Determining the Rydberg Constant and the First Ionization Energy of Hydrogen
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EXPERIMENT 15
THE HYDROGEN EMISSION SPECTRUM
 Abstract
 To determine the Rydberg constant and the first ionization energy of hydrogen using the positions of the lines in the Balmer series.
 A direct vision spectrometer was calibrated using sodium and mercury lamps. Measurements was then taken for the Balmer series of hydrogen.
 The Rydberg constant (R_{H}) was 109797 cm^{1}. The first ionization energy of hydrogen was 1313 kJmol^{1}.
 Introduction
The history of the hydrogen emission spectrum dates back to the early 1900’s. Niels Bohr modified Rutherford’s model of the atom and proposed a new model in 1913. “The Bohr Model is a planetary model in which the negativelycharged electrons orbit a small, positivelycharged nucleus similar to the planets orbiting the Sun.”^{[1]} Bohr explained that an electron relaxation or excitation is observed when there is an alteration in the angular momentum of an atom.
When an atom is supplied with a source of energy including heat, electricity and light (photons), the electrons in the orbital absorb this discrete energy and become excited. Due to this excitation, electrons ‘jump’ from a lower energy orbital to a higher energy orbital, which is known as upward transition. The electrons then release the absorbed energy (photons) to become deexcited and reach ground state (minimum energy level). Photons are defined as discrete energy packets of electromagnetic radiation.
Bohr was able to explain the above theory relative to a hydrogen atom using the Rydberg formula. The equation [1] describes the wavelength produced during an electron excitation or deexcitation. Z is the nuclear charge of the atom, where in the case for hydrogen is 1, R_{H}is the Rydberg constant with value 13.61 eV, n is the principal quantum number, in which n_{1}> n_{2} and λ is the wavelength of the photon released.
${\frac{1}{\lambda}=R}_{H}{Z}^{2}(\frac{1}{{n}_{1}^{2}}\u2013\frac{1}{{n}_{2}^{2}})$
[1]
The hydrogen emission spectrum gives rise to three different types of electromagnetic radiation. The high frequency ultra violet waves are formed in the Lyman series where electrons deexcite themselves to the ground state energy level (n=1). The visible spectrum of wavelengths “656.28 nm, 486.13 nm, 434.05 nm” and 410.17 nm is within the Balmer series where electrons deexcite to n = 2.^{ [2]} IR radiation consist of the Paschen, Brakette, Pfund and Humpheries series relevant to electron downward transitions of n=3, 4, 5 and 6 respectively.
“The wave number refers to the number of complete wave cycles of an electromagnetic field (EM field) that exist in one meter (1 m) of linear space.”^{[3]}
$\stackrel{\u0305}{\nu}=\frac{1}{\lambda}$
[2]
By relating equation [2] with [1], the relationship below can be derived;
$\stackrel{\u0305}{\nu}={\frac{1}{\lambda}=R}_{H}{Z}^{2}(\frac{1}{{n}_{1}^{2}}\u2013\frac{1}{{n}_{2}^{2}})$
[3]
 Experimental Procedure
Note: Hydrogen, sodium and mercury is used in this experiment. Hydrogen is explosive, sodium is highly reactive and mercury is considered toxic and harmful for the environment if released. Ensure that all lamps are handled carefully during this experimental procedure and be vigilant about the use of lamps as lamps can heat up beyond a third degree burn.
All appliances including lamps and the direct vision spectrometer was used as supplied by the undergraduate chemistry laboratory. The method was carried out as described on pages 142 to 143 of the first year laboratory manual.
A white light source was initially placed near the entry slit of the direct vision spectrometer. The cross hair was then focused against the rainbow background by moving the end of the eye piece. The white light source was then replaced by a sodium lamp (figure 1) and the spectrum was focused. Six readings were taken for each yellow line observed whilst using the sodium lamp. Two yellow lines were observed.
The sodium lamp was then replaced with a mercury lamp. Six readings were taken for each of the four colours observed by the mercury lamp.
The mercury lamp was replaced by a hydrogen lamp finally and six readings was taken for each of three visible colours; red, turquoise and violet.
Figure 1. The direct vision spectrometer aligned with the sodium lamp
All lamps were switched off and kept away in their original places when the procedure with each lamp was completed.
 Analytical data
4.1 Spectrometer calibration using sodium and mercury lamp
Lamp 
λ / nm 
< λ > / nm 
λ / nm 
Δλ / nm 
< Δλ > / nm 

Sodium 
589.7 
589.6 
589.6 
589.8 
589.9 
589.7 
589.7 
589.592 
0.1 
0.4 
590.2 
590.1 
590.0 
590.1 
589.8 
589.9 
590.0 
588.995 
1.0 

