Chapter 23 Nuclei (Nuclear Physics) free study material by TEACHING CARE online tuition and coaching classes

 

Rutherford’s a-scattering experiment established that the mass of atom is concentrated with small positively charged region at the centre which is called ‘nucleus’.

Nuclei are made up of proton and neutron. The number of protons in a nucleus (called the atomic number or proton number) is represented by the symbol Z. The number of neutrons (neutron number) is represented by N. The total number of neutrons and protons in a nucleus is called it’s mass number A so A = Z + N.

Neutrons and proton, when described collectively are called nucleons.

Nucleus contains two types of particles : Protons and neutrons

 

Nuclides are represented as

 Neutron.

_Z X ^A; where X denotes the chemical symbol of the element.

 

 

Neutron is a fundamental particle which is essential constituent of all nuclei except that of hydrogen atom. It was discovered by Chadwick.

• The charge of neutron : It is neutral
• The mass of neutron : 6750 ´ 10^–27 kg

 

 

• It’s spin angular momentum :

1 ´ æ h ö J – s

 

2   ç 2p ÷

è      ø

• It’s magnetic moment : 9.57 ´ 10^–27 J/Tesla
• It’s half life : 12 minutes
• Penetration power : High
• Types : Neutrons are of two types slow neutron and fast neutron, both are fully capable of penetrating a nucleus and causing artificial

Thermal neutrons

Fast neutrons can be converted into slow neutrons by certain materials called moderator’s (Paraffin wax, heavy water, graphite) when fast moving neutrons pass through a moderator, they collide with the molecules of the moderator, as a result of this, the energy of moving neutron decreases while that of the molecules of the moderator increases. After sometime they both attains same energy. The neutrons are then in thermal equilibrium with the molecules of the moderator and are called thermal neutrons.

Note : @Energy of thermal neutron is about 0.025 eV and speed is about 2.2 km/s.

Nucleus.

(1)  Different types of nuclei

The nuclei have been classified on the basis of the number of protons (atomic number) or the total number of nucleons (mass number) as follows

• Isotopes : The atoms of element having same atomic number but different mass number are called All isotopes have the same chemical properties. The isotopes of some elements are the following

 

 

 

1 H 1 , 1 H 2 , 1 H 3

8 O16,

8O17,

8O18

₂ He ³ ,

₂ He ⁴

17 Cl35 , 17Cl 37

92 U 235 ,

92U 238

 

• Isobars : The nuclei which have the same mass number (A) but different atomic number (Z) are called Isobars occupy different positions in periodic table so all isobars have different chemical properties. Some of the examples of isobars are

 

₁ H ³ and

₂ He ³ ,

₆C¹⁴ and 7 N ¹⁴ , 8 O¹⁷ and

9 F 17

 

• Isotones : The nuclei having equal number of neutrons are called For them both the atomic number (Z) and mass number (A) are different, but the value of (A – Z) is same. Some examples are

₄ Be ⁹ and 5 B¹⁰ , 6 C¹³ and 7 N ¹⁴ , 8 O¹⁸ and 9 F ¹⁹ , 3 Li⁷ and 4 Be ⁸ , 1 H ³ and 2 He ⁴

• Mirror nuclei : Nuclei having the same mass number A but with the proton number (Z) and neutron number (A – Z) interchanged (or whose atomic number differ by 1) are called mirror nuclei for

₁ H ³ and 2 He ³ , 3 Li⁷ and 4 Be⁷

(2)  Size of nucleus

• Nuclear radius : Experimental results indicates that the nuclear radius is proportional to A^1/3, where A is the

 

mass number of nucleus i.e.

R µ A1 / 3

Þ R = R0 A^{1 / 3} , where R₀ = 1.2 ´ 10^–15 m = 1.2 fm.

 

Note : @Heavier nuclei are bigger in size than lighter nuclei.

