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Using Bohr's postulates, derive the expression for the frequency of radiation emitted when electron in hydrogen atom undergoes transition from higher energy state (quantum number n_(i)) to the lower state, (n_(f)). When electron in hydrogen atom jumps from energy state n_(i) = 4 to n_(f) = 3, 2, 1, identify the spectral series to which the emission lines belong. |
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Answer» Solution :In the hydrogen atom, Radius of electron orbit, `r=(n^(2)h^(2))/(4 pi^(2)kme^(2))""...(i)` Kinetic energy of electron, `E_(K)=(1)/(2)nv^(2)=(ke^(2))/(2r)` Using equation (i), we get `E_(K)=(de^(2))/(2)(4 pi^(2)kme^(2))/(n^(2)h^(2))=(2pi^(2)k^(2)me^(4))/(n^(2)h^(2))` Potential energy, `E_(p)=(-k(E)xx(e))/(r)=(-ke^(2))/(r)` Using `eq^(n)` (i), we get `E_(p)=-ke^(2)xx(4pi^(2)kme^(2))/(n^(2)h^(2))=-(4pi^(2)k^(2)me^(4))/(n^(2)h^(2))` Total enery of electron, `E=(2pi^(2)k^(2)me^(2))/(n^(2)h^(2))-(4pi^(2)k^(2)me^(4))/(n^(2)h^(2))=-(2pi^(2)k^(2)me^(4))/(n^(2)h^(2))=-(2pi^(2)k^(2)me^(4))/(h^(2))xx((1)/(n^(2)))`. Now, according to BOHR's frequency condition when electron in hydrogen atom undergoes transition from higher energy STATE to the lower state `(n_(f))` is, `hv = E_(ni)-E_(nf)` `or""hv=-(2pi^(2)k^(2)me^(4))/(h^(2))xx(1)/(n_(i)^(2))-((-2pi^(2)k^(2)me^(4))/(h^(2))xx(1)/(n_(f)^(2)))` `or hv=(2pi^(2)k^(2)me^(4))/(h^(2))xx((1)/(n_(f)^(2))-(1)/(n_(i)^(2)))rArr v = (2pi^(2)k^(2)me^(4))/(h^(3))xx((1)/(n_(f)^(2))-(1)/(n_(i)^(2)))` `v = (C2 pi^(2)k^(2)me^(4))/(Ch^(3))xx((1)/(n_(f)^(2))-(1)/(n_(i)^(2)))` `(2pi^(2)k^(2)me^(4))/(Ch^(3)) = R` = Rydberg constant `R = 1.097 xx 10^(7) m^(-1)` Thus, `v = R xx ((1)/(n_(f)^(2))-(1)/(n_(i)^(2)))` `{:("Now, higher state,",,),(,n_(i)=4,),("Lower state,",,),(,n_(f)="3,2,1",):}""{:("For the transition"),(n_(i)=4 "to n"_(f)=3","rarr "Paschen Seriers"),(n_(i)=4 "to n"_(f)=2"," rarr "Balmer Series"),(n_(i)=4 "to n"_(f)=1"," rarr "Lyman Series"):}` |
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