Question

What is the wavelength of, in meters, of an alpha particle with kinetic energy of $ 8.0 \times {10^{ - 13}}J $ . Mass of an alpha particle $ = 4.00154amu $ and $ 1amu = 1.67 \times {10^{ - 27}}kg $ ?

Answer 1

Hint: The relationship between the kinetic energy of a particle and its wavelength can be calculated by taking into account the dual nature of the alpha particle. The kinetic energy is directly linked to the momentum which is linked to the wavelength of the particle.

Complete answer:
De-Broglie was the scientist who presented a mathematical model of the dual nature shown by different particles. For a long time scientists debated on the view of particles having characteristic features of matter or electromagnetic radiations that behave as oscillating waves.
De-Broglie defined the wavelength of the wave-like path followed by particles which depends on their mass and speeds. Thus, he found a relation between the wavelength $ (\lambda ) $ and momentum $ (p) $ of the particles. The two attributes showed an inverse relation and the constant used to define an absolute relation was Plank’s constant $ (h) $ .
The relationship of De-Broglie’s wavelength and momentum can be written as follows:
 $ \lambda = \dfrac{h}{p} $
Further, the relationship between kinetic energy $ (K.E) $ and momentum can be used to derive the desired relationship between wavelength and kinetic energy.
 $ p = \sqrt {2mK.E} $
On inserting this formula in the above equation we get,
 $ \lambda = \dfrac{h}{{\sqrt {2mK.E} }} $
The given values of kinetic energy, mass of alpha particles, and the Plank’s constant can be used to calculate the wavelength of the particle.
 $ \lambda = \dfrac{{6.626 \times {{10}^{ - 34}}Js}}{{\sqrt {2 \times 4.00154amu \times 1.67 \times {{10}^{ - 27}}kgam{u^{ - 1}} \times } 8.0 \times {{10}^{ - 13}}J}} $
On solving this we get,
 $ \lambda = 6.4 \times {10^{ - 15}}m $
 $ \Rightarrow $ Thus, the wavelength of the alpha particle is $ \lambda = 6.4 \times {10^{ - 15}}m $

Note:
The wavelength of any particle is inversely proportional to its mass therefore only tiny particles like protons, alpha particles and electrons are capable of showing the dual behavior in a practical sense. Heavier objects have such low wavelengths that makes their wave nature almost insignificant.