• Nuclear volume : The volume of nucleus is given by V = 4p R³ = 4 p R³ A Þ V µ A

3          3     ⁰

• Nuclear density : Mass per unit volume of a nucleus is called nuclear

Nuclear density(r ) = Mass of nucleus =                   mA          

 

Volume of nucleus

4 p (R

3      0

A1 / 3 )3

 

 

where m = Average of mass of a nucleon (= mass of proton + mass of neutron = 1.66 ´ 10^–27 kg) and mA = Mass of nucleus

 

0

Þ r =

3m 4pR ³

= 2.38 ´ 10¹⁷ kg / m³

 

Note : @r is independent of A, it means r is same of all atoms.

@    Density of a nucleus is maximum at it’s centre and decreases as we move outwards from the nucleus.

(3)  Nuclear force

Forces that keep the nucleons bound in the nucleus are called nuclear forces.

• Nuclear forces are short range These do not exist at large distances greater than 10^–15 m.
• Nuclear forces are the strongest forces in
• These are attractive force and causes stability of the

At low speeds, electromagnetic repulsion prevents the collision of nuclei

At high speeds, nuclei come close enough for the strong force to bind them together.

 

 

• These forces are charge
• Nuclear forces are non-central

Nuclear forces are exchange forces

According to scientist Yukawa the nuclear force between the two nucleons is the result of the exchange of particles called mesons between the nucleons.

p – mesons are of three types – Positive p meson (p⁺), negative p meson (p ^–), neutral p meson (p⁰) The force between neutron and proton is due to exchange of charged meson between them i.e.

p ® p ⁺ + n,  n ® p + p ^–

 

The forces between a pair of neutrons or a pair of protons are the result of the exchange of neutral meson (p^o)

 

between them i.e.

p ® p‘+p ⁰

and

n ® n‘+p ⁰

 

 

Thus exchange of p meson between nucleons keeps the nucleons bound together. It is responsible for the nuclear forces.

Dog-Bone analogy

The above interactions can be explained with the dog bone analogy according to which we consider the two interacting nucleons to be two dogs having a common bone clenched in between their teeth very firmly. Each one of these dogs wants to take the bone and hence they cannot be separated easily. They seem to be bound to each other with a strong attractive force (which is the bone) though the dogs themselves are strong enemies. The meson plays the same role of the common bone in between two nucleons.

(4)  Atomic mass unit (amu)

The unit in which atomic and nuclear masses are measured is called atomic mass unit (amu)

 

1 amu (or 1u) = 1 th of mass of

12

₆ C¹² atom = 1.66 ´ 10^–27 kg

 

Masses of electron, proton and neutrons

Mass of electron (m_e) = 9.1 ´ 10^–31 kg = 0.0005486 amu, Mass of proton (m_p) = 1.6726 ´ 10^–27 kg = 1.007276 amu

Mass of neutron (m_n) = 1.6750 ´ 10^–27 kg = 1.00865 amu, Mass of hydrogen atom (m_e + m_p) = 1.6729 ´ 10^–27 kg = 1.0078 amu

 

Mass-energy equivalence

According to Einstein, mass and energy are inter convertible. The Einstein’s mass energy relationship is given by E = mc ²

If m = 1 amu, c = 3 ´ 10⁸ m/sec then E = 931 MeV i.e. 1 amu is equivalent to 931 MeV or 1 amu (or 1 u) = 931 MeV

 

 

 

(5)  Pair production and pair-annihilation

 

 

When an energetic g–ray photon falls on a heavy substance. It is absorbed by some nucleus of the substance and an electron and a positron are produced. This phenomenon is called pair production and may be represented

 

by the following equation

hn

(g -photon)

=        1 b ⁰       +

(Positron)

-1 b 0 (Electron)

+1b0

 

The rest-mass energy of each of positron and electron is

E₀ = m₀c² = (9.1 ´ 10^–31 kg) ´ (3.0 ´ 10⁸ m/s)²

= 8.2 ´ 10^–14 J = 0.51 MeV

hn

g-photon

+Ze

 

Nucleus

 

 

 

–1b0

 

Hence, for pair-production it is essential that the energy of g-photon must be at least 2 ´ 0.51 = 1.02 MeV. If the energy of g-photon is less than this, it would cause photo-electric effect or Compton effect on striking the matter.