Mercury 
579.9 
580.0 
579.8 
579.9 
579.8 
579.9 
579.9 
579.227 
0.7 

577.1 
577.3 
577.3 
577.3 
577.2 
577.5 
577.3 
577.140 
0.1 

546.4 
546.3 
546.5 
546.3 
546.5 
546.4 
546.4 
546.227 
0.2 

435.9 
435.9 
436.1 
436.1 
436.0 
436.0 
436.0 
435.956 
0.0 
Table 1. Calibration of spectrometer to calculate systematic error
< λ > = Average wavelength =
$\frac{\textcolor[rgb]{}{\mathrm{Total\; sum\; of\; the\; wavelengths\; measured}}}{\textcolor[rgb]{}{\mathrm{Number\; of\; wavelengths\; measured}}}$[4]
Δλ = Literature wavelength value – Average wavelength value [5]
< Δλ > = systematic error =
$\frac{\textcolor[rgb]{}{\mathrm{Total\; sum\; of\; random\; error}}}{\textcolor[rgb]{}{6}}$=
$\frac{\sum \textcolor[rgb]{}{\mathrm{\Delta \lambda}}}{\textcolor[rgb]{}{6}}$[6]
Colour 
λ / nm 
< λ > / nm 
Δλ / nm 
λ / nm 

