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In a hypothetical hydrogen like atom; the electric potential between a proton and an electron, distant r apart, is given by V = V0ln (r/r0)

where V0 and r0 are constants. Assuming that the atom still obeys Bohr’s postulates, the dependence of the radius of the nth allowed orbit, rn , on the principal quantum number (n) is as

(1) rn ∝ n

(2) rn ∝ n-1

(3) rn ∝ n2

(4) rn ∝ n-2

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(1) rn ∝ n

U = The electrostatic P.E of the electrons, at a distance r from the proton, = eV = eV0ln (r/r0)

∴ F = The magnitude of the force between the electron and proton = \(|-\frac{dU}{dr}|=\frac{eV_0r_0}{r}\)

The force, F, provides the centeripetal force for the electron, orbiting in its nth allowed orbit of radius rn .

From Bohr’s postulate, we also have mvnr= n h/2π  ......(2)

From Eqn. (1) and (2), we get, rn ∝ n

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