Laboratory #3
Analysis of the Spectrum of a Hydrogen-like Atom
CHEM 1100
Section 03
Due: September 28, 2020
Purpose:
The purpose of this experiment is to understand how to find the nuclear charge of a
hydrogen-like atom using wavelengths and in the atomic spectrum and the Rydberg equation.
This lab provides students with the opportunity to gain practice and acquire new knowledge
regarding all observed spectral lines that are due to the movement of electrons between the
energy levels that can be found in the atom. Such spectral series plays a very important role in
astronomy since they help detect the presence of hydrogen and redshifts. Aside from the series
for hydrogen other elements were discovered along the way when spectroscopy techniques were
developed.
Theories/ Principles:
In this experiment, we were asked to investigate the atomic spectrum of hydrogen both
theoretically which meant that we had to use Excel to make calculations and experimentally
since it also included a section in which we needed to observe the spectrum and compare the
observed result to our calculated one. When atoms are excited, either by electricity or by heat
they often give off light. Some examples of this would be when a flame test is conducted as well
as a firework. The light that is produced is known as a characteristic of the electronic structure of
the atom, plus it is specific to it. Not only was light released when an atom was excited since a
wavelength was also emitted as a result of this action an atomic spectrum was created. It is said
that electrons can only exist in certain states, and each state has a fixed or quantized amount of
energy which can be supported by the quantum theory. Aside from this when an electron goes
from a lower state which typically means that they are closer to the nucleus to a higher state
which is farther from the nucleus, it must absorb the energy required; this may differ based on
the states. Lastly when an electron goes from a higher state to a lower state it emits energy, often
in the form of light which can also be considered a photon then must be equal to the change in
energy between the states.
In order to begin understanding the Rydberg equation, we need to break it down into its
components. The equation is as follows:
1
λ
= R H ( m12 −
1
n2 )
The values in the equation are the wavelength defined as λ , R as the Rydberg constant for the
atom which is 1.09678 × 10 7 m−1 , and m and n are positive non-zero ranges:
m= 1, 2, 3, …
n= m+1, m+2, m+3, …
The Rydberg constant for infinite separation, R​∞​, can be calculated by using a certain
fundamental constant of nature which happens to be given in the expression below:
R∞=
m e ·e 4
8ε 2o ·h 3 ·c
The values below are the components of the equation above which are needed to figure it out:
m​e​ = 9.1094 x 10​-31​ kg
e = 1.609 x 10​-19​ C
ε o​ = 8.8542 x 10​-12​ C​2​/N×m​2
h = 6.626 x 10​-34​ J×s
c = 2.9979 x 10​8​ m/s
electron mass
electron charge
vacuum permittivity
Planck’s constant
speed of light
Although equation 1 gives us the wavelength for the hydrogen atom spectrum. We are able to see
below that Z is the charge of the nucleus about which the electron is orbiting. Aside from this, a
general expression for any hydrogen-like atom is:
1
λ
=R ∞ Z
2
( m1
2
−
1
n2
)
An electron has two electrostatic forces constantly tugging and pushing on it which keep it in a
state of equilibrium caused by a positively charged proton and a negatively charged electron.
This then creates centrifugal force which prevents them from flying away.The two forces
mentioned above must be equal in magnitude but opposite in direction in order for the electron to
remain in its circular orbit. Based off this assumption the Bohr derived the energy of the electron
to be:
E
n
= −
m e ·e 4 1
8ε o ·h 2 n 2
=-2.179 × 10 −18 J n12
Experimental Procedures:
The first step would be to transfer the spectral data into the second column in a Microsoft
Excel spreadsheet. We do this in order to leave the first column for the calculated values of
m2 n2
(n2 −m2 )
. If we assume that our first value of m=1 and that subsequent values for n increase by 1
for each entry, then we can calculate the value for the equation mentioned and put it into the first
column. Using this information, we can create a scatter plot in the same program by setting the
info in column 1 as our values of x and values of column 2 as those for y. Then, if the data has
been input correctly, there will be a straight line.
