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 .

## Binding Energy

### 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 ;

## Wavelength of Electron

 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**

t1/2= 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 - λ1) / λ1

**Secular Equilibrium**

N1 / N2  = λ2 / λ1

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

**Disequilibrium**