Assume that the de Broglie wave associated with an electron can from a standing wave between the atom arrange in a one dimensional array with nodes at each of the atomic sites. It is found that one such standing wave if the distance d between the atoms of the array is `2 A^0` A similar standing wave is again formed if d is increased to `2.5Å` . Find the energy of the electrons in electron volts and the least value of d for which the standing wave type described above can from .

Explanation

For standing waves, the separation 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) implies 2.5-2=(lambda)/(2)`
`lambda=1`Å
de Broglie wavelength
`lambda=(h)/(sqrt(2mK))`
`K=(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)=0.5` Å