Experiment 9

Line Spectra (Rydberg’s Constant)

Objective:

The objective is to use the emission spectrum of hydrogen atom in order to verify the relation between energy levels and the photon wavelength as well as to calculate Rydberg’s constant R = 1.097x10⁷m^-1.

Equipment:

            A prism spectrometer, a low pressure hydrogen tube, a low pressure helium tube, a high voltage source, and a calculator         

Theory:

When an electron of an atom receives some energy by any means, it moves to a bigger orbit which energy level fits that electron’s energy.  Such atom is then said to be in an excited state.  The excited state is unstable however, and the electron returns to lower levels by giving off its excess energy in the form of electromagnetic radiation.  Max Planck showed that the frequency of occurrence ( f ) of a particular transition between two energy levels in an atom depends on the energy difference between those two layers.

E_n - E_m = hf

In this formula E_n is the energy of the n-th level, E_m, the energy of the m-th level (lower than the n-th level), and h = 4.14x10^-15 eV-sec is the Planck’s constant.  f is the frequency of the released photon.

Possibilities for the occurrence of electron jump from one level to other levels are numerous.  It depends on the amount of energy an electron receives.  An electron can get energized when a photon hits it, or is passed by another more energetic electron that repels it, or by any other means.  The electron return can occur in one step or many steps depending on the amount of energy it loses.  In the above figure, look at the 3 shown possibilities.  In each possibility, the red arrow shows the electron going to a higher energy level, and the black arrows show possible return occurrences.

Hydrogen is the simplest atom.  It has one proton and one electron.

For hydrogen atom, possible transitions from the ground state (E₁) to 2^nd state (E₂), 3^rd state (E₃), and 4^th state (E4) are shown in Fig. 1.   The possibilities for electron return are also shown.  The greater the energy difference between two states, the more energetic the released photon is when an excited electron returns to lower orbits.  If the return is very energetic, the wavelength may be too short to fall in the visible range and cannot be seen in the spectroscope.  Some transitions are weak and result in larger wavelengths in the infrared region that cannot be seen either.  However some intermediate energy transitions fall in the visible range and can be seen.

Grouping of the Transitions for Hydrogen:

Transitions made from higher levels to the first orbit form the Lyman Series.

Transitions made from higher levels to the second orbit form the Balmer Series.

Transitions made from higher levels to the third orbit form the Paschen Series.

Transitions made from higher levels to the fourth orbit form the Pfund Series.

Emission and Absorption Spectra

A hot gas emits light because of the energy it receives by any means to stay hot.   As was mentioned earlier, the received energy by an atom sends its electrons to higher levels, and in their returns, the electrons emit light at different wavelengths.  The emitted wavelengths can be observed in a prism spectrometer in the form of a few lines of different colors.  Each element has its own unique spectral lines that can be used as an ID for that element.  Such spectrum coming from a hot gas is called emission spectrum.  For a host gas spectral lines are discrete.

For white light entering a spectrometer the spectrum is a continuous band of rainbow colors.  This continuous band of colors in a spectrometer ranges from violet to red and gives the following colors:  violet, blue, green, yellow, orange, and red.  Light emitted from the Sun contains so many different colors (or electronic transitions) that its continuous spectrum gives variety of colors changing gradually from violet to red.  It contains so many different violets, blues, greens, yellows, oranges, and reds that it appears continuous. 

When white light passes through a cold gas, that cold gas absorbs only a certain energy-level photons and lets the remaining photons pass through.  The cold gas absorbs only those photons that it can emit when it is hot.  Therefore, if you are observing the continuous spectrum of a streak of sunlight into a spectrometer, and then place a cold tube of a certain gas in the way of sunlight before entering the spectrometer, the cold gas will absorb those photons that it can normally emit when it is hot and therefore some dark lines will appear inside the continuous spectrum.  The dark lines are called absorption spectrum.  It is very interesting to see that the absorption (dark) lines happen exactly at wavelengths that the emission lines occur when the gas is hot and emissive itself.

Procedure:

1. Place a spectrometer on the table at a height that you can comfortably look into it in sitting position.

2. Place a lit light bulb that emits bright white light nearby the spectrometer slit and adjust the spectrometer angle to where you can observe the rainbow colors in its eyepiece.  Explain why a wide and continuous range of all colors is present.  Write down the maximum and minimum wavelength you can read at the extreme ends of the band.

3. Place a cold hydrogen tube very near to the spectrometer slit such that the white light passes through hydrogen before entering the spectrometer.  Watch the spectrum again to see if a few dark lines appear in the previous continuous spectrum.  These dark lines are the absorption spectrum of hydrogen atom.  Write down the wavelengths at which these dark lines appear.

4. Remove the white light source. Place the hydrogen tube in the appropriate high-voltage source designed for it.   Turn the high voltage source on.  Adjust the position of the spectrometer and its entrance gap such that the hydrogen light going through it can easily be seen in the spectrometer.  The gas gets hot and excited due to the application of the high voltage that causes numerous ionizations and collisions between atoms, molecules, electrons and protons.  The collision between the ions (electrons and protons) means their repeated separation (ionization) and recombination.  Recombination of electrons and protons form excited or energized hydrogen atoms that emit several photons during stabilization.  The separation and recombination speed is of the order of a thousand trillion per second.  This is not happening for a single atom only.  The number of atoms in the tube could easily be of the order of 10¹⁸.  The effect of such numerous transitions appears as spectral emission lines in the spectroscope.

5. How many lines do you see and at what wavelengths?  Read the wavelengths and record them.

6. Compare your readings in steps 3 and 4.  Should they be the same?  Why?

7. Repeat steps 1 through 6 for a helium tube.

8. Use each wavelength you obtained for hydrogen in the formula for Balmer Series in order to calculate Rydberg’s constant.

9. Average the 3 or 4 values for R you obtained in the previous step.  This is your measured value of the constant.

10. Compare it with the accepted value of R = 1.097x10⁷m^-1 by calculating a %error.

Data:

Given:

            R_accepted = 1.097x10⁷m^-1

Measured:

The hydrogen visible wavelengths are:

                        λ₆₂ =   ?   nm,   λ₅₂ =   ?   nm,   λ₄₂ =    ?   nm,  λ₃₂ =   ?    nm,

Calculations:

Use the Balmer Series equation to calculate R for each of the measured wavelengths.  Next, find the average value of R.  It gives the measured value for R.

Comparison of the Results:  Calculate a %error on R using the usual %error formula.

Conclusion:        To be explained by students

Discussion:        To be explained by students