Question

RH = 2π2μZ2e4 (4πε0)2h2

1/μ = 1/me + 1/mnucleus , where me = mass of electron = 5.4858 x 10–4 u and mnucleus = mass of nucleus.

RH =2.17868891 x 10–18 J = 1.09677759 x 107 m–1

Note that the value of RH in m-1 is the energy in wavenumbers; this what you get when you divide RH in Joules by h and c; it corresponds to 1/λ, the number of waves per meter.

1. Using the equation E=RH (1/nl2 – 1/nh2), calculate the energy (in m–1) and wavelength of a bright line in the hydrogen spectrum. This line should correspond to the energy emitted when an electron in a hydrogen atom goes from nh=4 to nl=2. What range of electromagnetic spectrum is this in?

2. What is the emission wavelength for an electron falling from the vacuum level (n=∞) to the lowest energy level in a hydrogen atom? Use the Rydberg equation and the numerical values for the Rydberg constant RH given above. In which part of the electromagnetic spectrum is this?

3. Use Bohr’s expression for the Rydberg constant to estimate the n=∞ to n=1 transition for 7Li. (Estimate that mnucleus = 7 u; this is not exactly right, but close enough.) You must adjust Z and μ (the reduced mass) in the expression compared to the last calculation; everything else should be the same. Of course, there is an easy way and a hard way to do this. In which part of the electromagnetic spectrum is this?

Answer 1

Line spectra of H-atom: - delta E = E_2-E_1 = -13.6z^2/n^2_2-[-13.6z^2/n^2_1] delta E = 13.6z^2[1/n^2_1-1/n^2_2] lambda = hc/(delta E)_n_2 rightarrow n_1 lambda = hc/13.6z^2[1/n^2_1-1/n^2_2] 1/ lambda = 13.6z^2/hc[1/n^2_1-1/n^2_2] 1/ lambda = v = Rz^2[1/n^2_1-1/n^2_2] n_2 > n_1 R = Rydberg constant = 1.09678 times 10^7m^-1 R = 13.6ev/hc Rhc = 13.6ev where 1/ lambda = Reciprocal of wavelength v = wave number. n_1 & n_2 = energy levels (n_2 > n_1) z = Atomic Number(in case of Li, z = 3) \r\n I_n = H-spectra TE = KE + PE TE = 1/2Kze^2/r + (-kze^2/r) E_n = TE = -1/2kze^2/r and r_n = n^2h^2/4 pi^2 mkze^2 E_n = 2 pi^2 m k^2z^2e^4/n^2h^2 E_n = -13.6 z^2/n^2 ev atom^-1 where k = columbic constant = 9 times 10^-9Nm^2/c^2 e = charge on electron = 1.6 times 10^-19c pi = 3.14 h = 6.62 times 10^-34J sec = 6.62 times 10^-34kg m^2 sec^-1 m = Mass of e^- = 9.1 times 10^-31kg z = atomic Number n = Energy lavel in spectra.