# De-Broglie's Hypothesis | Chemistry Assignment Help

## De-Broglie's Hypothesis

The French physicist **Louis de Broglie** made a bold assertion in 1923. Considering Einstein's relationship of wavelength lambda to momentum p, de Broglie proposed that this relationship would determine the wavelength of any matter, in the relationship:

**lambda = h / p** Where ** h ** is **Planck's constant** and **p** is ** momentum**.

This wavelength is called the **de Broglie wavelength**.

According to de Broglie, an electron of mass ‘m’ moving with a velocity ‘v’ should be associated with a wave having wave length (lambda). This wave length and velocity are related by the following mathematical equation =>

…

**(1)**

Where h is the Planck's constant and mv is the momentum of the moving particle. Equation is known as de Broglie relationship and can be written as

or mv µ …

**(2)**

Equation (2) is another form of de Broglie relationship and this can be stated in words as “The momentum of a moving particle is inversely proportional to the wave-length of the waves associated with it.”

### *Proof of de Broglie Equation*

Let us first consider the case of a photon. If we consider it to be a wave of frequency n, its energy is given byE = hn …

**(3)**

If we now consider it as a particle of mass m, its energy is given by

E = mc2 …

**(4)**

From equation (3) and (4), we get

hn = mc2 …

**(5)**

As the photon travels in free space with velocity of light c, its momentum p is given by

P = mass ´ velocity = mc …

**(6)**

On dividing equation (5) by (6), we get

or [ c = vl = Frequency ´ wave length]

or …

**(7)**

de-Broglie assumed that the above relation holds good for material particles like electrons, and hence for electrons eq. (7) becomes as

…

**(8)**

Where m is the mass, v the velocity, l the wave length and p the momentum of an electron. Equation (8) is same as equation (1).

*De-Broglie Relationship and Bohr's Theory*

Application of the de-Broglie's relationship to a moving electron around a nucleus puts some restrictions on the size of orbits. It means that electron is not a mass particle moving in a circular path but is instead a standing wave train (non-energy, radiating motion) extending around the nucleus in the circular path as shown below:For the wave to remain continually in phase, the circumference of the orbit should be an integral multiple of wavelength l i.e.,

2pr = nl …

**(9)**

Where r is the radius of the orbit and n is a whole number.

From equation (8), we get

l = h/mv …

**(10)**

Substituting the value of l in equation (9), we get

or …

**(11)**

Which is the same as Bohr's second postulate. From equation

**(11)**it follows that “Electron can move only in such orbits for which the angular momentum must be an integral multiple of h/2p. If the circumference is bigger or smaller than the value as given by equation (11), it means that the wave is not in the phase as shown in figure(2). Thus, de-Broglie relation provides a theoretical basis for the Bohr’s second postulate.

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