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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: me = 9.1094 x 10-31 kg e = 1.609 x 10-19 C ε o = 8.8542 x 10-12 C2/N×m2 h = 6.626 x 10-34 J×s c = 2.9979 x 108 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