Question

Wavelength of an Alpha Particle. An alpha particle (m = 6.64 * 10^-27 kg) emitted in the radioactive decay of uranium-238 has an energy of 4.20 MeV. What is its de Broglie wavelength?

Short answer

The wavelength of alpha-particle is 7.01 femtometer.

Step-by-step solution

Given Data

According to de Broglie, Light has dual nature as in some situations it behaves like waves and in others like particles. Electrons, which sometimes exhibits particle nature may also behave like wave in some situations.

When a particle acts like a wave, it carries a wavelength which is known as de Broglie wavelength.

Wavelength of alpha-particle

Wavelength, λ = 2.80 x 10^-10 m

Mass of alpha-particle, m_a = 6.64 x 10^-27 Kg

Energy of alpha-particle, E = 4.20 MeV = 6.72 x 10^-13 J

De Broglie wavelength is given by

$$\begin{gathered} \lambda =\frac{h}{p} \\ =\frac{h}{\sqrt{2mE}} \\ =\frac{6.626*{10}^{-34}Js}{\sqrt{2\left(6.64*{10}^{-27}Kg\right)\left(6.72*{10}^{-13}J\right)}} \\ =7.01*{10}^{-15}m \\ =7.01fm \end{gathered}$$

Therefore, the wavelength of alpha-particle is 7.01 femtometer.