Atom and Nuclei

Bohr Radius
Radius is directly proportional to the square of n. Velocity is inversely proportional to n.

Energy of a Bohr Orbit
The energy of nth Bohr orbit = 13.6 / n2

Energy is inversely proportional to square of n.

Ionization Energy
It is directly proportional to the square of Atomic Number.

Einstein's Mass - Energy Equation
E = mc2

1 amu = 931 MeV

Nuclear Binding Energy
N.B.E. = { ZMp +(A-Z)MN - Mo } c2

where A-Z is the number of neutrons.

Z is the atomic number ;

Nuclear Coulombic Barrier
The force of repulsion between the nucleus and approaching α particle is called as Nuclear Coulombic Barrier.

Repulsive Potential Energy = V = (Z1e)(Z2e) / d = (Z1Z2)e2/ d

The Nucleus
Nuclear Size

r = r0A1/3{where r = nuclear size, A = Atomic mass number}

r0= 1.4 x 10-15 m= 1.4 fm

Nucleus Proton Ratio

Nuclei with even number of protons and neutrons are abundantly found.

Nuclei with even - odd combination are intermediately found.

Nuclei with odd - odd combination are rare.

Nuclear Stability

The Nucleus Proton Ratio gives us an idea of the stability of a nucleus.

As atomic Number increases, the N/P ratio increases. (This is because the more number of neutrons are added than protons .)

All nuclei above atomic number 82, are beyond stability belt and are radioactive.

Radioactive Decay
Radioactive Decay takes place in order to stabilize the nucleus. In the process the atom may gain or loss it's entities.

α Emission

It is equivalent to a Helium Atom. An α emission eliminates 4 amu and 2 Z from an atom.

Almost all elements above atomic number 82 decay by α emission. It leads to the formation of isodiapher i.e. difference in number of Neutrons and Protons remains same.

β Emission

It is equivalent to conversion of a neutron to proton. Thus, A β emission adds 1 Z to an Atom. A neutron is converted into a proton by removal of negative charge from it.

Elements having a very high N/P ratio and that lie above the stability belt, undergo β emission , as neutrons are converted into protons are added.

Positron Emission

It is equivalent to a particle having no mass but positive charge. This particle is called as a positron.

It is actually the conversion of proton to neutron by removing it's positive charge.

A positron emission or β+ emission eliminates 1 Z from the atom.

Electron Capture (K Capture)

It is equivalent to an electron from K shell. In this emission, an electron from K shell is captured by the nucleus. AS an effect, a proton is converted into a positron. Electron capture mostly occurs with Heavier elements.

γ Emission

It is equivalent to a neutron. γ Emission causes no change.

Nuclear DisIntegration
Rate of DIsintegration

Rate of DIsintegration = - dN/dt

Rate α N

Rate = λ N

Integrated Rate Law

λ = [2.303/t] log10 [N0/N]

N0/N gives the ratio of number of atoms, or the ratio of number of moles.

Simultaneous Decay

Decay pf an equimolar binary mixture.

2.303 log10[N1/N2] = (λ2 - λ1) t

Half Life Period

t1/2undefined= 0.693 / λ  Also ; at any point ; N = 2-n N0

Average Life Period

λ' = 1/λ = t1/2 / 0.693 = 1.44 t1/2

Radioactivity
Unit

The standard unit of radioactivity is Curie.

Parallel Radioactive Disintegration

Suppose a Nuclei A splits into nuclei X and Y, then

λA = λX + λY

Fractional Yield of X = λX / λA

Fractional Yield of Y = λY / λA

Successive Radioactive Disintegration

Daughter Nuclei, formed by parent nuclei also undergoes disintegration.

Suppose A forms B and B forms C with λ1 and λ2 as disintegration constants ;

N1/N2 = (λ2- undefinedλ1) / λ1

Secular Equilibrium

N1 / N2 = λ2 / λ1

i.e. radioactivity is not affected through the half-lives

Disequilibrium