Let us assume that the de-Broglie wave asssociated with an electron froms a standing wave between the atoms arranged in a one-dimensional array with nodes at each of the atomic sites. If is found that one such standing wave is formed, if the distacne between theatoms of the array is `3.X36Å` similar standing waave is again fromed, if d is increased to `3.5 Å` but not for any intermediate value of d. Find the energy of the electrons in electron volts and the least value of d for which the standing wave of the type described above can from.

Explanation

For standing waves, the spearation between consecutive nodes or antinodes.
`d=n(lambda)/(2)" " ....(i)`
`n=1,2....`
For next node,
`d'=(n+1)(lambda)/(2) " " ....(ii)`
`d'-d=(lambda)/(2)rArr-3=(lambda)/(2)rArrlambda=1 Å`
de-Broglie wevelength `lambda=(h)/(sqrt(2mK))`
`rArrK=(h^(2))/(2mlambda^(2))=((6.6xx10^(-34))^(2))/(2(9.1xx10^(-31))xx(10^(-10))^(2))=2.415xx10^(-17) J`
`=(2.415xx10^(-17))/(1.6xx10^(-19))=150 eV`
For minimum value of d, n=1
`:. " " d=n(lambda)/(2)=(lambda)/(2)=(1)/(2)=0.5Å`