Question

Question: Calculate the de Broglie wavelength for each of the following.

a. an electron with a velocity$\mathbf{10}\mathbf{\%}$of the speed of light

b. a tennis ball$\left(55g\right)$served at$\mathbf{35}\mathit{m}\mathbf{/}\mathit{s}\left(~80mi/h\right)$

Short answer

a) For electrons, the De Broglie wavelength is $2.424\times {10}^{-11}m$.

b) For a tennis ball, the De Broglie wavelength is $3.44\times {10}^{-34}m$.

Step-by-step solution

Concept used

De-Broglie relation:

The wave associated with matter can be linked to particle characteristics. This fact was observed by Louis de Broglie. The relationship amongst mass, velocity, and wavelength is established as follows:

$\lambda =hmv$

Where,

$\lambda$denotes wavelength.

$m$denotes mass.

$v$denotes velocity.

$h$denotes Planck's constant.

Calculate De Broglie wavelength for electron

a)

The velocity of an electron is$10\%$that of the velocity of light.

Value of $c$is$3\times {10}^{8}m/s$so the velocity of the electron is calculated as follows:

velocity of electron = $\left(10\%\right)\left(3\times {10}^{8}m/s\right)$

$=3\times {10}^{7}m/s$

The formula to calculate wavelength as per de Broglie relation is as follows:

$\lambda =hmv$

Value of $h$is $6.626\times {10}^{34}Js$.

Value of$m$is$9.11\times {10}^{-31}kg$.

Value of $v$is $3\times {10}^{7}m/s$.

Substitute the value in the above equation

$$\begin{gathered} \lambda =hmv \\ =6.626\times {10}^{-34}Js\left(9.11\times {10}^{-31}kg\right)\left(3\times {10}^{7}m/s\right) \\ =2.424\times {10}^{-11}m \end{gathered}$$

Thus, de Broglie wavelength for an electron is $2.424\times {10}^{-11}m$.

Calculate De Broglie wavelength for tennis ball

b)

Mass of tennis ball is $55g$.

The velocity of a tennis ball is $35m/s$.

To convert to use the following conversion factor:

$1g={10}^{-3}kg$

Thus, $55g$is converted to as follows:

Mass =$$\begin{gathered} \left(55g\right)\left({10}^{-3}kg1g\right) \\ =55\times {10}^{-3}kg \end{gathered}$$

The formula to calculate wavelength as per de Broglie relation is as follows:

$\lambda =hmv$

Value of $h$is $6.626\times {10}^{-34}Js$.

Value of $m$is $55\times {10}^{-3}kg$.

Value of $v$is $35m/s$.

Substitute the value in the above equation

$\lambda =hmv$

$$\begin{gathered} =6.626\times {10}^{-34}Js\left(55\times {10}^{-3}kg\right)\left(35m/s\right) \\ =3.44\times {10}^{-34}m \end{gathered}$$

Thus, de Broglie wavelength for tennis ball is $3.44\times {10}^{-34}m$.