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Developments Leading To Quantum Mechanical Model Of The Atom

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Developments Leading To Quantum Mechanical Model Of The Atom - Lesson Summary

Two important developments that questioned the rationality of Bohr's model are:

  • Dual behaviour of matter
  • Heisenberg uncertainty principle.


De Broglie proposed in 1924 that all matter has wave- and particle-like properties.

According to this theory every moving object be it an electron, a ball or a planet has a wave nature.

This theory gave the relation between wavelength and momentum of a material particle based on the energy equations of photons in their wave and particle forms.

λ = h/mc

Since the momentum of a particle is the product of its mass and velocity, the relation between wavelength and momentum of a material particle can be given by de Broglie's equation.

De Broglie's theory and equation was later proved to be correct when It was found that an electron beam undergoes diffraction, a property that is unique to waves.


Heisenberg uncertainty principle:
In 1927, a German physicist, Werner Heisenberg took forward the concept of dual behavior of matter and radiation to give a principle about the uncertainties in simultaneous measurements of position and momentum of small particles.

According to this theory "It is impossible to measure simultaneously and accurately the exact position and momentum of a small particle like an electron".

Mathematically,

         Δx × ΔPx ≥ h/4π

  Where, Δx = Uncertainty in position
            ΔPx = Uncertainty in momentum
               H = Planck's constant

The uncertainty in the position and velocity are inversely proportional to each other. So, if the position of the electron is known with high degree of accuracy then the velocity of the electron will be uncertain and vice versa.


Δx × ΔVx ≥ h/4πm

        ΔVx ≥ h/4πmΔx

The uncertainty in the position and velocity are inversely proportional to the mass of the object. So, that uncertainty in the position and velocity decreases as the mass of the object increases and vice versa.

The position of an electron can be determined experimentally by illuminating it with "electromagnetic radiation having a wavelength smaller than the dimensions of an electron.

However, the velocity of an illuminated electron cannot be detected as the high momentum photons of the light increase the energy of electrons by collisions.

For an electron,

Δx* Δv = 10⁻⁴sq.meters/second

The uncertainty in velocity can be calculated by substituting the value for Δx= 10-8 meters in the Heisenberg's equation.

This means when try to find the exact location of the electron to an uncertainty of only 10-8 meters, the uncertainty in velocity becomes positive and enormously significant.

So, it is impossible to determine simultaneously and accurately the position and velocity for an electron at any given instant.

Thus, Bohr's model not only ignores dual nature of matter but also contradicts the Heisenberg's Uncertainty Principle.

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