Question

H-like atom: Bohr's model part 1

You carry out a theoretical work on absorption of alpha rays, passing on to a study of the structure of atoms on the basis of Rutherford's discovery of the atomic nucleus. By introducing conceptions borrowed from Quantum Theory established by Planck, you succeeded in working out and presenting a picture of atomic structure that, with later improvements, still fitly serves as an elucidation of the physical & chemical properties of the elements.

You started with H-like atom (ie - 1 electron circulating around nucleus carrying +Z charge). You realize Coulomb force of attraction on the electron, Fcoul, provides the acceleration a towards the nucleus: F_coul = Ze²/(4*pi*epsilon₀*r²) = m_ea = m_ev²/r

In order to quantize the energy level of the electron to generate discrete spectra to fit the experimental observation, you propose 2nd Bohr condition, quantization of angular momentum:

angular momentum = m_evr = nh/2pi where n = 1,2,3,...

a) Given the above 2 equations, please show by algebra that the radius of the electron (ie - the distance between the electron and nucleus) is quantized (ie - only some specific values are allowed):

r_n = (epsilon₀*n²h²)/(nZe²m_e) = (n²/Z)a₀

b) Calculate the value of the constant, ao, in the SI units. (This constant is later called Bohr radius)

c) Since radius is quantized, following the 2nd Bohr condition, the velocity of the electron must be quantized, too. Please show by algebra that v_n=nh/(2pi*m_er_n) = Ze²/(2*epsilon₀*nh)

Answer 1

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