Question

Show that Bohr's second postulate can be explained on the basis of de Broglie hypothesis of wave nature of electrons.

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Solution

Bohr’s 2nd postulate tells us that electrons orbit the nucleus only in those orbits for which the angular momentum is an integral multiple of nh/2(pi). However the question arises is why? why should an electron only orbit a nucleus depending on the angular momentum of the orbit. The answer to that question was answerd by de broglie. By that time de broglie established his wave mater duality principle. From it we come to know of matter waves and how there wavelengths are inversly proportional to the mass of the body and the velocity of the body, by the relation:

(wavelength) = h /mv

thus we come to know about why massive macro objects dont show wave nature. the wave length is too small and negligable.

what de broglie did was he included his wave nature principle in bohrs 2nd prostulate. He imagined the circumference of the orbit as a string and extending it in a straight line applied his concept and derived the same equation of angular momentum as bohr. I provided the maths and the explaination below.

2(pi)r = n(wavelength)

2(pi)r = nh/mv

*here 2(pi)r is the circumference of the orbit = the length of the string and h/mv is the wavelength equation from debroglie (which I provided abover) and n is the number of wavelengths. What he wanted to see was how many wavelengths can fit inside a orbit of circumference 2(pi)r.

mvr = nh/2(pi)

L = nh/2(pi)

Solving the equation we get to Bohr’s 2nd postulate equation.

The explaination concluded by de broglie was, that wavelengths of mater waves was quantised. This means The electrons can exists in those orbits which had a complete set of n number of wavelengths (matter wave wavelengths depend on the mass and velocity of the electron) where n is a whole number (and not an integer like 1.5 or 2.7 etc). And since each of those orbits will have a constant angular momentum, hence the phenomenon can also be explained as, the electrons will orbit the nucleus in those orbits for which the angular momentum is nh/2(pi). where n is again a whole number.

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