Answer:
Broglie wavelength: electron 1.22 10⁻¹⁰ m
, proton 2.87 10⁻¹² m
, hydrogen atom 7.74 10⁻¹² m
Explanation:
The equation given by Broglie relates the momentum of a particle with its wavelength.
p = h /λ
In addition, kinetic energy is related to the amount of movement
E = ½ m v²
p = mv
E = ½ p² / m
p = √2mE
If we clear the first equation and replace we have left
λ = h / p =
λ = h / √2mE
Let's reduce the values that give us SI units
1 ev = 1,602 10⁻¹⁹ J
E1 = 100 eV (1.6 10⁻¹⁹ J / 1eV) = 1.6 10⁻¹⁷ J
We look in tables for the mass of the particle and the Planck constant
h = 6,626 10-34 Js
me = 9.1 10-31 Kg
mp = 1.67 10-27 Kg
Now let's replace and calculate the wavelengths
a) Electron
λ1 = 6.6 10⁻³⁴ / √(2 9.1 10⁻³¹ 1.6 10⁻¹⁷) = 6.6 10⁻³⁴ / 5.39 10⁻²⁴
λ1 = 1.22 10⁻¹⁰ m
b) Proton
λ2 = 6.6 10-34 / √(2 1.67 10⁻²⁷ 1.6 10⁻¹⁷) = 6.6 10⁻³⁴ / 2.3 10⁻²²
λ2 = 2.87 10⁻¹² m
c) Bohr's first orbit
En = 13.606 / n2 [eV]
n = 1
E1 = 13.606 eV
E1 = 13,606 ev (1.6 10⁻¹⁹ / 1eV) = 21.77 10⁻¹⁹ J
λ3 = 6.6 10⁻³⁴ /√(2 1.67 10⁻²⁷ 21.77 10⁻¹⁹) = 6.6 10⁻³⁴ / 8.52 10⁻²³
λ3 = 0.774 10⁻¹¹ m = 7.74 10⁻¹² m