The converse phenomenon pair-annihilation is also possible. Whenever an electron and a positron come very close to each other, they annihilate each other by combining together and two g–photons (energy) are produced. This phenomenon is called pair annihilation and is represented by the following equation.

 

 

+1 b 0 (Positron)

+         -1 b ⁰ (Electron)

=           hn

(g -photon)

+       hn

(g -photon)

 

 

(6)  Nuclear stability

Among about 1500 known nuclides, less than 260 are stable. The others are unstable that decay to form other nuclides by emitting a, b-particles and g – EM waves. (This process is called radioactivity). The stability of nucleus is determined by many factors. Few such factors are given below :

• Neutron-proton ratio æN Ratioö

Z

ç             ÷

è             ø

The chemical properties of an atom are governed entirely by the number of protons (Z) in the nucleus, the stability of an atom appears to depend on both the number of protons and the number of neutrons.

For lighter nuclei, the greatest stability is achieved when the number of protons and neutrons are

approximately equal (N » Z) i.e. N = 1

Z

Heavy nuclei are stable only when they have more neutrons than protons. Thus heavy nuclei are neutron rich compared to lighter nuclei (for heavy nuclei, more is the number of protons in the nucleus, greater is the electrical repulsive force between them. Therefore more neutrons are added to provide the strong attractive forces necessary to keep the nucleus stable.)

 

 

Figure shows a plot of N verses Z for the stable nuclei. For mass number upto about A = 40. For larger value of Z the nuclear force is unable to hold the nucleus together against the electrical repulsion of the protons unless the number of neutrons exceeds the number of protons. At Bi (Z = 83, A = 209), the neutron excess in N – Z = 43. There are no stable nuclides with Z > 83.

Note : @The nuclide 83 Bi ²⁰⁹ is the heaviest stable nucleus.

 

@          A nuclide above the line of stability i.e. having excess neutrons, decay through b ^– emission

(neutron changes into proton). Thus increasing atomic number Z and decreasing neutron number N.

 

In b ^–

emission,

N ratio decreases.

Z

 

A nuclide below the line of stability have excess number of protons. It

decays by b ⁺ emission, results in decreasing Z and increasing N. In b ⁺

 

 

emission, the

N ratio increases.

Z

Proton number (Z)

 

• Even or odd numbers of Z or N : The stability of a nuclide is also determined by the consideration whether it contains an even or odd number of protons and

It is found that an even-even nucleus (even Z and even N) is more stable (60% of stable nuclide have even Z

and even N).

An even-odd nucleus (even Z and odd N) or odd-even nuclide (odd Z and even N) is found to be lesser sable while the odd-odd nucleus is found to be less stable.

Only five stable odd-odd nuclides are known : 1 H ² , 3 Li⁶ , 5 Be¹⁰ , 7 N ¹⁴ and 75 Ta¹⁸⁰

• Binding energy per nucleon : The stability of a nucleus is determined by value of it’s binding energy per In general higher the value of binding energy per nucleon, more stable the nucleus is

Mass Defect and Binding Energy.

(1)  Mass defect (Dm)

It is found that the mass of a nucleus is always less than the sum of masses of it’s constituent nucleons in free state. This difference in masses is called mass defect. Hence mass defect

Dm = Sum of masses of nucleons – Mass of nucleus

= {Zmp   + (A – Z)mn }- M  = {Zmp   + Zme   + (A – Z)mz }- M‘

where m_p = Mass of proton, m_n = Mass of each neutron, m_e = Mass of each electron

M = Mass of nucleus, Z = Atomic number, A = Mass number, M¢ = Mass of atom as a whole.

Note : @The mass of a typical nucleus is about 1% less than the sum of masses of nucleons.

