Using Rutherford model of the atom, derive the expression for the total energy of the electron in hydrogen atom. What is the significance of total negative energy possessed by the electron?

Explanation

As per Rutherford model of the atom, an electron revolves around the nucleus of hydrogen atom such that the requisite centripetal force is provided by the coulombian attraction force acting on it due to nucleus. Thus
`(mv^(2))/(r) = (1)/(4pi in_(0)).(e^(2))/(r^(2)) rArr mv^(2) =(1)/(4 pi in_(0)) .(e^(2))/(r)`
`therefore ` Kinetic energy of orbiting electron `k = (1)/(2) mv^(2) = (1)/(2) xx (1)/(4 pi in_(0)) .(e^(2))/(r) = (e^(2))/(8 pi in_(0)r)` and potential energy of orbiting electron `U = (q_(1)q_(2))/(4 pi in_(0) r) = (e^(2))/(4pi in_(0) r)`
`therefore ` Total energy of electron in hydrgoen atom ` E = K + U = (e^(2))/(8 pi in_(0) r) - (e^(2))/(4pi in_(0)r) = - (e^(2))/(8 pi in_(0)r)`
The negative sign of energy signifies that the electron is bound to the nucleus and cannot escape from its orbit at its own.