InterviewSolution
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InterviewSolution
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Find the number of ratational levels per unit energy interval, `dN//dE`, for a diatomic molecule as a function of rotational energy `E`. Calculate of an magnitude for an iodine molecule in the state with rotational quantum number `J=10`. The distance between the nuclei of that molecule is equal to `267 p m`. |
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Answer» From the formula `J(J+1)( ħ^(2))/(2I)= E` we get `J(J+1)- 2IE// ħ^(2)` or `(J+(1)/(2))^(2)-(1)/(4)= (2IE)/( ħ^(2))` Hence `J=-(1)/(2)+sqrt((1)/(4)+(2IE)/( ħ^(2)))` writing `J+1= -(1)/(2)+sqrt((1)/(4)+(2I)/( ħ^(2))(E+DeltaE))` we find `1=sqrt((1)/(4)+(2I)/(ħ^(2))E+(2I)/(ħ^(2))DeltaE)-sqrt((1)/(4)+(2IE)/(ħ^(2)))` `sqrt((1)/(4)+(2I)/(ħ^(2))E)[(1+(DeltaE)/(E+(ħ^(2))/(8I)))^(1//2)]` `sqrt((1)/(4)+(2I)/(ħ^(2))E).(DeltaE)/(2(E+(ħ^(2))/(8I)))` `sqrt((2I)/(ħ^(2)))(DeltaE)/(2sqrt((E+(ħ^(2))/(8I))))` The quantity `(dN)/(dE)is (1)/(DeltaE)`. For large `E` it is `(dN)/(dE)= sqrt((I)/(2 ħ^(2)E))` For an iodine molecule `I= M_(I)d^(2)//2= 7.57xx10^(-38)gm cm^(2)` Thus for `J= 10` `(dN)/(dE)=sqrt((I)/(2ħ^(2).(ħ^(2))/(2I)J(J+1)))=(I)/(sqrt(J(J+1)ħ^(2)))` Substitution gives `(dN)/(dE)= 1.04xx10^(4)` levels per `eV`. |
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