(2)  Packing fraction

Mass defect per nucleon is called packing fraction

 

 

 

Packing fraction (f ) = Dm = M – A

where M = Mass of nucleus, A = Mass number

 

A             A

Packing fraction measures the stability of a nucleus. Smaller the value of packing fraction, larger is the stability of the nucleus.

• Packing fraction may be of positive, negative or zero

(iii) At A = 16, f ® Zero

(3)  Binding energy (B.E.)

The neutrons and protons in a stable nucleus are held together by nuclear forces and energy is needed to pull them infinitely apart (or the same energy is released during the formation of the nucleus). This energy is called the binding energy of the nucleus.

or

The binding energy of a nucleus may be defined as the energy equivalent to the mass defect of the nucleus. If Dm is mass defect then according to Einstein’s mass energy relation

Binding energy = Dm × c² = [{m_pZ + m_n(A – Z)} – M]× c²

(This binding energy is expressed in joule, because Dm is measured in kg)

If Dm is measured in amu then binding energy = Dm amu = [{m_pZ + m_n(A – Z)} – M] amu = Dm ´ 931 MeV

(4)  Binding energy per nucleon

The average energy required to release a nucleon from the nucleus is called binding energy per nucleon. Binding energy per nucleon =                            Total binding energy                = Dm ´ 931    MeV

 

Mass number (i.e. total number of nucleons)

Binding energy per nucleon µ Stability of nucleus

 Binding Energy Curve.

A          Nucleon

 

 

It is the graph between binding energy per nucleon and total number of nucleons (i.e. mass number A)

 

 

 

 

 

 

 

 

 

 

• Some nuclei with mass number A < 20 have large binding energy per nucleon than their neighbour

 

For example

₂ He ⁴ , 4 Be ⁸ , 6 C¹² , 8 O¹⁶ and 10 Ne ²⁰ . These nuclei are more stable than their neighbours.

 

 

• The binding energy per nucleon is maximum for nuclei of mass number A = 56 (₂₆ Fe ⁵⁶ ). It’s value is 8

MeV per nucleon.

• For nuclei having A > 56, binding energy per nucleon gradually decreases for uranium (A = 238), the value of binding energy per nucleon drops to 5 MeV.

Note : @When a heavy nucleus splits up into lighter nuclei, then binding energy per nucleon of lighter nuclei is more than that of the original heavy nucleus. Thus a large amount of energy is liberated in this process (nuclear fission).

@ When two very light nuclei combines to form a relatively heavy nucleus, then binding energy per nucleon increases. Thus, energy is released in this process (nuclear fusion).

 

B.E. A

 

 

 

A

Nuclear Reactions.

The process by which the identity of a nucleus is changed when it is bombarded by an energetic particle is called nuclear reaction. The general expression for the nuclear reaction is as follows.

 

X          +         a

¾¾®         C

¾¾®

Y             +         b           +    Q

 

(Parent nucleus)

(Incident particle)

(Compound nucleus)

(Compound nucleus)

(Product particles)

(Energy)

 

 

Here X and a are known as reactants and Y and b are known as products. This reaction is known as (a, b) reaction and can be represented as X(a, b) Y

(1)  Q value or energy of nuclear reaction

The energy absorbed or released during nuclear reaction is known as Q-value of nuclear reaction.

Q-value = (Mass of reactants – mass of products)c² Joules

= (Mass of reactants – mass of products) amu

If Q < 0, The nuclear reaction is known as endothermic. (The energy is absorbed in the reaction) If Q > 0, The nuclear reaction is known as exothermic (The energy is released in the reaction)

(2)  Law of conservation in nuclear reactions

• Conservation of mass number and charge number : In the following nuclear reaction

2 He 4 + 7 N 14  ® 8 O17 + 1 H 1

 

Mass number (A) ®	Before the reaction	After the reaction
Charge number (Z) ®	4 +14 = 18 2 + 7 = 9	17 + 1 = 18 8 + 1 = 9

 

 

• Conservation of momentum : Linear momentum/angular momentum of particles before the reaction is equal to the linear/angular momentum of the particles after the That is Sp = 0
• Conservation of energy : Total energy before the reaction is equal to total energy after the Term

Q is added to balance the total energy of the reaction.

