In 1885, through a feat of inductive reasoning, Balmer discovered, in a totally empirical way, the formula for calculating the visible wavelengths present in the hydrogen spectrum. That is, Balmer did not derive his formula from a theory like Quantum Mechanics; he simply manipulated the numbers until he found a pattern. A glance at the visible hydrogen wavelengths in the table below will indicate the magnitude of his feat! Subsequently, after much work attempting to explain why Balmer's formula gives the visible hydrogen wavelengths so precisely, in 1913 Bohr provided a mathematical model from which Balmer's formula could be derived. It is interesting to note that 28 years passed between Balmer's formula and the theoretical explanation for it.

At first, Bohr's model was difficult to believe because one feature of the model was so strange: the orbital electron in the hydrogen atom had certain "permitted" orbits in which it would not emit light (even though electric charges moving in circles must emit light according to the classical electromagnetic theory). Bohr's electron would emit light, however, if it "jumped" from one permitted orbit to another permitted orbit of lower energy. (To jump from a low-energy orbit to a higher-energy orbit, the electron must absorb light.)

Balmer's formula is

where R is called the Rydberg constant with value 109,677.58 cm

- CAUTION: The difraction grating is a photographic reproduction and
**should NOT be touched**. The deeper recess in the holder is intended to protect it from damage. Therefore, the glass is on the shallow side of the holder and the grating is on the deep side.

- Place the grating on the center of the table with its scratches running
vertically, and with the base material (glass) facing the light source.
In this way, one can study diffraction without the complication of
refraction (recall from the previous lab how light behaves when traveling
through glass at other than normal incidence). Fix the grating in place
using masking tape.

- Rotate the table to make the grating perpendicular to the incident beam
by eye. This is not critical since the average of
_{R}and_{L}accommodates a minor misalignment.

- Affirm maximum brightness for the straight through beam by adjusting the
source-slit alignment. At this step, the slit should be narrow, perhaps a
few times wider than the hairline. Search for the spectrum by moving the
telescope to one side or the other.

- For each of the four colors in the hydrogen spectrum, measure the
angles
_{R}and_{L}to the nearest tenth of a degree by placing the hairline on the stationary side of the slit.

Wavelength (angstroms) Color Intensity _{R}_{L}6562.8 Red Strong 4861.3 Cyan Strong 4340.5 Indigo Strong 4101.7 Violet Weak

- Average the right and left angles for each color.

- Use the grating equation with d = 1 / N to find the wavelength
for each color, where
"N" is the grating constant determined experimentally in the previous
laboratory exercise.

- Calculate the percent deviation for each wavelength using
% deviation = (data-theory)/theory x 100% where "theory" is the tabulated wavelength. Do not ignore the sign; it contains information. A positive % deviation means that the value is above the theory; a negative % deviation means that the value is below the theory.

- Use your experimental values of
and the theoretical
value of the Rydberg constant (109,677.58 cm
^{-1}) in the Balmer formula to determine n_{2}for each color. Remember that n_{2}must be an integer greater than 2. Round your experimental n_{2}to the nearest integer.If all four of your n

_{2}values are very nearly 2, you have performed the unit conversion from angstroms to centimeters incorrectly. Remember that 10^{8}angstrom = 1 cm.

- Use your experimental values of
and the integer
n
_{2}values just found above in the Balmer formula to determine an experimental value of the Rydberg constant for each of the four colors.

- Report the best value of the Rydberg as the average of the four values
found above.

- Report the error on the best value of the Rydberg as the
standard deviation of the mean (SDOM). See
*Taylor*section 4.4 if you are confused.

- Does your best experimental value of the Rydberg agree or disagree with the
theoretical value? See
*Taylor*page 5 if you are confused.

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