3.10 Sspd_chapter1_part 7_hydrogen atom analysis

SSPD_CHAPTER 1_Part 7_ ANALYSIS OF HYDROGEN ATOM.

Consider a Hydrogen Atom where an electron is orbiting around the proton in circular path.

From Neil Bohr’s first postulate:

m e .v.r = n(h/(2π)) 1.23

where n is the principal quantum number and an integral number.

Squaring both sides of Eq.(1.23) and reshuffling the terms,

v 2 = n 2 .ћ 2 /(m e 2 .r 2 ) 1.24

From rotational motion we know that:

Centripetal Force = m _e .v ² /r = electrostatic force between electron and proton.= q ² /(4πε _0. r ² );

Therefore v 2 = q 2 /(4πε 0. r.m e ) 1.25

Dividing Eq(1.25) by Eq.(1.24) we get:

q ² /(4πε _0. r.m _e ) = n ² .ћ ² /(m _e ² .r ² )

Simplifying the expression:

r n = n 2 .h 2 .ε 0 /(m e .q 2 .π) = the Bohr Radius of nth Orbit; 1.26

h=6.62×10 ^-34 (J-sec) = Plank’s Constant;

m _e = 9.1×10 ^-31 Kg;

q = 1.6 ×10 ^-19 Coul;

ε ₀ = absolute permittivity= (1/(36π×10 ⁹ )) F/m;

ε _r = relative permittivity of vacuum= 1;

a ₀ = ћ ² /(m _e .q ² )= 0.4756 = Bohr Radius;

By Dimensional Balance the correctness of the Equation(1.26) is established.

Eq.(1.26) tells us that only spheres of orbital radius r _n = n ² .h ² .ε ₀ /(m _e .q ² .π) are permitted.

The radius of the first and innermost spherical orbit = 0.529Å;

The radius of the second spherical orbit = 2.116 Å because n=2;

The radius of the third spherical orbit = 4.761 Å because n=3;

Thus we see that only a given number of spherical orbits having Bohr radii are permitted around the nucleus of an Atom. Also in these permissible orbits the standing wave boundary conditions are fulfilled as shown in Fig.(1.14)

i.e. 2π r n = n.λ n 1.27

From Eq.(1.26) r n = n 2 .0.529Å 1.28

Substituting Eq.(1.28) in Eq.(1.27)

2π.n ² . 0.529 Å = n.λ _n

Simplifying the expression:

λ n = (n.π.1.058) Å 1.29

Figure 1.14. Standing Wave Pattern in permitted spherical orbits around the nucleus of an Atom.

From the two Equations (1.28) and (1.29), the radii of the permissible spherical orbits as well as the wavelength of the standing wave patterns are calculated as shown in Table(1.2).

Table 1. 2. The radii of the orbits and the wavelength of the standing wave pattern in first three permissible orbits.

The bounded electrons in an Atom are in stationary and non-radiating orbits unlike Betatron.

The bounded electrons orbital radii are quantized as shown by Eq.(1.28) as well as the energy of these bounded electrons in the spherical potential well are also quantized as we will show in the following derivation.

Total Energy = Kinetic Energy + Potential Well

E= (1/2)m e v 2 – (q e .q p )/( 4πε 0. r n ) 1.30

In bringing a positive charge, work will be done against the repulsive field due the positive nucleus and hence potential energy will increase as we bring the positive charge nearer and nearer the nucleus. So a positive charge sees a potential hump at the positive nucleus. By similar logic , electron which is a negative charge sees a potential well at the positive nucleus hence potential energy is more and more negative as it falls deeper and deeper into the potential well.