Question

Calculate the de Broglie wavelength for each of the following.

a. an electron with a velocity 10%of the speed of light

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

Short answer

Answer:

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

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

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:

$\lambda \mathit{=}hmv$

Where,

$\lambda$denotes wavelength.

m denotes mass.

v denotes velocity.

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

$$\begin{gathered} \mathrm{Velocity}\mathrm{of}\mathrm{eletron}=\left(10\%\right)\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right) \\ =3\times {10}^7\mathrm{m}/\mathrm{s} \end{gathered}$$

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

$\lambda \mathit{=}hmv$

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

Value of m is $9.11\times {10}^{-35}\mathrm{kg}$.

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

Substitute the value in the above equation

$$\begin{gathered} \lambda \mathit{=}hmv \\ \mathit{}\mathit{}\mathit{=}\mathit{6}\mathit{.}\mathit{626}\mathit{\times }{\mathit{10}}^{\mathit{-}\mathit{34}}Js\left(9.11\times {10}^{-31}kg\right)\left(3\times {10}^{7}m/s\right) \\ \mathit{}\mathit{=}\mathit{2}\mathit{.}\mathit{24}\mathit{\times }{\mathit{10}}^{\mathit{-}\mathit{11}}m \end{gathered}$$

Thus, de Broglie wavelength for an electron is $\mathit{2}\mathit{.}\mathit{24}\mathit{\times }{\mathit{10}}^{\mathit{-}\mathit{11}}m$.

Calculate De Broglie wavelength for tennis ball

b)

Mass of tennis ball is 55g.

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

To convert to use the following conversion factor:

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

Thus, 55g is converted to as follows:

$$\begin{gathered} \mathrm{Mass}=\left(55\mathrm{g}\right)\left({10}^{-3}\mathrm{kg}1\mathrm{g}\right) \\ =55\times {10}^{-3}\mathrm{kg} \end{gathered}$$

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

$\lambda \mathit{=}hmv$

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

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

Value of v is $35\mathrm{m}/\mathrm{s}$.

Substitute the value in the above equation

$$\begin{gathered} \lambda \mathit{=}hmv \\ \mathit{}\mathit{}\mathit{=}\mathit{6}\mathit{.}\mathit{626}\mathit{\times }{\mathit{10}}^{\mathit{-}\mathit{34}}Js\mathit{(}\mathit{55}\mathit{\times }{\mathit{10}}^{\mathit{-}\mathit{3}}kg\mathit{)}\mathit{(}\mathit{35}m\mathit{/}s\mathit{)} \\ \mathit{}\mathit{=}\mathit{3}\mathit{.}\mathit{44}\mathit{\times }{\mathit{10}}^{\mathit{-}\mathit{34}}m \end{gathered}$$

Thus, de Broglie wavelength for tennis ball is $\mathit{3}\mathit{.}\mathit{44}\mathit{\times }{\mathit{10}}^{\mathit{-}\mathit{34}}m$.