The Rydberg constant (i.e. the transition wavelength) for hydrogen-like atoms de
ID: 1033765 • Letter: T
Question
The Rydberg constant (i.e. the transition wavelength) for hydrogen-like atoms depends slightly on the mass of the nucleus due to the effective mass. The observation of splitting in the first spectroscopic line of the Balmer series (ni-2 to n2-3 transition, the so-called Balmer-a line) for hydrogen vs. deuterium was the first experimental confirmation of the existence of deuterium by Harold Urey, for which he received Nobel prize in 1934 Calculate precisely (to at least 5 digits) the effective masses uH and uo, the transition energies ??? and AED, and the transition wavelengths in nm (1 nm 109 m) AH and Av, for the ni-2 to n2 3 transition in hydrogen and in deuterium. Then, calculate the isotopic shift AA-A that Urey measured experimentally. Is this a large or a small shift in relative terms? Evaluate ????? and comment on how easy/difficult would it be to measure it. Use the following exact values for the masses of the particles: electron hydrogen nucleus mp=1.672648x10-27 kg deuterium nucleus mD-3.343467x1027 kg me-9.109534x10-31 kg ?? (energy in Joules) (4m0)22A2?Explanation / Answer
1. Reduced mass or effective mass :- U = me / (1 + me/mn) , where me is mass of electron and mn is mass of nuclues.
UH = 9.10457 * 10-31 kg
UD = 9.10705 * 10-31 kg
2. delta(EH) = 3.020575 * 10-19 J
delta(ED) = 3.021398 * 10-19 J
3. delta(lambda) = (657.63830396 - 657.45916956) nm = 0.1791344 nm . Since ratio of lambdaH - lambdaD = 1.000272465 and In astronomical observation, this corresponds to a blue Doppler shift of about 0.000272 times the speed of light, or 81.6 km/s [observation took from wikipedia] which indicates that the shift (0.1791344 nm) is not that small as it appears to be relatively.
4. ratio comes out to be :- 0.00027239045. As you can see that the ratio is very very small than 1 and from this you can inference that how precisely Urey measured it. The value is affected by the precision of our calculation.
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