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Question
On the basis of Bohr's theory, derive an expression for the radius of the nth orbit of an electron of hydrogen atom.
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Solution
Let r be the orbit's radius and e, m, and v represent the electron's charge, mass, and velocity. Ze represents the nucleus's positive charge. For an atom of hydrogen, Z = 1. The electrostatic force of attraction provides centripetal force.
Therefore,
`(mv^2)/r = 1/(4piε_0) (Ze xx e)/ r^2`
`mv^2 = (Ze^2)/(4piε_0r)` ...(i)
By first postulate:
`mvr = (nh)/(2pi)` ...(ii)
Where n is the quantum number.
Squaring equation (ii) and dividing by equation (i), we get:
`(m^2v^2r^2)/(mv^2) = (n^2h^2)/((4pi^2)/((Ze^2)/(4piε_0r)))`
Then, `r = (n^2h^2ε_0)/(piZe^2m)`
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