(3)  Common nuclear reactions

The nuclear reactions lead to artificial transmutation of nuclei. Rutherford was the first to carry out artificial transmutation of nitrogen to oxygen in the year 1919.

2 He 4 + 7 N 14 ® 9 F 18 ® 8 O17 + 1 H 1

It is called (a, p) reaction. Some other nuclear reactions are given as follows.

 

 

(p, n) reaction Þ (p, a) reaction Þ (p, g) reaction Þ (n, p) reaction Þ (g, n) reaction Þ

1 H 1 + 5 B11 ® 6 C12  ® 6 C11 + 0 n1

₁ H ¹ + 3 Li¹¹ ® 4 Be ⁸ ® 2 He ⁴ + 2 He ⁴

1 H 1 + 6 C12 ® 7 N 13 ® 7 N 13 + g

0 n1 + 7 N 14 ® 7 N 15 ® 6 C14 + 1 H 1

g + 1 H 2 ® 1 H 1 + 0 n1

 

Nuclear Fission and Fusion.

Nuclear fission

The process of splitting of a heavy nucleus into two lighter nuclei of comparable masses (after bombardment with a energetic particle) with liberation of energy is called nuclear fission.

The phenomenon of nuclear fission was discovered by scientist Ottohann and F. Strassman and was explained by N. Bohr and J.A. Wheeler on the basis of liquid drop model of nucleus.

(1)  Fission reaction of U²³⁵

• Nuclear reaction :

 

 

92 U 235 + 0 n1 ®

92 U 236

(unstable nucleus)

® 56 Ba141 + 36 Kr 92 + 30 n1 + Q

 

 

• The energy released in U²³⁵ fission is about 200 MeV or 8 MeV per nucleon.

 

 

• By fission of

₉₂U ²³⁵ , on an average 2.5 neutrons are liberated. These neutrons are called fast neutrons

 

and their energy is about 2 MeV (for each). These fast neutrons can escape from the reaction so as to proceed the chain reaction they are need to slow down.

• Fission of U²³⁵ occurs by slow neutrons only (of energy about 1eV) or even by thermal neutrons (of energy about 025 eV).
• 50 kg of U²³⁵ on fission will release » 4 × 10¹⁵ J of This is equivalence to 20,000 tones of TNT

explosion. The nuclear bomb dropped at Hiroshima had this much explosion power.

 

 

• The mass of the compound nucleus must be greater than the sum of masses of fission

 

 

• The

Binding energy A

of compound nucleus must be less than that of the fission products.

 

 

₃₆ Kr

• It may be pointed out that it is not necessary that in each fission of uranium, the two fragments ₅₆ Ba

are formed but they may be any stable isotopes of middle weight atoms.

and

 

 

 

Same other U ²³⁵

fission reactions are

92 U 235 + 0 n1 ® 54 Xe140 + 38 Sr 94 + 20 n1

® 57 La148 + 35 Br 85  + 3 0 n1

® Many more

 

• The neutrons released during the fission process are called prompt
• Most of energy released appears in the form of kinetic energy of fission

 

 

 

 

Slow Neutron

92U235                 92U236

 

 

(2)  Chain reaction

In nuclear fission, three neutrons are produced along with the release of large energy. Under favourable conditions, these neutrons can cause further fission of other nuclei, producing large number of neutrons. Thus a chain of nuclear fissions is established which continues until the whole of the uranium is consumed.

In the chain reaction, the number of nuclei undergoing fission increases very fast. So, the energy produced takes a tremendous magnitude very soon.

