THE
BALMER SERIES AND THE RYDBERG CONSTANT
Introduction
The different spectra of
atoms were very confusing before 1900, because there seemed to be no
mathematical relationship between the lines produced by different atoms, nor
any relationship between one atom’s spectra and another’s. In 1885 J.J. Balmer
discovered a mathematical relationship between the line spectra of the smallest
atom with the simplest spectrum:
hydrogen. But it wasn’t until
1905 that neils Bohr was able to provide an acceptable theory for the hydrogen
spectrum.
The purpose of this activity
is to verify Balmer’s formula:
f
= RH (1 / n2 - 1/4)
where f is the frequency of
light, RH is called the Rydberg constant, and n is an integer larger
than 2; n = 3, 4, 5, ...
The units of the Rydberg
constant will be the unit of frequency, Hz.
However, since frequency is proportional to energy, according to
Planck’s relation:
E
= h f
multiplying both sides of
the equation by Planck’s constant, 6.626 x 10-34 J-s, will express the
energy form of the relationship and give the Rydberg constant in units of
Joules, so that
E
= RH (1 / n2 - 1/4)
On the other hand, since
light is a wave, the relation can also be expressed in the familiar units of
wavelength:
f λ = c
or
f
= c / λ
where c = 3.00 x 108
m/s. Therefore, dividing the Rydberg
constant by the speed of light will give an alternate form of the constant
based on wavelength. However, this form
of the equation is not used here because the correct relationship will be in
terms on inverse wavelength, or wave number:
1
/ λ = RH (1 / n2
- 1/4)
Wavelength is a measure of
the length of one wave; wavenumber is a statement of how many complete waves
put in sequence (back to back) are needed to match a certain length, like a
nanometer.
Instructions
Use a very accurate
spectrometer to determine the wavelengths of light produced by the hydrogen
atom to complete the following table.
Assume that the red line corresponds to n=3, the green line is n=4, and
the violet line is n=5.
|
wavelength (nm) |
frequency (Hz) |
n |
n2 |
1/n2 |
|
|
|
3 |
|
|
|
|
|
4 |
|
|
|
|
|
5 |
|
|
Make a graph of frequency
vs. 1/n2. Find the value of
the Rydberg constant form both the slope of the straight line and the
Y-intercept.
Rydberg constant from slope: ______________
Rydberg constant from intercept: ______________
Accepted value of the Rydberg constant: ______________
Percent error: ______________
QUESTIONS
1) Calculate the frequency and wavelength of the
Balmer series spectral line corresponding to n=6. In what region of the spectrum - ultraviolet,
visible, or infrared - is this line located?
Given this fact, where are all the rest of the lines of the Balmer
series (n = 7,8,9,...) located in the electromagnetic spectrum?
2) The Rydberg constant may be given in units of
frequency, energy, or inverse wavelength, and all three values may be found in
the text book. Show that all three
values of the Rydberg constant are equivalent.
Multiply the constant in Hz by Planck's constant to get its value in
Joules. Divide the constant in Hz by
the speed of light to get its value in m-1.