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Question -

Classically, an electron can bein any orbit around the nucleus of an atom. Then what determines the typicalatomic size? Why is an atom not, say, thousand times bigger than its typicalsize? The question had greatly puzzled Bohr before he arrived at his famousmodel of the atom that you have learnt in the text. To simulate what he mightwell have done before his discovery, let us play as follows with the basicconstants of nature and see if we can get a quantity with the dimensions oflength that is roughly equal to the known size of an atom (~ 10−10 m).

(a) Constructa quantity with the dimensions of length from the fundamental constants eme,and c. Determine its numerical value.

(b) You will find that thelength obtained in (a) is many orders of magnitude smaller than the atomicdimensions. Further, it involves c. But energies of atoms aremostly in non-relativistic domain where is not expected toplay any role. This is what may have suggested Bohr to discard andlook for ‘something else’ to get the right atomic size. Now, the Planck’sconstant h had already made its appearance elsewhere. Bohr’sgreat insight lay in recognising that hme,and will yield the right atomic size. Construct a quantitywith the dimension of length from hme,and and confirm that its numerical value has indeed thecorrect order of magnitude.



Answer -

(a) Chargeon an electron, e = 1.6 × 10−19 C

Mass of an electron, me =9.1 × 10−31 kg

Speed of light, c =3 ×108 m/s

Letus take a quantity involving the given quantities as 

Where,

0 = Permittivity of free space

And, 

The numerical value of the takenquantity will be:

Hence, the numerical value of thetaken quantity is much smaller than the typical size of an atom.

(b) Chargeon an electron, e = 1.6 × 10−19 C

Mass of an electron, me =9.1 × 10−31 kg

Planck’s constant, h =6.63 ×10−34 Js

Letus take a quantity involving the given quantities as 

Where,

0 = Permittivity of free space

And, 

The numerical value of the takenquantity will be:

Hence, the value of the quantitytaken is of the order of the atomic size.

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