Question

Problem 4.54

ducl a calelülly scaled energy- for Het similar to that in Figure 4-16, showing the levels for n - 1, 2, 3, el daz (b) What is the ionization energy of He? (c) Compute the difference in wave- each of the first two lines of the Lyman series of hydrogen and the first of the He Balmer series. Be sure to include the reduced mass correction for both Show that for every spectral line of hydrogen, He has a spectral line of very 4, 5, and the same wavelength. (Mass of He* = 6.65 × 10-27 kg.) in the table are the L x-ray wavelengths for several elements. Construct a lot from these data. Compare the slope with the appropriate one in Figure 4-19 and interpret the intercept on your graph, using a suitably modified version of 4-53 Determine a Equation 4 4-35. P Ca Co Kr Mo I 15 20 27 36 42 53 Wavelength (nm) 10.41 4.05 1.79 0.73 0.51 0.33 Element 454. In this problem you are to obtain the Bohr results for the energy levels in hydrogen thout using the quantization condition of Equation 4-17. In order to relate Equation 4-14 to the Balmer-Ritz formula, assume that the radii of allowed orbits are given by r no here n is an integer and ro is a constant to be determined. (a) Show that the frequency of radiation for a transition to n,- 1 is given by f kZe /hron for large n. (b) Show that the frequency of revolution is given by e Use the correspondence principle to determine ro and compare with Equation 4-19 A-S5. Calculate the energies and speeds of electrons in circular Bohr orbits in a hydrogen- like use atom using the relativistic expressions for kinetic energy and momentum. iase tao Write a computer program for your personal computer or programmable calcu or that will provide you with the spectral series of H-like atoms. Inputs to be included n. Z, and the nuclear mass M. Outputs are to be the wavelengths and frequencies lines and the series limit for the specified ny, Z, and M. Include the reduced corection.(b) Use the program to compute the wavelengths and frequencies of the 100, name the series the [your name] series, and use your pute the wavelengths and frequencies of the first three lines and the limit. gure 4-26 shows an energy loss spectrum for He measured in an apparatus such as moen in Figure 4-24a. Use the spectrum to construct and draw carefully to scale an of the first six and program to com energy-level diagram for He

Answer 1

The electron orbiting around nucleus experience as force given by coulomb's law = kze^2/r^2_n (pointing radically inward) where = radius of nth orbit z = atomic number (number of protons) k = 1/4 phi epsilon_0 = 9 times 10^9 N m^2/c^2 e = charge of electron = 1.6 times 10^- 19 c this force is balanced by centrifugal force mv^2_n/r_n = k z e^2/r_n mv^2_n = k z e^2/r_n now the energy in n^th orbit is E_n = 1/2 mv^2_n - k z e^2/r_n putting mv^2_n from 1 E_n = 1/2 k ze^2/r_n - k ze^2/r_n = -kze^2/2v_n = -k ze^2/2r_0 n^2 {r_n = r_0 n^2 now the transition n rightarrow n - 2 Vertical-bar delta E Vertical-bar = Vertical-bar E_n - 2 - E_n Vertical-bar = + kze^2/2 r_0 [1/(n - 1)^2 - 1/n^2] = + kze^2/2r_0 [n^2 - (n - 1)^2/n^2 (n - 1)^2] = + kze^2/2r_0 [n^2 - n^2 - 1 + 2n/n^2 (n - 1)^2
For part C equation 4-19 is not provided