Understanding the Concepts
* De Broglie Wavelength: The De Broglie hypothesis states that all matter exhibits wave-like properties. The wavelength of a particle is inversely proportional to its momentum:
λ = h / p
where:
* λ is the De Broglie wavelength
* h is Planck's constant (6.63 x 10^-34 Js)
* p is the momentum of the particle
* Kinetic Energy and Momentum: The kinetic energy (KE) of a particle is related to its momentum:
KE = p^2 / 2m
where:
* m is the mass of the particle
Calculations
1. Convert Electron Volts to Joules:
1 eV = 1.602 x 10^-19 J
Therefore, 10 eV = 10 * 1.602 x 10^-19 J = 1.602 x 10^-18 J
2. Calculate the Momentum:
KE = p^2 / 2m
p^2 = 2mKE
p = √(2mKE)
* The mass of an electron (m) is 9.11 x 10^-31 kg.
* Substitute the values and calculate p.
3. Calculate the De Broglie Wavelength:
λ = h / p
Substitute the values of h and p you calculated.
4. Calculate the Frequency:
The relationship between wavelength (λ), frequency (f), and the speed of light (c) is:
c = λf
Since the electron is non-relativistic, its speed is not the speed of light. We need to use the velocity of the electron.
* First, calculate the velocity (v) of the electron using the kinetic energy:
KE = 1/2 * mv^2
v = √(2KE / m)
* Then, calculate the frequency:
f = v / λ
Let's do the calculations:
1. Momentum (p):
p = √(2 * 9.11 x 10^-31 kg * 1.602 x 10^-18 J) ≈ 1.92 x 10^-24 kg m/s
2. De Broglie Wavelength (λ):
λ = (6.63 x 10^-34 Js) / (1.92 x 10^-24 kg m/s) ≈ 3.46 x 10^-10 m
3. Velocity (v):
v = √(2 * 1.602 x 10^-18 J / 9.11 x 10^-31 kg) ≈ 1.88 x 10^6 m/s
4. Frequency (f):
f = (1.88 x 10^6 m/s) / (3.46 x 10^-10 m) ≈ 5.43 x 10^15 Hz
Therefore, the frequency of the matter wave associated with a 10 eV free electron is approximately 5.43 x 10^15 Hz, and its wavelength is approximately 3.46 x 10^-10 m.
