of the most accurately known constants of nature. The Rydberg formula is related to the Balmer expression by choosing n_{1} = 2 and multiplying the term in parentheses by 4.
The fact that this single expression could explain every line in the atomic hydrogen emission spectrum, using only two integers (quantum numbers) was a tremendous success and incentive to continue to search for a theory that would give rise to such a simple and elegant expression.
In the early years of the “quantum mechanical revolution”, the solution to the Schrödinger equation for the hydrogen atom was indeed obtained, and yielded the following expression for the energy levels:
- E n= 8−2 2 2h ne4 n =1,2,3,K
In this expression, the value is the reduced mass of the proton and the electron in the hydrogen atom, = m_{p}*m_{e}/(m_{p}+m_{e}) ~ m_{e} (approximately). Using this expression, it is possible to predict that the difference in energy between any two electronic energy levels will be
- =E En1 – En2 =8−2eh42 n112 − n122
which clearly has the same form as the Rydberg formula given above as Equation 2), and shows that the Rydberg constant (which is equivalent to the collection of constants in the left-hand set of parentheses) may be obtained from first-principles theory as well as from experimental data. The close agreement between the two was a major triumph for the quantum theory.
The geometric progression in the atomic hydrogen spectrum converges at a limit associated with a value of n_{2} = ∞. At this point in the absorption spectrum, the electron has been promoted beyond the last bound or discrete energy level into the unbound continuum corresponding to a free electron leaving a proton. We typically consider such an event as atomic ionization, corresponding to the reaction: A + energy (light) → A^{+} + e^{–}. The reverse reaction where an electron recombines with the atomic ion results in a release of light as emission. Thus, the Rydberg formula, or corresponding experimental data can be used to predict the ionization energy for atomic hydrogen. This may be obtained directly from the Lyman series (where n_{1} = 1, i.e., transitions occur from the true electronic ground state), where
8−e4 1− 1 =−82eh42 = IE H( )
5) =E E1 −En = 2h2
In this exercise, you will use a hydrogen discharge lamp to obtain the atomic emission spectrum of hydrogen. The hydrogen lamp is filled with a low pressure of molecular hydrogen, and relies on excitation from the electrical discharge which occurs within the tube to dissociate molecular hydrogen, yielding highly electronically excited hydrogen atoms. These atoms “relax” to the ground state by emission of light, in accordance with the Planck equation ΔE = hυ = hc/λ, such that the emission contains characteristic lines associated with the atomic hydrogen spectrum. An unfortunate feature of the lamp is that emission associated with other processes, such as electronH_{2}^{+} recombination, atom-atom recombination to reform molecular hydrogen, and other processes simultaneously occur. This provides a continuous background emission which results in a dispersed white light spectrum under the line spectrum of hydrogen. This cannot be avoided, and must be tolerated. The chief problem with this continuous “rainbow-like” emission is that it makes observation of weaker atomic emission lines very difficult, however, the most distinct and most intense lines of hydrogen are readily evident.
Procedure:
You will use the Virtual Experiment (simulation) of prism spectrograph from the website:
Virtual Amrita Laboratories: http://vlab.amrita.edu/?sub=1&brch=195&sim=359&cnt=4
to identify and measure the wavelengths of a portion of the Balmer or visible emission lines of the hydrogen atom.
Please note you will need to register for a FREE access to this experiment BEFORE you arrive at the lab.
Turn on the Classroom Laptop and login to the desktop (ask your demonstrator for the password, if needed).
Click on the tab for “Simulator”. You should see the prism spectrograph:
Move the “Calibrate Telescope” toggle until a clear picture comes into view (the “start” tab will become be functional). Hit “start”.
Select the mercury lamp from the pull down menu on the Right Hand side (under “Variables”). Click on the button to turn on the lamp and then the button to place grading.
