Question

Answer for 50 points


29. Using the relevant Bohr's postulates, derive the expression for the speed of the electron in the nth orbit.
​

Answer 1

Let the electron of mass m revolves around a nucleus (H-atom) in an orbit of radius rn with linear velocity vn.

As the electron revolves in an stationary orbit, thus centrifugal force acting on the electron is balanced by the coulombic force ie F centrifugal = F coulombic.

∴ mv²n / rn = ke² / rn²

Where k = 1/ 4πϵ = 9 x 10⁹

⟹ rn = k² / mv²n

Also we use mvnrn = nh / 2π

Eliminating vn from both equations, we get rn = k² / m x 4πm²r²n / n²h²

⟹ rn = h²n² / 4π²mke²

Putting m = 9.1 × 10 -³¹ kg , h = 6.626×10 -³⁴ Js and e = 1.6 × 10-¹⁹C

We get radius of n th bohr orbit rn

= 0.529 n² Ao

Answer 2

The Orbital period as per Bohr's Postulates is given as T = 2πrn / v, where r is radius of nth orbit, v is the velocity of electron is nth orbit.

The expression for the orbital period of the electron is proof from the expression of- angular momentum of an electron moving in an atom As given by Bohr's postulate is given by mvr equals to nh/2π.