Q12 of 18 Page 436

The gravitational attraction between electron and proton in a hydrogen atom is weaker than the coulomb attraction by a factor of about 10–40. An alternative way of looking at this fact is to estimate the radius of the first Bohr orbit of a hydrogen atom if the electron and proton were bound by gravitational attraction. You will find the answer interesting.

The radius of first Bohr orbit is given by the relation:

……………(1)


Where ε0 is the absolute permittivity of free space and is given as = 8.854 × 10-12 N-1C2m-2


h is Planck’s constant and is given as 6.64 × 10-34 Js


e is charge on an electron and is equals to 1.6 × 10-19C


me is the mass of an electron, and is equals to 9.1 × 10-31 Kg


mp is the mass of proton and is equals to 1.67 × 10-27 Kg


The forces of coulomb attraction between an electron and proton is given by the relation:


Gravitational forces of attraction between an electron and proton is given by the relation



Where G is the Gravitational constant and is equal to G = 6.67 × 10-11 N m2 Kg2


When coulomb force and Gravitational forces of attraction are equal



Or
…………(1)


Substituting the values of equation (1) in equation (2), we get



Substituting the values, we get



On calculating, we get


∴ r1= 1.21 × 1029 m


Since Universe is 150billion light years wide, which concludes that the radius of first Bohr orbit is much greater than the size of the universe.


More from this chapter

All 18 →
10

In accordance with the Bohr’s model, find the quantum number that characterises the earth’s revolution around the sun in an orbit of radius 1.5 × 1011 m with orbital speed 3 × 104 m/s. (Mass of earth = 6.0 × 1024 kg.)

 11

Answer the following questions, which help you understand the difference between Thomson’s model and Rutherford’s model better.

(a) Is the average angle of deflection of α-particles by a thin gold foil predicted by Thomson’s model much less, about the same, or much greater than that predicted by Rutherford’s model?


(b) Is the probability of backward scattering (i.e., scattering of α-particles at angles greater than 90°) predicted by Thomson’s model much less, about the same, or much greater than that predicted by Rutherford’s model?


(c) Keeping other factors fixed, it is found experimentally that for small thickness t, the number of α-particles scattered at moderate angles is proportional to t. What clue does this linear dependence on t provide?


(d) In which model is it completely wrong to ignore multiple scattering for the calculation of average angle of scattering of α-particles by a thin foil?

 13

Obtain an expression for the frequency of radiation emitted when a hydrogen atom de-excites from level n to level (n–1). For large n, show that this frequency equals the classical frequency of revolution of the electron in the orbit.

 14

Classically, an electron can be in any orbit around the nucleus of an atom. Then what determines the typical atomic size? Why is an atom not, say, thousand times bigger than its typical size? The question had greatly puzzled Bohr before he arrived at his famous model of the atom that you have learnt in the text. To simulate what he might well have done before his discovery, let us play as follows with the basic constants of nature and see if we can get a quantity with the dimensions of length that is roughly equal to the known size of an atom (~ 10–10m).

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


(b) You will find that the length obtained in A is many orders of magnitude smaller than the atomic dimensions. Further, it involves c. But energies of atoms are mostly in non-relativistic domain where c is not expected to play any role. This is what may have suggested Bohr to discard c and look for ‘something else’ to get the right atomic size. Now, the Planck’s constant h had already made its appearance elsewhere. Bohr’s great insight lay in recognising that h, me, and e will yield the right atomic size. Construct a quantity with the dimension of length from h, me, and e and confirm that its numerical value has indeed the correct order of magnitude.