Difficulties in chain reaction

 

 

 

• Absorption of neutrons by U ²³⁸ , the major part in natural uranium is the isotope U²³⁸ (99.3%), the isotope

 

U 235

is very little (0.7%). It is found that U ²³⁸

is fissionable with fast neutrons, whereas U ²³⁵ is fissionable with slow

 

neutrons. Due to the large percentage of

U 238 ,

there is more possibility of collision of neutrons with

U ²³⁸ . It is

 

found that the neutrons get slowed on coliding with

U 238 ,

as a result of it further fission of U²³⁸ is not possible

 

(Because they are slow and they are absorbed by U²³⁸). This stops the chain reaction.

 

Removal : (i) To sustain chain reaction 92 U ²³⁵

is separated from the ordinary uranium. Uranium so obtained

 

(92 U 235 )

is known as enriched uranium, which is fissionable with the fast and slow neutrons and hence chain

 

reaction can be sustained.

• If neutrons are slowed down by any method to an energy of about 3 eV, then the probability of their

 

absorption by U ²³⁸

becomes very low, while the probability of their fissioning U ²³⁵

becomes high. This job is done

 

by moderators. Which reduce the speed of neutron rapidly graphite and heavy water are the example of moderators.

• Critical size : The neutrons emitted during fission are very fast and they travel a large distance before being slowed down. If the size of the fissionable material is small, the neutrons emitted will escape the fissionable material before they are slowed Hence chain reaction cannot be sustained.

Removal : The size of the fissionable material should be large than a critical size.

The chain reaction once started will remain steady, accelerate or retard depending upon, a factor called neutron reproduction factor (k). It is defined as follows.

k = Rate of production of neutrons Rate of loss of neutrons

® If k = 1, the chain reaction will be steady. The size of the fissionable material used is said to be the critical size and it’s mass, the critical mass.

® If k > 1, the chain reaction accelerates, resulting in an explosion. The size of the material in this case is super critical. (Atom bomb)

® If k < 1, the chain reaction gradually comes to a halt. The size of the material used us said to be sub-critical. Types of chain reaction : Chain reactions are of following two types

 

Controlled chain reaction	Uncontrolled chain reaction
Controlled by artificial method	No control over this type of nuclear reaction
All neurons are absorbed except one	More than one neutron takes part into reaction
It’s rate is slow	Fast rate
Reproduction factor k = 1	Reproduction factor k > 1
Energy liberated in this type of reaction is always less	A large amount of energy is liberated in this type of
than explosive energy	reaction
Chain reaction is the principle of nuclear reactors	Uncontrolled chain reaction is the principle of atom bomb.

 

 

Note : @The energy released in the explosion of an atom bomb is equal to the energy released by 2000 tonn

of TNT and the temperature at the place of explosion is of the order of 10^{7 o}C.

Nuclear Reactor.

A nuclear reactor is a device in which nuclear fission can be carried out through a sustained and a controlled chain reaction. It is also called an atomic pile. It is thus a source of controlled energy which is utilised for many

 

useful purposes.

 

Coolant

 

 

Concrete wall

 

 

Moderator

Cadmium

rods                    Core

 

Coolant out Turbine

 

 

Water

Heat

 

 

 

 

To electric generator

 

Condenser

 

 

 

(1)  Parts of nuclear reactor

 

Fuel elements

Coolant in exchanger

 

• Fissionable material (Fuel) : The fissionable material used in the reactor is called the fuel of the reactor. Uranium isotope (U²³⁵) Thorium isotope (Th²³²) and Plutonium isotopes (Pu²³⁹, Pu²⁴⁰ and Pu²⁴¹) are the most commonly used fuels in the
• Moderator : Moderator is used to slow down the fast moving neutrons. Most commonly used moderators are graphite and heavy water (D₂O).
• Control Material : Control material is used to control the chain reaction and to maintain a stable rate of This material controls the number of neutrons available for the fission. For example, cadmium rods are inserted into the core of the reactor because they can absorb the neutrons. The neutrons available for fission are controlled by moving the cadmium rods in or out of the core of the reactor.
• Coolant : Coolant is a cooling material which removes the heat generated due to fission in the Commonly used coolants are water, CO₂ nitrogen etc.
• Protective shield : A protective shield in the form a concrete thick wall surrounds the core of the reactor to save the persons working around the reactor from the hazardous

Note : @It may be noted that Plutonium is the best fuel as compared to other fissionable material. It is because fission in Plutonium can be initiated by both slow and fast neutrons. Moreover it can be obtained from U ²³⁸ .