E
Move the mouse over to the end of the eyepiece (E, in the above diagram). A double-headed arrow should appear allowing you to move the eyepiece by holding the mouse button and moving the mouse. When you move the eyepiece, you should see several coloured line appear in the magnified circle at the top of the simulation box. These coloured lines are the characteristic atomic emission lines of excited-state Hg. The three most intense lines will be in the blue-violet (435.8 nm), green (546.1 nm) and yellow (in the true emission spectrum for Hg this line would be a doublet, 579.1 and 577.0 nm – however, this simulation shows just one line at 578.0 nm). There is also a red line at 623.4 nm, and a violet line at 404.7 nm.
Move the eyepiece until the green line is in the centre of the cross hairs.
Use the two Vernier scales to record the position of the instrument (the diffraction angle) for each of the visible lines listed in Table 1. You can view each of the Vernier scales by place the mouse over the scale and looking at the position in the “zoom” window.
You will repeat these measurements for each visible spectral line.
In your lab notebook, create a chart with columns for line colour (helpful for keeping track), wavelength, and each of the two angle readings. The wavelengths for the Hg lines are given above, and those for the Ne and H lines are in Table 1. Record the angle reading of both Vernier scales for a given line – to read a Vernier scale you first look at the 0 line on the top scale. Whichever number this line matches up with on the bottom scale is your degrees reading. The bottom scale is divided into ½ degrees, and the number that you want is the line on the outer scale that is to the LEFT of the inner scale 0 line. Therefore, if the inner O line falls between 108.5° and 109°, you would record 108.5° (the line to the left of the inner O line).
The Vernier measurement can have more significant accuracy than one decimal point. Thus, the next step is to find the line on the top scale that exactly matches up with a line on the bottom scale – a number of lines will appear to match up, but there will only be one (maybe two) that match the best. This line is on the top scale, so it has a value between 0 and 30, and its units are in minutes. To calculate your total angle measurement, you take the degrees reading and add to it the minutes reading divided by 60:
minutes
total angle = angle+
60
You must make the angle measurement at two points, because there is a systematic error in the instrument. However, by making the measurement at two points, the error can be cancelled by symmetry. The second measurement will be greater than 180 degrees, since it is measured in the other hemisphere. From the larger measurement, subtract 180 degrees. With both measurements obtained, average the two readings, to get an angle associated with the wavelength of light. This procedure eliminates error in the position of the scales on the instrument very effectively.
When you have finished recording the Hg lines, switch the lamp setting to the Neon lamp. In the same way as suggested above, measure the angle of diffraction for each visible line in the Ne spectrum (see Table 1) and note its colour. You may not be able to see lines for all of the wavelengths listed in Table 1, but assign wavelengths to those you can see using the colour of the line and what you know about the wavelength range associated with that colour – there is a poster of emission spectra in the lab WHICH YOU MUST USE TO IDENTIFY COLOUR WITH EXACT WAVELENGTH. You MUST consult this before leaving the lab, to associate the exact wavelength with the colour.
When finished with the Ne lamp emission, switch to the hydrogen emission lamp. Search for and record the angular position data needed for calculation of the wavelength of each of the lines you find. You can again consult the poster in the CHEM 2500 laboratory to verify that you are seeing the spectrum you expect. When you have recorded data for at each line, close the program and turn off the laptop computer.
