Question

Calculate the velocities of electrons with de Broglie wavelengths of $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{2}}\mathit{n}\mathbf{m}$and$\mathbf{1}\mathbf{.}\mathbf{0}\mathit{n}\mathit{m}$respectively.

Short answer

Answer

Velocities of the electron that have de Broglie wavelengths as $1.0x{10}^{2}n\mathrm{m}$and 1.0nm are $7.273\times {10}^{3}m/s$and $7.273\times {10}^{5}m/s$respectively.

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.

Given information from the question 

De Broglie wavelength for an electron is $1.0\times {10}^{2}nm$and $1.0nm$respectively.

To convert to use the following conversion factor:

$1nm={10}^{-9}m$

Thus, $1.0\times {102}^{2}nm$is converted to m as follows:

Wavelength$=\left(1.0\times 10n{m}^2\right)\left({10}^{-9}m1nm\right)=1.0\times {10}^{-7}m$

Calculate the velocities from wavelength as per the de Broglie relation 

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

$v=hm\lambda$ ……. (1)

Value of h is $6.626\times {10}^{-32}Js$.

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

Value of $\lambda$is ${10}^{-9}m$

Substitute the value in the above equation to calculate v.

localid="1656397941625" $\begin{array}{ll}v=hm\lambda & \\ =6.626x{10}^{-34}Js\left(9.11\times {10}^{-31}kg\right)& \left({10}^{-8}m\right)\\ =7.273x{10}^{5}m/s& \end{array}$

Value of $\lambda$is $1.0\times {10}^{-7}m$.

Substitute the value in equation (1) to calculate v.

localid="1656397731997" $$\begin{gathered} v=hm\lambda \\ =6.626\times {10}^{-34}Js\left(9.11\times kg\right)\left(1.0\times {10}^{-7}m\right) \\ =7.273\times {10}^{3}m/s \end{gathered}$$

Thus, velocities of the electron that have de Broglie wavelengths as $1.0\times {10}^{2}nm$and $1.0nm$are $7.273\times {10}^{3}m/s$and $7.273\times {10}^{5}m/s$respectively.