(a) Find the radius of `Li(++)` ions in its grounds state assuming Bohr's model to be valid.
(b) Find the maximum angular speed of the electron of a hydrogen atom in a stationary orbit.
(c ) Average lifetime of a H atom excited to n=2 state is `10^(-8)` sec. Find the number of revolutions made by the electron on the average before it jumps to the ground state.
(d) Calculate the magnetic dipole moment corresponding to the motion of the electron in the ground state of a hydrogen atom.
(e) Using the known values for hydrogen atom, calculate :
(i) radius of third orbit for `Li^(+2)`
(ii) speed of electron in fourth orbit for `He^(+)`

Explanation

(a) `r_(n)=0.53(n^(2))/(Z)`Å
`n=1, Z=3`
`=(0.53)/(3)=0.18` Å
(b) `omega_(n)=(v_(n))/(r_(n))=(2.2xx10^(6)(Z)/(n)m//sec)/(0.53(n^(2))/(Z)`Å)
`=(2.2xx10^(6))/(0.53xx10^(-10))(Z^(2))/(n^(3)) rad//sec`
For `._(1)H^(1),Z=1, omega_(n)` will be maximum for n=1
`omega_(max)=(2.2xx10^(6))/(0.53xx10^(-10))=4.1xx10^(16)rad//sec`
(c) Time period of revolution of electron in n=2, Z=1
`T_(n)=(2pir_(n))/(v_(n))=(2pi)/(omega_(n))=(2pi)/((2.2xx10^(6))/(0.53xx10^(-10))(Z^(2))/(n^(3)))=(2pi.n^(3))/(4.1xx10^(16)Z^(2))`
`T_(2)=(2pi(2)^(3))/(4.1xx10^(16))=(16 pi)/(4.1xx10^(16))=12.25xx10^(-16) sec`
Number of revolutions `=(t)/(t_(2))`
`=(10^(-8))/(12.25xx10^(-16))=8.2xx10^(6)`
(d) Magnetic dipole moment `M or mu =i A`
`i=(q)/(T)=(e)/(2pir//v)=(ev)/(2pir)`
`A=pi r^(2)`
`M=(ev)/(2pir) pi r^(2)=(evr)/(2)`
`v_(n)=2.2xx10^(6)(Z)/(n)m//sec`
`r_(n)=0.53(n^(2))/(Z)` Å `=0.53xx10^(-10)(n^(2))/(Z)m`
For hydrogen Z=1, ground state n=1
`M=(ev_(1)r_(1))/(2)=(1.6xx10^(-19)xx2.2xx10^(6)xx0.53xx10^(-10))/(2)`
`=9.2xx10^(-24) A.m^(2)`
(e)(i) `r_(n)=0.53(n^(2))/(Z)`Å
`n=3,Z=3`
`r_(3)=0.53xx3=1.59` Å
(ii) `v_(n)=2.2xx10^(6)(Z)/(n)m//sec`
`n=4, Z=2`
`=2.2xx10^(6)xx(2)/(4)=1.1xx10^(6)m//sec`