Table 1
The strongest emission lines of various elements in selected wavelength regions
Element | Wavelength | Peaks |
Helium (He) | 380-700 nm | 382, 389, 396, 403, 414, 439, 447, 471, 492, 502, 588, 668 |
Neon (Ne) | 530-700 nm | 540, 594, 610, 614, 633, 638, 640, 651, 660 |
Sodium (Na) | 460-700 nm | 467, 498, 515, 569, 589, 616 |
Argon (Ar) | 400-500 nm | 404, 416, 420, 427, 434 |
Krypton (Kr) | 400-600 nm | 427, 432, 438, 446, 557, 587 |
Mercury (Hg) | 400-700 nm | 405, 436, 546, 577, 579 |
Calculations:
Following completion of the experiment, there is an Excel template provided to you, into which you should enter your collected wavelength and angle data. The template will then use your He and Hg data to create two graphs – angle vs. wavelength, and angle vs. 1/(wavelength)^{2}. These are calibration curves, and a best fit line with equation will be displayed for each plot. The best fit line for the reciprocal wavelength plot will be linear, and the best fit line for the wavelength plot will have a polynomial equation. Using the linear equation you can now convert the measured angles from the hydrogen emission into wavelengths:
1
angle = (slope)(wavelength _{2 }) +intercept
For your hydrogen emission data, use the calculated wavelength values to determine the wavenumber value for each emission line. You should also look at the H-atom spectrum and assume a value of n_{2} for each emission line knowing that the visible transitions of the H-atom spectrum occur between upper states of n_{2}=3,4,5,… and n_{1} = 2 (the Balmer series). Next, calculate 1/n_{2}^{2} for each line, and enter all of this data into your template. If you choose not to use the template, you are required to perform all of these plots yourself. If you have trouble using the template, consult your teaching assistant.
If you are using the template, this will automatically construct a plot of wavenumber vs. 1/n_{2}^{2} for the H-atom data, and display the best fit line and equation. From the slope of this plot, calculate the Rydberg constant (expand the parentheses in Equation 2 to recast it as a linear equation relating the wavenumber to 1/n_{2}^{2 }to help do this). From the intercept obtained, confirm that the value of n_{1} for the visible lines of hydrogen is 2, that is, the emission is to the first excited state of hydrogen. Use this information to establish the ionization energy for the ground (n = 1) and first excited state (n = 2) of hydrogen, employing your value for the Rydberg constant.
Questions:
- Compare your value of the Rydberg constant to the accepted value of 109,677.57 cm^{-1}. Is the difference within the expected uncertainty of your experimental value? If not, how might you account for it?
- Use your results to calculate the first and second ionization energies of the hydrogen atom.
Present your answer in J mol^{-1}. (An atom has been ionized when the final state n_{2} = infinity)
- Suppose we want to view the emission spectra for transitions to the n = 1 state of hydrogen. What part of the spectrum would this be in? What about the transition to n = 3? In general, what can you say about the energy and frequencies of photons needed to promote electrons from successfully “further away” or higher energy orbitals? In which order of n = 1, 2 and 3, is the electron more tightly bound to the nucleus?
Instructions for Lab Report
A formal report is required in accordance with on the Guidelines in the “Guidelines for Laboratory Reports” provided in the Laboratory Information Section in Black Board.
Please make your writing concise and organize the presentation of your results. Your lab report should be around 6 pages! A 15 page lab report that rambles on and on will be penalized!
Title
{Name, CHEM 2500H Lab Section, Due Date)
Introduction: Start the report by stating, in one or two sentences of your own words, what was the main purpose of the exercise.
In a few lines, state either (1) the main equation or equations that will be used in the analysis (without reproducing the derivation!) and summarize how it will be used, or (2) the main approach taken in the exercise, and the reason why this is being done.
Results: Present your data table, and the 3 graphs here. Feel free to use the template provided.
Data Analysis: and Calculations: Provide a sample calculation for:
i) A calculation for ONE of the average angles for Graph 1. ii) A calculation for ONE of the wavelengths for hydrogen in Graph 3 iii) A calculation for ONE of the wavenumbers for hydrogen in Graph 3 iv) Your calculation of the Rydberg Constant.
Discussion: Concise and relevant discussion around your results. You may refer to your figures here i.e, “As seen in Figure 2, …..”
Please answer the 3 questions here. You may list them 1, 2, 3 if you prefer, or include them in the body of your discussion.
Conclusions: A final paragraph. summarizing key results, and the accuracy of the technique. Do not repeat your discussion
References: Please provide a list of references at the end of your report.