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Verification of the Balmer’s Formula and measuring the Rydberg Constant for Helium and Deuterium Jaythan Edrick S. Salazar,* Gelli Mae P. Gloria, and Ma. Janelle G. Manuel National Institute of Physics, University of the Philippines, Diliman, Quezon City, Philippines * Corresponding author: [email protected] Abstract The Rydberg-Ritz formula and Balmer series describe the relationship of light emitted or absorbed by electrons of hydrogen-like atoms and their energy levels. This experiment aims to obtain the emission spectrum of hydrogen, deuterium, and helium and evaluate their applicability to the Balmer series by observing the constants. Using a student spectrometer and diffraction grating, the angular displacement of emission lines from the three discharge lamps were recorded. For hydrogen and deuterium, the calculated Rydberg constants had minimal deviations from the theoretical value. For helium, the calculated Rydberg constant deviated from the theoretical by 74.69% and 75.94% for the first order and second order diffraction, respectively. The deviation found in Hydrogen and Deuterium can be caused by interference from external light sources while the deviation found in Helium can be attributed to its non-applicability to the Balmer series and Rydberg-Ritz formula. Keywords: Balmer, Hydrogen, Deuteurium, Helium, Rydberg, emission spectrum 1 Introduction Deuterium is an isotope of hydrogen, with a neutron and a proton bound together in its nucleus and is known as ”heavy hydrogen”. Both Hydrogen and Deuterium have one valence electron. Actually, singly ionized atoms are all known as ”Hydrogen-like”, with Deuterium as an example. Helium has 2 protons and 2 neutrons in its nucleus, and has 2 valence electrons. As a noble gas, it is difficult to get it to share its electrons, even though a singly ionized He+ is Hydrogen-like. [1] The Rydberg formula is used to calculate the wavelength of energy emitted or absorbed when an electron in a hydrogen-like atom transitions from one quantum state to the other.It is given by 1 1 1 2 = RZ − 2 λ n22 ni (1) where R is the Rydberg constant, Z is the atomic number of the hydrogen-like atom, n2 is the final quantum state, and ni is the excited state levels of the atom. The Balmer series is a set of spectral emission lines for Hydrogen that are a result of electron transitions from higher levels (ni =3,4,5,6) to low energy levels (n2 = 2) [Figure 1]. The formulation is a special case of the Rydberg formula. 1 1 1 =R 2 − 2 (2) λ n2 ni The experiment was conducted to obtain the emission spectra of deuterium, helium, and hydrogen discharge source. From these spectra, the applicability of the spectrum of each gas to the Balmer series of Hydrogen and the Balmer’s equation were compared and evaluated by observing their experimental Rydberg constant. 2 Methodology 2.1 Materials A student spectrometer was the main apparatus used in the experiment, and a diffraction grating (300 lines/mm) was used. A magnifying glass was used to read the measurements on the Vernier scale of the spectrometer. 2 lamps, a desk lamp for illumination while reading measurements, and a discharge lamp as the main light source of the experiment, were used. Different discharge lamps were used, namely Hydrogen, Deuterium, and Helium [4]. 1 (a) (b) Figure 1: (a)Different spectral series of a hydrogen atom [2] and(b) Emission spectra of different gases[3] 2.2 Procedure After calibrating the student spectrometer using the first red line and making sure that the diffraction grating is perpendicular to the optical axis (absolute difference should be less than 10’), the angular displacements of each visible bright line in the first order and second order for all discharge lamps were measured and recorded. In the calculation, the direct image angle was subtracted. The wavelength from the angular displacements was calculated for every 1st and 2nd order color [4]. 