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Question

The ratio of the de Broglie wavelengths of a proton and an α-particle will be 1 : 2 if their:
  1. velocities are in the ratio 1 : 8
  2. velocities are in the ratio 8 : 1
  3. kinetic energies are in the ratio 1 : 64
  4. kinetic energies are in the ratio 1 : 256


Solution

The correct option is B velocities are in the ratio 8 : 1
de Broglie equation is:
λ = hmv
where m is the mass and v is the velocity
λpλα = mαvαmpvp        eq.1
 Here,
λp = wavelength of the proton
λα = wavelength of the alpha particle
mp = mass of the proton
mα = mass of the alpha particle
vp = velocity of the proton
vα = velocity of the alpha particle

We know,
mα = 4mp
 given λpλα=12
Substituting the value of mα in eq. 1, we get:

λpλα = (4mp)vαmpvp
12 = 4×vαvp
vpvα=81


 

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