Calculate the de Broglie wavelength in picometers (1 picometer =10^−12 meters) of an electron that has kinetic energy of 10 keV. The mass of an electron is 9.11 x 10^−31 kg
To calculate the de Broglie wavelength of an electron, we can use the de Broglie wavelength formula:
λ = h / p
Where:
λ is the de Broglie wavelength
h is Planck's constant (6.63 x 10^-34 J·s)
p is the momentum of the electron
To find the momentum of the electron, we can use the equation:
p = √(2mK)
Where:
p is the momentum of the electron
m is the mass of the electron
K is the kinetic energy of the electron
Given:
m = 9.11 x 10^-31 kg
K = 10 keV = 10 x 10^3 eV
First, we need to convert the kinetic energy from electron volts (eV) to joules (J). We know that 1 eV = 1.6 x 10^-19 J, so:
K = 10 x 10^3 eV x 1.6 x 10^-19 J/eV = 1.6 x 10^-15 J
Now we can substitute the values into the momentum equation:
p = √(2mK)
p = √(2 x 9.11 x 10^-31 kg x 1.6 x 10^-15 J)
p ≈ 4.24 x 10^-23 kg·m/s
Finally, substituting the momentum into the de Broglie wavelength equation:
λ = h / p
λ = 6.63 x 10^-34 J·s / 4.24 x 10^-23 kg·m/s
λ ≈ 1.56 x 10^-11 meters
To convert this to picometers, we use the conversion factor of 1 picometer = 10^-12 meters:
λ ≈ 1.56 x 10^-11 meters x (10^12 picometers/meter)
λ ≈ 1.56 x 10 picometers
Therefore, the de Broglie wavelength of an electron with a kinetic energy of 10 keV is approximately 1.56 x 10 picometers.