Niels Bohr used classical orbits (as in the description of planetary motion around the sun) to quantize the motion of the electron in the Hydrogen atom. He started by assuming a circular orbit of radius r. He also knew
the mass of the electron, m=9.11x10-31 kg, the charge on the proton, e=1.602x10-19 C, the charge on the electron, -e= -1.602x10-19 C.
Bohr considered that the forces pulling the electron in towards the nucleus had to balance the force pulling the electron out, if the orbit were to be stable. Coulomb force(k=8.99x109 J-m/C2) Centrifugal force
The condition of balance, Fin + Fout = 0 implies that Fout = -Fin which yields:
force balance equation Bohr also considered that the angular momentum, L=mvr, must be quantized in units of
= 1.055x10-34 J.s: mvr = n
(n=1,2,3, ...) quantization equation These last two equations contain two unknowns, v and r. If we can solve for these, we can get the radius, r, of the orbit from which we can calculate the potential energy (of an electron at a fixed distance from a proton) and the velocity, v, from which we can calculate the kinetic energy, mv2/2. The quantization equation can be solved for v to give
which can be plugged back into the force balance equation to give:
or
or Solving this last equation for r, we get:
so rn = n2 x 0.529x10-10 m or r for n=1 is 0.529x10-10 m or 0.529 Å. This distance is called the Bohr radius. We can now use the equation for r to get v:
The total energy of this electron is now given as the sum of the kinetic energy (KE) and the potential energy (PE).
or
This last quantity RH = 2.18x10-18 J is the Rydberg constant and the energy of the nth state of hydrogen is
.
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