Question

An electron is moving with a speed of 8.00 x 10⁶ m/s. What is the speed of a proton that has the same de Broglie wavelength as this electron?

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.

Velocity of proton

Speed of electron, V = 8 x 10⁶ m/s

Mass of electron, m_e = 9.11 x 10^-31 Kg

Mass of proton, m_p = 1.67 x 10^-27 Kg

De Broglie wavelength is given by

$$\begin{gathered} {\lambda }_e=\frac{h}{p} \\ =\frac{h}{m_e{v}_e} \\ =\frac{6.626*{10}^{-34}Js}{\left(9.11*{10}^{8}Kg\right)\left(8*{10}^{6}m/s\right)} \\ =9.09*{10}^{-11}m \end{gathered}$$

The velocity proton is given by

$$\begin{gathered} {\lambda }_e={\lambda }_p \\ {\lambda }_e=\frac{h}{p_p} \\ {\lambda }_e=\frac{h}{m_p{v}_p} \\ {v}_p=\frac{h}{m_p{\lambda }_e} \\ =\frac{6.626*{10}^{-34}Js}{\left(1.67*{10}^{-27}Kg\right)\left(9.09*{10}^{-11}m/s\right)} \\ =4.36*{10}^{3}m \end{gathered}$$

Therefore, the velocity of proton is 4.36 x 10³ m/s.