Nuclear Chemistry, a key topic for GATE chemistry coaching and Chemistry JAM coaching

The Atomic Nucleus:

The protons and Neutrons are the constituent particles of the Nucleus. There nuclear constituent particular are in general called nucleons which follow the Fermi – Dirac statistics.

Ø The nucleus contains Z protons (Z denotes the atomic Number) and N neutrons to give the mass number A (=Z + N). For a particular value of Z, because of the different numbers of N, the isotopes appear. The neutron- proton ratio and their oddness and evenness are very much important in determining the stability of the nucleus.

Ø The emission of particles i.e. electrons, from the nucleus does not contradict the proton – neutron model of the nucleus. It is now believed that the trans formation of a neutron into a proton leads to the ejection of a  particle.(Important topic for Chemistry JAM Coaching)

In this proton – neutron model, it is now believed that they are continuously changing one into another through the exchange of common property. Thus, within the nucleus, it is not appropriate to define a particle either as a proton or as a neutron. They are really indistinguishable within the nucleus due to the exchange phenomenon. This is why, within the nucleus, the constituent particles are reasonably referred to as nucleons, and they can be characterized as protons neutrons when they emerge form the nucleus. Important topic for Chemistry JAM Coaching

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Fermions and Bosons

Ø Maxwell –Boltzmann statistics is well documented to explain the properties (e.g. velocity distribution, energy distribution, etc) for the assemblies (i.e. large number) of molecules of a gas. However, this classical mechanics is not applicable for the elementary particles like electron, protons, neutrons, meson, neutrinos, etc and atomic nuclei. To deal with these particles, two quantum mechanical statistics, namely, Fermi-Dirac statistics and Bose –Einstein statistics are developed. The particles having half integral spins (e.g. electrons , protons, neutrons, positrons, etc) which  follow the Fermi –Dirac statistics  are called fermions  while the particular having the integral spins (e.g. photons, mesons, α particles, etc) which follow the Bose –Einstein statistics  are called bosons.

The fermions follow the Pauli’s exclusion principle while the bosons do not

Nuclear Forces: àWhat is the nature of the forces which hold the protone and neutrons together in such a small nucleus?

If electrostatic forces (Coulombic forces) alone were involved, the repulsion between the protons would have rendered the nucleus highly unstable and labile to disintegration. Since this does not happen except in the case of heavier elements which are radioactive it is evident that there must be some attractive force between protone and neutron, neutron and neutron and even between proton and proton. This must be stronger than the repulsive forces due to positive charges on the protone. These attractive forces are called nuclear forces. These forces are short range forces (i.e. 10^-16m). There are not follow the inverse square law. 

Hideki Yukawa suggested that pi mesons oscillate between neighboring nucleons (protons and neutrons) with a velocity close to that of light.

Proton + negative pi meson Neutron

Neutron + positive pi meson  Proton

There is thus an exchange of pi mesons back and forth between neighboring nucleons. This results in an allraction between neutron and proton. Important topic for Chemistry JAM Coaching

Nuclear Size:

Some indication about the size of the nucleus was obtained from Ruther ford’s experiments on scattering of alpha particles. More accurate information has now been obtained from experiments on scallering of neutrons.

The results can be summarized by the equation.

                                   Where r = radius of a nucleus                                                                          A = Mass Number                                                                    

                                                   R₀ = Constant =

Nuclear radius is generally measured in Fermi units

Example:  Estimate the radius of nucleus.

Solution   The radius                              

Where R = 1.5 f         and A = 27

Then 

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Atomic mass Unit (a.m.u)

According to the latest convention, the mass of the standard carbon atom  is taken to be 12 a.m.u. Thus one amu, is exactly 1/12^th of the mass of an  atom.

1amu = where is Avogadro’s number

=  

Nuclear Density: à

As mentioned above, the radius (r) of a nucleus is extremely small being of the order of . Since the nucleus is believed to be spherical, its volume  would be for too smaller, being to the order of . As a result, the density (mass/ volume) of the nucleus would be tremendously high. It is independent of the nature and size of the nuclei.