We continue this process until the next line added is no longer part of the straight line.
This value will require us to increase the value of m by 1 and calculate a new value for x. Now
that we have added all of the spectral lines, we must add a trendline to the plot and determine its
slope which equals 1/C. Now that we have the slope, we can find the value charge of the nucleus
or Z and given the value of R ∞ .
Data Table Summary:
Table # 1
Wavelength (nm)
Wavelength (nm)
(Wavelength (nm)
10.257
45.588
208.402
10.292
48.241
216.121
10.342
54.03
240.68
10.42
72.941
291.763
10.553
102.573
319.222
10.806
106.096
337.689
11.397
111.691
366.332
13.508
121.568
415.617
42.628
142.463
450.252
43.224
192.965
517.086
44.125
201.99
828.873
This table contains the wavelengths on the handout which was given to us.
Table #2
m
n
m^2n^2/(n^2-m^2)
λ
1
2
1.3333
10.257
1
3
1.125
10.292
1
4
1.0667
10.342
1
5
1.04167
10.42
1
6
1.0287
10.553
1
7
1.0208
10.806
1
8
1.01587
11.397
1
9
1.0125
13.508
2
3
7.2
42.628
2
4
5.333
43.224
2
5
4.7619
44.125
2
6
4.5
45.588
2
7
4.3556
48.241
2
8
4.2667
54.03
2
9
4.0278
72.941
3
4
20.5714
102.573
3
5
14.0625
106.096
3
6
12
111.691
3
7
11.025
121.573
3
8
10.4727
142.463
3
9
10.125
192.965
4
5
44.4444
201.99
4
6
28.8
208.402
4
7
23.7576
216.121
4
8
21.3333
240.68
4
9
19.9385
291.763
5
6
81.8182
319.222
5
7
51.04167
337.689
5
8
41.0256
366.332
5
9
36.1607
415.617
6
7
135.6923
450.252
6
8
82.2857
517.086
6
9
64.8
828.873
The data table above shows the original information that I was able to obtain by substituting in
the values for M and N into the equation found in the third column. Meanwhile, the 4th column
represents the data table of wavelength as demonstrated in the previous table.
Figure # 1
These are all the points mentioned in table 1 in the context of a scatterplot with a trendline.
Figure # 2
This graph above contains information about the relationship between the wavelength and
absorption/emissions of hydrogen atoms. The curve also curves up as the atom absorbs or emits
more.
Results and Discussion:
Using the given values placed in the rightmost column of our table, we can begin to
analyze the information when we manipulate the Rydberg equation into a slope-intercept form.
In figure one, we can see that there are clusters of points that get close to the trendline however,
they are not very precise. In order to refine our results, we need to go through different points
and find which ones lie closest to the line. These points include (10.257, 1.3333), (42.628, 7.2),
(102.573, 20.5714), and (201.99, 44.444). The slope of this line then gives us the frequency of
the hydrogen atom provided in the laboratory handout. The Rydberg equation helps us
understand how it’s constant of separation can be broken up into different components such as an
electron’s mass, its charge, vacuum permittivity, Planck’s constant, and the speed of light. This
is then an essential step into solving for the wavelength. Each atom emits its own light frequency
and therefore helps us to identify them.
Conclusion:
In conclusion, this lab allowed us to understand how increasing the n value causes the
number of frequencies to decrease proportionally. This indicates that the relationship between
this value and the frequency is reduced, meaning that it has less of an electrostatic force which is
mentioned in Bhor’s theory. This then helps us understand how atoms can be different but still
fall under the same name as their frequency is not that different. There are also many constants
that allow us to reduce the number of errors by eliminating the potential of misreading a
measurement.
Reference:
Department of Chemistry. (2018). Experiments in General Chemistry. Los Angeles, CA:
California State University, Los Angeles