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Rydberg Constant from Balmer Series Jordan Brown 4/20/11 Rydberg Constant from Balmer Series Jordan Brown 4/20/11 “The most accurately measured fundamental physical constant.” [2] Background ● 1885: Johann Balmer experimentally determined 4 spectral lines of hydrogen in the visible spectrum. ● 1888: Johannes Rydberg determined a formula to predict the position of spectral lines in hydrogen-like atoms. ● Rydberg's formula is a modification of the Balmer formula with a constant Rm. ● 1913: Niels Bohr went on to present an updated model of the atom. He also empirically determined Rm from more fundamental constants. [1] Method ● Calibrate the apparatus ● Went on to measure alpha, beta, and gamma spectral ● Use Hg to determine lines. diffraction width. ● Determined Rm from these spectral lines. Apparatus Calibration ● We observed spectral lines of Hg to determine diffraction width with known wavelength of 546.074 nm. ● Determined diffraction slit width d of 3.276 +/- .005 um ● This value of slit width was determined from the diffracted angle theta which could only be determined within about 5 arc-minutes. ● Width d was determined from the slope of the next graph which has the relation d=(1/a) in this case, a is m_2=.00030416 => d=3276 nm. Hg Calibration Results ● We determined diffraction grating width d=3276 nm. ● For each data set, the value of the angles between the +/- n-values differed at most by about 0.4° but usually around 0.1° ● With d, we calculated wavelength of the alpha line to be 654.0 +/- 0.8 nm where the accepted value is 656.3 nm. Chi2- 18.4, Chi2/dof- 3.07, p-value of .006 ● Determined wavelength of Beta line to be 484.4+/- 0.8 nm. Accepted value is 486.1 nm Chi2- 4.04, Chi2/dof- .673, p-value of .67 ● Determined wavelength of Gamma line to be 433.4+/- 0.9 nm. Accepted value is 434.1 nm Chi2- 5.68, Chi2/dof- .947, p-value of .43 ● From these values of wavelengths, we plotted and found the value of Rm to be (1.097+/- 0.003) *10^7 (1/m). Accepted value: 1.0967* 10^7 (1/m). Chi2-1.60, Chi2/dof- 0.80. for 2 dof, p-value was .45 [3] Alpha Beta Gamma Rm Discussion ● We determined a good value for Rm that was only 0.02% off, achieving the 0.05% goal. ● Took a look at the values for the wavelengths for alpha, beta, and gamma and noticed a trend that they were all lower than the accepted value by about 2nm ● We suspected some sort of systematic error for this discrepancy; most likely to be the original calibration with Hg. ● To investigate the discrepancy, we considered altering the number of n-values to include in the fit for Hg. Unfortunately, this brought Rm down further. Conclusion ● We may have had a bit of error in original calibration of Hg. ● Given percent probability of 45 for Rm, it's a good model and the data supports the linear fit ● We calculated Rm of 1.097+/-0.003*10^7 (1/m). which is close to the accepted value of 1.0967*10^7 (1/m). References [1] Bohr, N. (1985). “Rydberg’s discovery of the spectral laws”. In Kalckar, J.. Collected works. 10. Amsterdam: North-Holland Publ. Cy.. pp.373/200/223379. [2] Mohr, P. J.; taylor, B. N.; Newell, D. B. (2008). “CODATA recommended values of the fundamental physical constants: 2006”. Reviews of Modern Physics 80: 633/200/223730. doi:10.1103/RevModPhys.80.633. [3] J.R. Taylor, An Introduction to Error Analysis, 2nd Ed. University Science Books, Suasalito CA (1997). .
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