  

Where m = mass (in kg)

      r  = Radius (in m)

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Shape of the Nucleus: à

In our previous consideration, the spherical nature of the nucleus has been considered. But the actual shape of the nucleus depends on the total angular quantum number or simply, the nuclear spin I and the electric quadrupole moment (=Q) of the nucleus under consideration.

The parameter, electric quadrupole moment (Q) describes the interaction of the nucleus with the electric field gradient . Thus the term Q can be expressed as follows.

Where e is the unit of electrostatic charge, (r) gives the electric charge density at a distance r from the centre of the nucleus with x,y,z as its coordinates whose origin also lies at the centre of the nucleus –Q give a measure of deviation of nuclear charge distribution from the spherical symmetry. If the charge is spherically and symmetrically distributed then it leads to.

   

In such case (i.e. spherically symmetrical) e Q = 0, i.e. Q= 0. The positive value of Q indicates that the nucleus is elongated in the Z- direction and negative value indicates that the nucleus is contracted in the z – direction.

It has been found that the systems having I 1 and eQ are deviated from the spherical shape. On the other hand, the nuclei having I = 0. ½ and e Q = 0, are perfectly spherical. In this parameter e is the unit of electrostatic charge and Q gives the measure of deviation from sphericality of the nucleus. If eQ is positive, ie eQ >0, then the nucleus becomes a prelate spheroid (ie egg shaped) while for eQ <0, the nucleus becomes an oblate spheroid (ie. discus or pumpkin shaped). These are shown in Fig.

     

The dimension of quadrupole moment (Q) is cm² and it is very often expressed in the unit of (=1barn). The value for deuteron is m barn (milli barn)

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Equivalence of a.m.u. and Energy: à

According to Einstein’s mass energy relationship, E = mc² where m is mass in kg, C is velocity of light in ms^-1 and E is energy in joules. Therefore. energy equivalent to one a.m.u is given by

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Packing fraction:  

It has been observed that isotopic masses of element are close to but not exactly equal to whole numbers. The mass numbers, however, are invariably whole number. Aston expressed the variation of isotopic mass from mass number of the isotope in terms of a quantity called packing fraction.

Packing fraction may have a negative an a positive sign. The negative sign of the packing fraction means that the isotopic mass is less than the mass number of the isotope. In such cases, some mass gets transformed into energy in the formation of that nucleus. In accordance with Einstein’s equation, E = mc². Such nuclei therefore are more stable. A positive packing fraction. On the other hand would imply a tendency towards instability. But this is not correct especially for elements of low atomic masses.

Mass Defect:    Binding Energy of Nucleus:à

It has been found that the actual isotopic mass of an element is almost invariably less than the sum of masses of protons. Neutrons and electrons present in it. The difference between the two is known as mass defect. 

Suppose A is the mass number and Z is the atomic Number (nuclear charge) of an element. Evidently its atom contains Z protons, (A-Z) neutrons and Z electrons. Let m_p be the mass of a proton, m_n that of a neutron and m_e that of an electron. The atomic mass M’ of the isotope, therefore should be given by.

 

Where  is mass of Hydrogen atom.

The actual atomic mass of the element, however, is found to be less  than M’ suppose it is M. Then

Mass defect = M’ – M = ΔM = Z  + (A-Z) M_n – M

The quantity ΔM represents the loss of mass in the formation of the nucleus of the atom. The loss of mass is equivalent of the energy released in the formation of the given nucleus from individual protons and neutrons. The release of energy results in the stability of the nucleus.

The energy released in the formation of a nucleus from its constituent nucleons is called the binding energy of the nucleus. it is a important topic for GATE chemistry coaching

This binding energy of a nucleus when divided by the number of nucleons gives the means binding energy per nucleon.  The binding energy per nucleon is a measure of stability of the nucleus. The greater the binding energy per nucleon is a measure of stability of the nucleus. it is a important topic for GATE chemistry coaching

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