To calculate the velocity of an electron based on its wavelength, we can use the de Broglie wavelength formula, which relates the wavelength (λ) to the momentum (p) of a particle:
λ = h / p
Where:
- λ (lambda) is the wavelength,
- h is Planck’s constant (6.626 x 10^-34 Js),
- p is the momentum of the electron.
Momentum (p) can also be expressed as:
p = m * v
Where:
- m is the mass of the electron,
- v is the velocity of the electron.
Combining these formulas, we can express the wavelength in terms of velocity:
λ = h / (m * v) ➔ v = h / (m * λ)
Now we can plug in the known values:
- λ = 0.265 nm = 0.265 x 10^-9 m
- m = 9.11 x 10^-28 kg
- h = 6.626 x 10^-34 Js
Substituting these values into the velocity formula:
v = 6.626 x 10^-34 Js / (9.11 x 10^-28 kg * 0.265 x 10^-9 m)
Calculating the denominator:
9.11 x 10^-28 kg * 0.265 x 10^-9 m = 2.41515 x 10^-36 kg·m
Now calculating the velocity:
v = 6.626 x 10^-34 Js / 2.41515 x 10^-36 kg·m ≈ 274.5 x 10^6 m/s
Thus, the velocity of the electron is approximately:
v ≈ 2.745 x 10^8 m/s
This value is significant as it illustrates the high speed at which an electron can move when its wavelength is relatively small, highlighting its wave-particle duality nature described in quantum mechanics.