where did they come up with 2.19 x 10 ^6 Can an electron in a hydrogen atom have

Question

where did they come up with 2.19 x 10 ^6

Can an electron in a hydrogen atom have a speed of 3.60 times 10^5 m/s? If so, what are its energy and the radius of its orbit? What about a speed of 3.65 times 10^5 m/s? To be in a stationary state, the electron must have speed v_R = v_1/n, with n = v_1/v an integer. Only if v_1/v is an integer is v an allowed electron speed. solve A speed of 3.60 times 10^5 m/s would require quantum number n = v_1/v = 2.19 times 10^6 m/s/3.60 times 10^5 m/s = 6.08 This is not an integer, so the electron cannot have this speed. But if v = 3.65 times 10^5 m/s, then n = 2.19 times 10^6 m/s/3.65 times 10^5 m/s = 6 This is the speed of an electron in the n = 6 excited state. An electron in this state has energy E_6 = -13.60 eV/6^2 = -0.378 eV and the radius of its orbit is r_6 = 6^a_B = 6^2 (0.0529 nm) = 1.90 nm

Explanation / Answer

according to Bohrs second posulate

mvr = nh/2pi

v = nh/2pi*mr

Now

mv^2/r = kZe^2/r^2

Because necessary centripetal force is provided by the electrostatic force of attraction between electron and nucleus] whose charge is Ze where Z is the atomic number of the atom.

(m/r)(nh/2pimr)^2 = kZe^2/r

for hydrogen atom Z =1

r = n^2h^2/4pi^2 mke^2 ..(1)

v = nh/2pimr

from

v = 2pike^2/nh

for n =1 , k = 9*10^9 N.m^2/C^2

e = 1.6*10^-19 C, h = 6.63*10^-34 J.s

v = 2.19*10^6 m/s

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