3 Results and Discussion Using a diffraction grating of grating constant equal to 300 lines/mm, the student spectrometer was first aligned. Using the Hydrogen discharge lamp, the direct image and the visible first order red were used as a basis for the alignment. The angular displacements recorded is shown by Table 1. Table 1: Angular displacement of a hydrogen emission line separated using grating Trial 1 Direct Image θ0,cw θ0,ccw 176.5o 356.5o Color line θcw θccw 165o 8o ∆θcw ∆θccw 11o 30’ 11o 30’ Absolute Difference 0’ The determination of the experimental Rydberg constants of the different gases were done by locating the first- and second-order visible spectral lines. Tables 2, 3, and 4 show the different measurement for Hydroen, Deuterium, and Helium, respectively. Table 2: Balmer series for the 1st and 2nd order diffraction of Hydrogen ni 3 (Red) 4 (Blue-green) 5 (Blue-Violet) ∆θ 11o 30’ 8o 15’ 7o 23’ First Order −1 λexp (nm) λexp (mm−1 ) 645.5 1549 478.3 2091 428.4 2335 (a) (ni )−2 0.1111 0.0625 0.0400 ∆θ 23o 16o 44’ 15o Second Order −1 λexp (nm) λ−1 ) exp (mm 651.2 1536 479.9 2084 431.4 2318 (ni )−2 0.1111 0.0625 0.0400 (b) Figure 2: (a) First and (b) Second-order spectra of Hydrogen From the data shown in Table 2, the Rydberg constant for Hydrogen was determined as the slope 2 from a graph of λ−1 vs n2i of the first and second-order diffraction. Based on the slope of the graphs from Figure 2, the experimental R for the first and second-order were calculated to be 1.1061 x 107 m− 1 (relative deviation: 0.85 %) and 1.1050 x 107 m− 1 (relative deviation of 0.75 %), respectively Hydrogen has a proton number, Z, equal to 1. The experimental n2 for both orders were also calculated and were found to be both equal to 2, with no deviation from the theoretical. Only three out of four emission lines were obtained for Hydrogen. Moreover, only three emission lines were seen from the two orders of diffraction of Hydrogen. Table 3: First- and second-order-diffraction series of Deuterium ni 3 (Red) 4 (Blue-green) 5 (Blue-Violet) ∆θ 11o 13’ 8o 19’ 7o 30’ First Order −1 λexp (nm) λexp (mm−1 ) 648.4 1542 482.1 2074 435.1 2298 (ni )−2 0.1111 0.0625 0.0400 (a) ∆θ 22o 54’ 16o 44’ 14o 50’ Second Order −1 λexp (nm) λ−1 ) exp (mm 648.5 1542 479.9 2084 426.7 2344 (ni )−2 0.1111 0.0625 0.0400 (b) Figure 3: (a) First and (b) Second-order spectra of Deuterium The experimental R for the first and second-order were calculated to be 1.0682 x 107 m− 1, with a relative deviation of 2.61 %, and 1.1255 x 107 m− 1 (relative deviation of 2.62 %), respectively. Since Deuterium is an isotope of Hydrogen, its proton number, Z, is also 1. The experimental n2 for both orders were also calculated. 1.98, with a relative deviation of 1.00 %, was obtained from the first-order emission spectra while n2 =2 with no deviation from the theoretical was obtained for the second-order spectral lines. Only three out of four emission lines were obtained for Deuterium. Table 4: First- and second-order-diffraction series of Helium ni 3 (Red) 4 (Blue-green) 5 (Blue-Violet) ∆θ 11o 30’ 8o 30’ 7o 30’ First Order −1 (mm−1 ) λexp (nm) λexp 664.6 1505 492.7 2030 435.1 2298 (a) (ni )−2 0.1111 0.0625 0.0400 ∆θ 23o 24’ 17o 20’ 15o 22’ Second Order −1 ) λexp (nm) λ−1 exp (mm 661.9 1511 496.6 2014 441.7 2264 (ni )−2 0.1111 0.0625 0.0400 (b) Figure 4: (a) First and (b) Second-order spectra of Helium The experimental R for the first and second-order were calculated to be 2.7755 x 106 m− 1 (relative deviation: 74.69%) and 2.639 x 106 m− 1 (relative deviation of 75.94 %), respectively. The experimental n2 was also calculated for helium. For this case, Z=2. The values obtained were 2.01 with relative deviation of 0.50% and 1.98 with relative deviation of 1.00% for the first and second order, respectively. 3 The same trend was also made in the emission spectra for Helium where only 3 emitted wavelengths were observed. The deviation found in Hydrogen and Deuterium can be caused by interference from external light sources. In addition, only three out of four emission lines were observed. The deviation found in Helium can be attributed to its non-applicability to the Balmer series and Rydberg-Ritz formula. 4 Conclusion For hydrogen and deuterium, the calculated Rydberg constants had minimal deviations from the theoretical value. For helium, the calculated Rydberg constant deviated from the theoretical by 74.69% and 75.94% for the first order and second order diffraction, respectively. From this, we can verify that the Balmer series is not applicable for non-Hydrogen like atoms like Helium. References [1] [2] [3] [4] R. Harris, Modern Physics, 2nd Edition (University of California, Davis, 2008). Langsam (Retrieved from http : //eilat.sci.brooklyn.cuny.edu/cis15 /OldHW s/HW 2dC .htm, nd). M. Bolte (Retrieved from http : //astronomy.nmsu.edu/geas/lectures/lecture19/slide02.html, nd). Physics 104.1 Laboratory Manual: Rydberg Constant (National Institute of Physics, University of the Philippines Diliman, 2018). 4