Assume a hypothetica hydrogen atom in which the potential energy between electron and proton at separation r is given by `U = [k in r - (k/2)],` where k is a constant. For such a hypothetical hydrogen atom, calcualete the radius of nth Bohr orbit and energy levels.

Answer

B

Explanation

Force of interaction between electron and
proton is
`F = -(dU)/(dr) = (-k)/(r ) `
Force is nagetive. It means there is an attraction
between the particles and they are bound to each
other. This force provides the necessary centripetal
force for the electron.
`:. (mv)^2/(r ) = (k)/(r) ...(i)`
According to Bohr's assumption,
`mvr =n(h)/(2pi).....(ii)`
Solving Eqs. (i) and (ii) we get
`r = (nh)/(2pisqrt(mk)`
and `v= sqrt((k)/(m))`
`:. E =U +(1)/(2) mv^2 = k1nr-(k)/(2) + (k)/(2) = k1nr`
Thus, `r_n = (nh)/(2pisqrt(mk))`
and `E_n = k1n{(nh)/(2pisqrt(mk))}`