de Broglie relation

Particle and wave nature of an electron

In 1924 de Broglie's relation pointed out that just as a light electron also has both particle and wave nature. 
    According to de Broglie, this dual nature - wave and particle - should not be confined to radiations alone but should also be extended to matter.
    He suggested that electrons travel in waves, analogous to light waves. His idea could be fitted to drive the same relation that Bohr arrived at from his particle treatment of electrons.

Momentum and wavelength of a particle

de Broglie relation
de Broglie relation
    de Broglie proposed a relation between momentum and wavelength of a particle in motion. He considered the light of frequency ν, the energy is given by
E = hν = h (c/λ)
    Where λ = wavelength, c = velocity of light, and h = Plank constant = 6.627× 10⁻²⁷.
    Again from the famous mass-energy equivalence relation from Einstien,
E = m c²

The momentum of the photon is
p = mv = mc
    Combining these two relations, we have
mc = h/λ
or, p = h/λ
∴ λ = h/p
    de Broglie extended this relationship to the dynamics of a particle and proposed that a wavelength λ is associated with a moving particle and is related to its momentum as
λ = h/p = h/mv
    Where m is the total mass of the particle and v is its velocity.

Bohr's model and de Broglie relation

    Bohr's model for the angular momentum of n-th orbital of moving electrons
mvr = n (h/2π)
or, mv = n (h/2πm)
    where m = mass of an electron, n = 1, 2, 3, 4, ..., and r = radius of the orbital of an atom.
    Again according to de Broglie relation
λ = h/mv
or, mv = h/λ
    where λ = wavelength of the moving electron.
    Combining these two relations
nh/2πr = h/λ
or, 2πr = n λ
    Thus a standing produces a stationary pattern, its profile being fixed within the space allowed to it. It does not travel beyond the allowed space.

Electron diffraction experiment

    de Broglie's suggestion of matter waves and its confirmation by Davisson and Germer's electron diffraction experiment conclusively proves that electrons are not is not an ordinary particle.
    From the evidence by the experiment of determination of mass and e/m electron has particle nature.
    Electron diffraction experiment by Davison and Gramer's given the evidence of the wave nature of the electron.

The kinetic energy of an electron

    If the particle is an electron and if it is subjected to the potential difference V so as to acquires a velocity v, then,
Kinetic energy of an electron = Ve = ½ mv²
where e is the charge of an electron.

Again from the de Broglie relation
λ = h/mv

From these two relations
λ = h/√(2mVe)

Particle and wave nature of an electron de Broglie relation, electron diffraction and kinetic energy of an electron for study chemistry online

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