is found experimentally that 13.6 eV energy is required to separate a hydrogen atom into a proton and an electron. Compute the orbital radius and the velocity of the electron in a hydrogen atom.

Explanation

Total energy of the electron in hydrogen atom is `-13.6eV=-13.6xx1.6xx10^(-19)J [ :. 1eV=1.6 xx10^(-19)J]`
`rArr` Mass of electron,
`m=9.1x10^(-31)kg, e=1.6xx10^(-19)C`
The energy of the electron in hydrogen atom,
`E=-(e^(2))/(8pi epsi_(0)r)`
`:.r=-(ke^(2))/(2E)` where `(1)/(4pi epsi_(0))=K=9xx10^(9)Nm^(2)C^(-2)`
`:.r=-(9xx10^(9)xx(1.6xx10^(-19))^(2))/(2xx(-13.6xx1.6xx10^(-19)))`
`:. r=(9xx10^(9)xx1.6xx10^(-19))/(2xx13.6)`
`:. r=0.52974xx10^(-10)m`
`:.r~~5.3xx10^(-11)m`
Now we know, that total energy of electron E=-k
Kinetic energy `:. K=-E`
`:.(1)/(2)mv^(2)=-E`
`:.v^(2)=-(2E)/(m)`
`:.v= sqrt(-(2E)/(m))`
`:.v= sqrt(-(2-13.6xx16xx10^(-19))/(9.1xx10^(-31)))`
`:.v=sqrt((43.52xx10^(-19))/(9.1xx10^(-31)))`
`:.v= sqrt(4.7824xx10^(12))`
`:.v=2.1868xx10^(6) :.v=2.2xx10^(6)m//s`
Second method :
`v=(e)/(sqrt(4pi m epsi_(0)r))`
`=(1.6xx10^(-19))/(sqrt(9.1xx10^(-31)xx4x3.14xx8.85xx10^(-12)xx5.3xx10^(-11)))=2.2xx10^(6)(m)/(s)`