(Chemistry Ch-2) 6. Dual Behaviour of Matter & Heisenbergs Uncertainty Principle

Dual Behaviour of Matter

• Matter, like radiation, exhibits dual behaviour (i.e., both particle and wave-like properties).
• Electrons should have momentum as well as wavelength, just as photon has momentum as well as wavelength.
• De Broglie gave the relationship between wavelength (λ) and momentum (p) of a material particle.

Where, m is the mass of the particle and v is its velocity

• According to de Broglie, every object in motion has a wave character.
• Wavelengths of objects having large masses are so short that their wave properties cannot be detected.

Example Let us try to calculate the wavelength of a ball of mass 0.01 kg, moving with a velocity of 20 ms−1. Using de Broglie equation ∴λ = 3.313 × 10−33 m The value of wavelength is so small that wave properties of the ball cannot be detected.

• Wavelengths of particles having very small masses (electron and other subatomic particles) can be detected experimentally.

Example If an electron is moving with a velocity of 6.0 × 106 m/s, then the wavelength of the electron can be calculated as follows: ∴λ = 1.21 × 10−10 m Thus, de Broglie’s concept is more significant for microscopic particles whose wavelength can be measured.

Heisenberg’s Uncertainty Principle

• Impossible to determine simultaneously the exact position and the exact momentum of an electron (microscopic particle) with absolute accuracy and certainty
• Mathematically, it can be represented as

Δx × Δp_x ≥ 

Or, Δx × Δ(mv_x) ≥ 

Or, Δx × Δv_x ≥ 

Where,

Δx is the uncertainty in position

Δv_x is the uncertainty in velocity

Δp_x is the uncertainty in momentum

• If the uncertainty in position (Δx) is less, then the uncertainty in momentum (Δp_x) would be large. On the other hand, if the uncertainty in momentum (Δp) is less, the uncertainty in position (Δx) would be large.

Significance of Uncertainty Principle

• Heisenberg’s uncertainty principle rejects the existence of definite paths or trajectories of electrons and other similar particles.
• The effect of Heisenberg’s uncertainty principle on the motion of macroscopic objects is negligible.

Example When an uncertainty principle is applied to an object of mass 10−6 kg, Δv ⋅ Δx =  =  ≈ 0.5275 × 10−28 m2 s−1 The value of Δv ⋅ Δx is extremely small, and hence, is insignificant. Thus, Heisenberg’s uncertainty principle has no significance for macroscopic bodies.

• However, this is not the case with the motion of microscopic objects.

Example For an electron whose mass is 9.11 × 10−31 kg, the value of Δv⋅ Δx can be calculated as follows: Δv⋅Δx =  =  ∴ Δv⋅Δx ≈ 0.0579 × 10−4 m2 s−1 The value of Δv⋅Δx is quite large, and hence, cannot be neglected.

Reasons for the failure of the Bohr’s Model

• Bohr’s model ignores the dual behaviour of matter − In Bohr’s model of hydrogen atom, electron is regarded as a charged particle moving around the nucleus in well-defined circular orbits. It does not account for the wave character of an electron.
• Bohr’s model contradicts Heisenberg’s Principle − In the Bohr’s model, the electron moves in an orbit. An orbit by definition is a clearly defined path. However, such a completely-defined path can be obtained only if the position and velocity of the electron are known exactly at the same time. This is not possible according to the Uncertainty Principle.