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

Explanation

`U=-(ke^2)/(3r^3)`
Force acting on the electron can be calculated as follows :
`F=-(dU)/(dr)=d/(dr)((ke^2)/(3r^3))=-(ke^2)/r^4`
Here negative sign just indicates that the force acting is towards the centre . And this force will act as centripetal force for electron.s motion.
`(mv^2)/r=(ke^2)/r^4`
`mv^2=(ke^2)/r^3`
`mv^2r^3=ke^2` .....(i)
As per Bohr.s theory we can write the following :
`mvr=n h/(2pi)` ...(ii)
`rArr m^2v^2r^2=n^2 h^2/(4pi^2)` ....(iii)
On dividing (i) by (iii) we get the following :
`(mv^2r^3)/(m^2v^2r^2)=(4pi^2ke^2)/(n^2h^2)`
`r/m=(4pi^2ke^2)/(n^2h^2)`
`r=(4pi^2mke^2)/(n^2h^2)`.....(iv)
Substituting value of r in equation (ii) we get the following :
`mv((4pi^2mke^2)/(n^2h^2))=n h/(2pi)`
`v=(n^3h^3)/(8pi^3m^2ke^2)`....(v)
Kinetic energy of electron can be written as follows :
K.E.=`1/2mv^2=(n^6h^6)/(128pi^6m^3k^2e^4)`.....(vi)
Potential energy can be written as follows :
`U=-(ke^2)/(3r^3)=-(ke^3)/3((n^2h^2)/(4pi^2mke^2))^3`
`U=-(ke^2)/3 ((n^6h^6)/(64pi^6m^3k^3e^6))`
`U=-(n^6h^6)/(192pi^6m^3k^2e^4)`....(vii)
Total energy will be sum of kinetic energy and potential energy .
`E=K.E.+U=(n^6h^6)/(128pi^6m^3k^2e^4)-(n^6h^6)/(192pi^6m^3k^2e^4)`
`E=(n^6h^6)/(pi^6m^3k^2e^4)[1/128-1/192]=(n^6h^6)/(pi^6m^3k^2e^4)[(192-128)/(192xx128)]`
`E=(n^6h^6)/(384pi^6m^3k^2e^4)`