Question

Compare the de Broglie wavelength, speed and momenta of a proton and alpha particle
having the same kinetic energy.

Answer 1

We have to compare the De - Broglie wavelength, speed and momentum of a proton and alpha particle having the same kinetic energy.

solution : we know, kinetic energy, K = P²/2m

where P is momentum and m is mass of particle.

if kinetic energy are same for both particles.

P₁²/P₂² = m₁/m₂

here m₁ = mass of proton = mp

m₂ = mass of alpha particle = 4mp

so, P₁²/P₂² = 1/4 = (1/2)²

⇒P₁/P₂ = 1/2

therefore the ratio of momentum of proton to alpha particle is 1 : 2.

De - Broglie wavelength equation is, λ = h/P

means, wavelength is inversely proportional to linear momentum.

so, ratio of wavelength must be reverse of the ratio of linear momentum.

so, ratio of wavelength of proton to alpha particle is 2 : 1.

kinetic energy , K = 1/2 mv²

here you see, if K remains constant.

v² is inversely proportional to m

so, v₁²/v₂² = m₂/₁ = 4mp/mp = 4/1

⇒v₁/v₂ = 2/1

Therefore the ratio of speed of proton to alpha particle is 2 : 1.