Atoms

Key Point The de-Broglie wavelength of a particle of mass

m

and moving with velocity

v

is given by

%5Clambda%3D%5Cfrac%7Bh%7D%7Bm%20v%7D%20(%5Cbecause%20p%3Dm%20v)

de-Broglie wavelength of a proton of mass

m_%7B1%7D

and kinetic energy

k

is given by

%5Clambda_%7B1%7D%20%3D%5Cfrac%7Bh%7D%7B%5Csqrt%7B2%20m_%7B1%7D%20k%7D%7D%20(%5Cbecause%20p%3D%5Csqrt%7B2%20m%20k%7D)

%3D%5Cfrac%7Bh%7D%7B%5Csqrt%7B2%20m_%7B1%7D%20q%20v%7D%7D%20%5Cldots%20%5Ctext%20%7B%20(i)%20%7D%5B%5Cbecause%20k%3Dq%20V%5D

For an alpha particle mass

m_%7B2%7D

carrying charge

q_%7B0%7D

is accelerated through potential

V

, then

%5Clambda_%7B2%7D%3D%5Cfrac%7Bh%7D%7B%5Csqrt%7B2%20m_%7B2%7D%20q_%7B0%7D%20V%7D%7D

%5Cbecause

For

%5Calpha

-particle

%5Cleft(%7B%20%7D_%7B2%7D%5E%7B4%7D%20He%20%5Cright)

%5Ctherefore%20q_%7B0%7D%3D2%20q

and

m_%7B2%7D%3D4%20m_%7B1%7D

%5Ctherefore%20%5Clambda_%7B2%7D%3D%5Cfrac%7Bh%7D%7B%5Csqrt%7B2%20%5Ctimes%204%20m_%7B1%7D%20%5Ctimes%202%20q%20%5Ctimes%20V%7D%7D

...(ii) The ratio of corresponding wavelength, from Eqs. (i) and (ii), we get

%5Cfrac%7B%5Clambda_%7B1%7D%7D%7B%5Clambda_%7B2%7D%7D%20%3D%5Cfrac%7Bh%7D%7B%5Csqrt%7B2%20m_%7B1%7D%20q%20V%7D%7D%20%5Ctimes%20%5Cfrac%7B%5Csqrt%7B2%20%5Ctimes%20m_%7B1%7D%20%5Ctimes%204%20%5Ctimes%202%20q%20V%7D%7D%7Bh%7D%3D%20%5Cfrac%7B4%7D%7B%5Csqrt%7B2%7D%7D%20%5Ctimes%20%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B2%7D%7D

%3D2%20%5Csqrt%7B2%7D

Chapter Questions
1 MCQs