Red 
657.1 
657.2 
657.3 
657.2 
657.2 
657.1 
657.2 
0.4 
656.8 
Turquoise 
486.6 
486.6 
486.7 
486.7 
486.6 
486.5 
486.6 
0.4 
486.3 
Violet 1 
434.3 
434.3 
434.5 
434.4 
434.4 
434.4 
434.4 
0.4 
434.0 
Violet 2 
No readings were taken 
410.2 
4.2 Wavelengths of the hydrogen emission spectrum
Table 2. Wavelengths of the Balmer series of hydrogen emission spectrum
λ = average value + systematic error = < λ > + Δλ [7]
4.3 Data Analysis and treatment
ṽ / cm^{1} 
n_{2} 
1/n_{2}^{2} 
15225 
3 
0.1111 
20565 
4 
0.0625 
23040 
5 
0.0400 
Table 3. Wavenumber ṽ for corresponding values of n
ṽ =
$\frac{1}{\mathrm{}\textcolor[rgb]{}{\mathrm{\lambda}}\mathrm{}}$[2]
Figure 2. Relationship between ṽ and 1/n_{2}^{2}
 Discussion of results
With reference to equation [3] the relationship between wavenumber ṽ and 1/n_{2}^{2} is observed.
$\stackrel{\u0305}{\nu}={\frac{1}{\lambda}=R}_{H}{Z}^{2}(\frac{1}{{n}_{1}^{2}}\u2013\frac{1}{{n}_{2}^{2}})$
[3]
If the equation [3] is further simplified for the hydrogen emission spectrum where Z = 1 and n_{1 }= 2, the equation of a straight line [8] can be obtained.
$\stackrel{\u0305}{\nu}=\u2013{R}_{H}\left(\frac{1}{{n}_{2}^{2}}\right)+\frac{{R}_{H}}{4}$
[8]
With respective to equation [8] and the general equation of a straight line graph y = mx + c,
${R}_{H}=\u2013\mathit{m}$
[9]
${R}_{H}=4\left(c\right)$
[10]
Therefore, two values of the Rydberg constant can be calculated using the gradient and the intercept.
Using the gradient; R_{H} = – (109864 cm^{1}) = 109864 cm^{1}
Using the intercept; R_{H} = 4 (27432 cm^{1}) = 109728 cm^{1}
R_{H (average)} =
$\frac{109864c{\mathrm{m}}^{\u20131}+109728c{\mathrm{m}}^{\u20131}}{2}$= 109797 cm^{1}
% error R_{H} =
$\frac{\mathit{Literature\; Value}\u2013\mathit{Calculated\; Value}}{\mathit{Literature\; Value}}\times 100$=
$\frac{109737\mathrm{c}{m}^{\u20131}\u2013109797\mathit{c}{m}^{\u20131}}{109737\mathit{c}{m}^{\u20131}}\times 100=$– 0.0542%
Electromagnetic radiation is a form of a wave with wavelength λ and frequency f and travels at the speed of light in a vacuum.
c = fλ [11]
f = c / λ [12]
Max Plank proposed the fact that energy of a photon is directly proportional to its frequency, with constant h (plank constant = 6.626 x 10^{34} Js)
E = h f [13]
By relating equations [12] and [13],
E = hc / λ [14]
By relating equations [2] and [14]
E = hc
$\stackrel{\u0305}{\nu}$[15]
By relating equations [3] and [15]
E = hc
${R}_{H}{Z}^{2}(\frac{1}{{n}_{1}^{2}}\u2013\frac{1}{{n}_{2}^{2}})$[16]
The first ionization energy of an atom is the amount of energy required to remove one mole of electrons from one mole of gaseous ‘X’ atoms to form gaseous ‘X^{+}’ ions under standard conditions of 1 atm pressure, 298 K.
The equation [16] gives the difference between energy levels for one atom. 1 mole of gas consists of N_{A} (Avogadro constant = 6.022 x 10^{23} mol^{1}) number of atoms. An electron is completely removed from the atom. Hence, n_{2}=
$\infty $.
By relating equation [16] and the definition of first ionization energy
First ionization energy (F.I.E.) = hc
${R}_{H}{Z}^{2}\left(\frac{1}{{n}_{1}^{2}}\right){N}_{A}$[17]
First ionization energy of hydrogen (F.I.E. H_{2} (g)) =
=6.626 x 10^{34} x 299792458 ms^{1} x 109797 cm^{1 }x 100 x 1 x 1 x 6.022 x 10^{23} mol^{1 }
=1313419 Jmol^{1} = 1313 kJmol^{1}
% error of F.I.E. H_{2} (g) =
$\frac{\mathit{Literature\; Value}\u2013\mathit{Calculated\; Value}}{\mathit{Literature\; Value}}\times 100$=
$\frac{1312.74982{\mathrm{kJmol}}^{\u20131}\u20131313\mathit{}{\mathrm{kJmol}}^{\u20131}}{1312.74982{\mathrm{kJmol}}^{\u20131}}\times 100=$– 0.0191%
“Ionization potential (I.P.) is the energy usually required to remove an electron from an atom, molecule, or radical, usually measured in electron volts (eV).”^{[4]}
1 eV = 96.4853329 kJ mol^{–1}
First I.P. of H_{2} (g) =
$\frac{1313\mathrm{}{\mathrm{kJmol}}^{\u20131}\times 1\mathrm{eV}}{96.4853329{\mathrm{kJmol}}^{\u20131}}$= 13.6 eV
% error of First I.P. H_{2}(g) = % error of F.I.E. H_{2}(g) = 0.0191%
Hence
R_{H} = 109797 cm^{1}
$\pm $0.0542%
F.I.E. of H_{2} (g) = 1313 kJmol^{1}
$\pm $0.0191%
First I.P. of H_{2} (g) = 13.6 eV
$\pm $0.0191%
The percentage uncertainty value of the calculated results is significantly low (<1.00%) The calculated values can be therefore considered accurate and is very close to the literature value. Six measurements was taken for each line using the direct vision spectrometer which ensures reliable results. The graph shows a value of R^{2}=1 which guarantees the consistency of the measurements taken using the direct vision spectrometer.
 Conclusions
The direct vision spectrometer calibrated had a systematic error of 0.4 nm. The Rydberg constant (R_{H}) is 109797 cm^{1} which was calculated using the balmer series of the hydrogen emission spectrum. The first ionization energy of hydrogen is 1313 kJmol^{1} which equivalent to a first ionic potential of 13.6 eV. Since energy for an orbital is given by
$E=\u2013\frac{{R}_{H}{Z}^{2}}{{n}_{1}^{2}}$
[18]
and since this energy is equivalent to the first ionization energy of an electron and Z and n_{1} is equivalent to 1, it can be concluded that
F.I.E. of H_{2} (g) = First I.P. of H_{2} (g) = Rydberg constant (R_{H})
Hence, the Rydberg constant (R_{H}) can be expressed in several units including 109797 cm^{1}, 13.6 eV and 1313 kJmol^{1}.
 Future work
The room in which the experiment was carried out was not dark enough. The room had some percentage of sunlight entering the room as well as other researchers were using high intensity white lamps during the conduct of the experiment. These light sources could have passed through the entrance slit along with the light from the lamps in which I conducted the experiment with. I did not take a reading for the second violet line of wavelength 410.2 nm since the cross hair was not visible in the line. The separation between the yellow lines in sodium emission spectrum was very small (0.5 nm approximately) and it was difficult to focus the cross hair on one line to take the measurement. The readings were only to one decimal place but literature values of the wavelengths were taken up to two decimal places. My readings would have been more accurate and reliable if more than six readings were taken for each line observed.
In the future, I would ensure that the experiment is handled in a very dark room with no interference of other light sources. A more accurate direct vision spectrometer measuring up to two decimal places will be used. More than six measurements will be taken to ensure reliability and accuracy of the results in the future.
 References
 https://www.thoughtco.com/bohrmodeloftheatom603815, (accessed 08^{th} October 2018, 02:20 pm)
 https://online.manchester.ac.uk/bbcswebdav/pid6107089dtcontentrid25707530_1/courses/I3022CHEM1060011811YR001700/YR1_EXP15_INTRO%281%29.pdf, (accessed 08^{th} October 2018, 02.30 pm)
 https://whatis.techtarget.com/definition/wavenumber, (accessed 08th October 2018, 02.32 pm)
 https://www.thefreedictionary.com/ionization+potential, (accessed 08^{th} October 2018, 08.23 pm)
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