Objective:

In this laboratory you will study the emission of light from a hydrogen discharge source.  You will measure the wavelengths of the visible lines in the Balmer series and and analyze your data to determine the Rydberg constant.

Background information:

When an electron changes from one energy level to another, the energy of the atom must change as well.  It requires energy to promote an electron from one energy level to a higher one.  This energy can be supplied by a photon whose energy E = hf = hc/λ.

Since the energy levels are quantized, only certain photon wavelengths can be absorbed.  If a photon is absorbed, the electrons will be promoted to higher energy levels and will then fall back down into the lowest energy state (ground state) in a cascade of transitions.  Each time the energy level of the electron changes, a photon will be emitted and the energy (wavelength) of the photon will be characteristic of the energy difference between the initial and final energy levels of the atom in the transition.  The energy of the emitted photon is just the difference between the energy levels of the initial (n_i) and final (n_f) states.

The set of spectral lines for a given final state n_f are generally close together.  The lines for which n_f = 2 are called the Balmer series and many of these spectral lines are visible.  Students will be measuring the wavelengths of the Balmer series lines in this laboratory.

The photon energies E = hf for the Balmer series lines are given by the formula

hf = -13.6 eV(1/n_i² - 1/2²) = 13.6 eV(1/4 - 1/n_i²).

We may write hc/λ = 13.6 eV(1/4 - 1/n_i²), or

1/λ = (13.6 eV(/hc))(1/4 - 1/n_i²) = R(1/4 - 1/n_i²).

The constant R is called the Rydberg constant.  Students will experimentally determine the Rydberg constant in SI units.

Equipment needed:

• Pasco Model SP-9268 precision student spectrometer

• a grating with 300 lines per mm

• a hydrogen discharge lamp and power supply

• a black cloth to black out stray light

Experiment: 

The different parts of the spectrometer are identified in the figure below.

The schematic diagram below illustrates its principle of operation.

The spectrometer consists of a collimator tube with an adjustable entrance slit, a rotatable table on which the diffraction grating can be mounted, a telescope for observing the diffracted light, and an accurate angular scale for measuring the relative directions of the various spectrometer components.  The diffraction grating disperses the light, so that the relationship between the diffraction angle θ and the wavelength λ of the light is given by the equation

dsinθ = mλ,   m = 0, 1, 2, ... .

The integer n is referred to as the diffraction order.
For a 300 lines per mm grating d = (1/300) mm. 
Students will measure the wavelengths for the n_i = 3, 4, 5, and 6 lines in the Balmer series of hydrogen by measuring the angle θ through which the light of the lines is deflected by the grating in first order (m = 1) and second order (m = 2).

Procedure:

• Place your hydrogen discharge source approximately 1 cm in front of the collimator slit, turn it on, and move the telescope until you can observe the illuminated slit through the telescope.  A wood block is provided, so that the light source can partially rest on the spectrometer platform.
  Caution: HIGH VOLTAGE!  Do not touch the tube, especially near the ends where the electrical contacts are made.
• Adjust the position of the light source to maximize the brightness of the image.  Adjust the slit width so that it is fairly narrow and adjust both the telescope focus and the telescope eyepiece so that both the cross-hairs and the illuminated slit are in good focus.  Make sure that the slit is vertical and the cross hairs are aligned vertically and horizontally.  (You can loosen the alignment ring and rotate the eyepiece.)
• Align the cross hairs with the left edge of the image of the light source.  Tighten the telescope arm rotation lock screw.  Then loosen the table rotating lock screw and rotate the table base until its zero mark lines up with the zero mark on the Vernier scale.  This sets your reference angle to zero.  Tighten the lock screw.
• Place the grating into the grating mount with the grating side of the glass against the vertical posts.  Loosen the spectrometer table lock screw and rotate the grating so that it is perpendicular to the axis formed by the collimator and telescope.  Tighten the lock screw. 
• Everything should be aligned and tightened down at this point.
• Loosen the telescope arm rotation lock screw, rotate the telescope to view the diffracted line of interest, align the left edge of the line with the vertical cross-hairs, tighten the lock screw and read the angular scale, estimating the angles to the nearest 0.1 degree on the scale.  Put the black cloth over the apparatus to shield the apparatus from stray light.  The nearest division on the main scale is 0.5 degrees, so the estimation involves 5 equal imaginary increments of 0.1 degree in each division.

  

  Record your angular readings (in degree) for the violet (6 --> 2),  violet-blue (5 --> 2), blue-green (4 --> 2), and red (3 --> 2) lines into columns D and E of the linked spreadsheet. This should be done for the first and second order right and left diffracted lines.  (Note:  The grating is strongly blazed.  The lines on one side are much brighter than on the other side.  You may not be able to see all the lines in second order on the weak side.)

Data analysis:

• Into cell F2 type =D2*PI()/180, and into cell G2 type =E2*PI()/180.  This gives the deflection angles in radians.  Copy the formula down to row 9.  Make sure that the left and right deflection angles are approximately the same (at least to within 0.01 radians).  If not, repeat the measurement, and if this does not help, check your alignment.
• Calculate the wavelength λ using the formula  dsinθ = mλ,  λ = dsinθ/m.  Into cell h2 type =(1/300000)*(SIN(F2)+SIN(G2))/(2*B2).  Copy this formula down to row 9.  You are averaging the right and left deflection angle by using (sin(θ_R) + sin(θ_L))/2, and you are using d = (1/300) mm = 1/(300000) m.  The units of the wavelength λ will be meters.
• Let column I contain 1/λ.  Let column J contain (1/4 - 1/n_i²).
• Plot 1/λ (y-axis) versus (1/4 - 1/n_i²) (x-axis).  The slope of this graph should be the Rydberg constant R.
• Add a trendline to find the slope.  Display the equation on the chart and set the intercept to be zero.
• Calculate the value for R = 13.6 eV/(hc) in SI units and compare your measured value (the slope) with your calculated value.  What is the percent difference?

  %difference = |calculated value - measured value|/calculated value