Explain the formulas of energy of electron in atom revolving around the nucleus in different orbits.

Explanation

According to the Rutherford nuclear model of the atom which is a electrical neutral sphere of a very small, massive and positively charged nucleus at the centre surrounding by the revolving electron in their respective stable orbits. The electrostatic force of attraction, `F_(e)` between the revolving electrons and the nucleus provides the requisite centripetal force to keep them in their orbits.
For a stable orbit in a hydrogen atom,
`F_(e)=F_(e)` where `F_(e)=` Electric force
`F_(e)=` Centripetal force
`:.(1)/(4pi epsi_(0))*(e^(2))/(r^(2))=(mv^(2))/(r)`
`:.r=(e^(2))/(4pi epsi_(0)) mv^(2)....(1)`
which is the relation between the velocity of electron to the orbital radius.
The kinetic energy of electron in hydrogen atom.
`(1)/(2)mv^(2)=(e^(2))/(2xx4pi epsi_(theta)r) [ :.` From equqtion (1)]
And potential energy
`tj=-(1)/(4pi epsi_(0))*(Zexxe)/(r)" "..(3)` [ `:.` For hydrogen Z=1]
In this formula, the negative sign in U signifies that the electrostatic force is in the -r.
Hence, the total energy of electron in a hydrogen atom is,
`E=K+U`
`=(e^(2))/( 8pi epsi_(0)r)-(e^(2))/(4pi epsi_(0)r)` [ `:.` From equation (1) and (2)]
`:.E=(e^(2)-2e^(2))/(8 pi epsi_(0)r)=-(e^(2))/(8 pi epsi_(0)r)`
`:. E=-(e^(2))/(8 pi epsi_(0)r)`
The total energy of the electron is negative. The implies the fact that the electron is bound to tl nucleus.
If total energy were positive, an electron will n follow a closed orbit around the nucleus.