Answer :
Answer: The electron moves [tex]0.73\%[/tex] less fast than light
Explanation:
The complete question is written below:
The mass of an electron is [tex]9.11(10)^{-31} kg[/tex] . If the de Broglie wavelength for an electron in a hydrogen atom is [tex]3.31(10)^{-10} m[/tex], how fast is the electron moving relative to the speed of light? The speed of light is [tex]3(10)^8 m/s[/tex].
The De Broglie wavelength equation is:
[tex]\lambda_{e}=\frac{h}{p_{e}}[/tex] (1)
Where:
[tex]\lambda_{e}=3.31(10)^{-10} m[/tex] is the de broglie wavelength for an electron
[tex]h=6.626(10)^{-34}J.s=6.626(10)^{-34}\frac{m^{2}kg}{s}[/tex] is the Planck constant
[tex]p_{e}[/tex] is the momentum of the electron
On the other hand, the momentum of the electron is given by:
[tex]p_{e}=m_{e}V_{e}[/tex] (2)
Where:
[tex]m_{e}=9.11(10)^{-31} kg[/tex] is the mass of the electron
[tex]V_{e}[/tex] is the velocity of the electron
Substituting (2) in (1):
[tex]\lambda_{e}=\frac{h}{m_{e}V_{e}}[/tex] (3)
Isolating [tex]V_{e}[/tex]:
[tex]V_{e}=\frac{h}{m\lambda_{e}}[/tex] (4)
[tex]V_{e}=\frac{6.626(10)^{-34}J.s}{(9.11(10)^{-31} kg)(3.31(10)^{-10} m)}[/tex]
Finally:
[tex]V_{e}=2,197,379.461 m/s \approx 2.20(10)^{6} m/s[/tex] This is the velocity of the electron
Calculating the ratio between the velocity of the electron and the velocity of a photon:
[tex](\frac{2.20(10)^{6} m/s}{3(10)^8 m/s})100\%=(0.0073)(100)=0.73\%[/tex]
Therefore, the electron moves [tex]0.73\%[/tex] less fast than the photon (light).