Understanding the Concepts
* De Broglie Wavelength: The de Broglie hypothesis states that all matter exhibits wave-like properties, and the wavelength (λ) of a particle is inversely proportional to its momentum (p).
* Momentum: The momentum of a particle is the product of its mass (m) and velocity (v): p = mv.
The Equation
The de Broglie wavelength equation is:
λ = h / p
where:
* λ = wavelength (in meters)
* h = Planck's constant (6.63 x 10⁻³⁴ J·s)
* p = momentum (in kg·m/s)
Calculations
1. Convert Wavelength to Meters:
0.1 nm = 0.1 x 10⁻⁹ m
2. Rearrange the de Broglie Equation to Solve for Velocity:
Since p = mv, we can substitute to get: λ = h / (mv)
Solving for velocity (v): v = h / (mλ)
3. Plug in the Values:
* v = (6.63 x 10⁻³⁴ J·s) / (9.11 x 10⁻³¹ kg)(0.1 x 10⁻⁹ m)
* v ≈ 7.28 x 10⁶ m/s
Answer:
The speed of an electron with a wavelength of 0.1 nm is approximately 7.28 x 10⁶ m/s.