@ Nuclear reactor is firstly devised by fermi.

@ Apsara was the first Indian nuclear reactor.

(2)  Uses of nuclear reactor

• In electric power
• To produce radioactive isotopes for their use in medical science, agriculture and

 

• In manufacturing of

PU 239

which is used in atom bomb.

 

 

• They are used to produce neutron beam of high intensity which is used in the treatment of cancer and nuclear

Note : @A type of reactor that can produce more fissile fuel than it consumes is the breeder reactor.

Nuclear fusion

In nuclear fusion two or more than two lighter nuclei combine to form a single heavy nucleus. The mass of single nucleus so formed is less than the sum of the masses of parent nuclei. This difference in mass results in the release of tremendous amount of energy

₁ H ² + 1 H ² ® 1 H ³ + 1 H ¹ + 4 MeV

₁ H ³ + 1 H ² ® 2 He ⁴ + 0 n¹ + 17.6 MeV

or                    1 H ² + 1 H ² ® 2 He ⁴ + 24 MeV

For fusion high pressure (» 10⁶ atm) and high temperature (of the order of 10⁷ K to 10⁸ K) is required and so the reaction is called thermonuclear reaction.

Fusion energy is greater then fission energy fission of one uranium atom releases about 200 MeV of energy.

 

But the fusion of a deutron

(1 H ² )

and triton

(1 H ³ )

releases about 17.6 MeV of energy. However the energy

 

released per nucleon in fission is about 0.85 MeV but that in fusion is 4.4 MeV. So for the same mass of the fuel, the energy released in fusion is much larger than in fission.

Plasma : The temperature of the order of 10⁸ K required for thermonuclear reactions leads to the complete ionisation of the atom of light elements. The combination of base nuclei and electron cloud is called plasma. The enormous gravitational field of the sun confines the plasma in the interior of the sun.

The main problem to carryout nuclear fusion in the laboratory is to contain the plasma at a temperature of 10⁸K. No solid container can tolerate this much temperature. If this problem of containing plasma is solved, then the large quantity of deuterium present in sea water would be able to serve as in-exhaustible source of energy.

Note : @To achieve fusion in laboratory a device is used to confine the plasma, called Tokamak. Stellar Energy

Stellar energy is the energy obtained continuously from the sun and the stars. Sun radiates energy at the rate of about 10²⁶ joules per second.

Scientist Hans Bethe suggested that the fusion of hydrogen to form helium (thermo nuclear reaction) is continuously taking place in the sun (or in the other stars) and it is the source of sun’s (star’s) energy.

The stellar energy is explained by two cycles

 

 

 

 

About 90% of the mass of the sun consists of hydrogen and helium.

Nuclear Bomb.

Based on uncontrolled nuclear reactions.

 

 

Example: 1        A heavy nucleus at rest breaks into two fragments which fly off with velocities in the ratio 8 : 1. The ratio of radii of the fragments is                                                                                                     [EAMCET (Engg.) 2001]

(a) 1 : 2                           (b) 1 : 4                              (c) 4 : 1                        (d) 2 : 1

 

Solution : (a)

v₁

By conservation of momentum m₁v₁ = m₂v₂

 

Þ v1 = 8 = m2

…… (i)

 

v2      1    m1

r      æ A

ö1 / 3

æ 1 ö^{1 / 3}     1

 

v₂         Also from r µ A^{1 / 3}

Þ 1 = ç   ¹ ÷

= ç   ÷      =

 

r2     è A2 ø

è 8 ø         2

 

13

Example: 2           The ratio of radii of nuclei ²⁷ Al

and ¹²⁵ Te

is approximately                                                  [J & K CET 2000]

 

52

(a) 6 : 10                         (b) 13 : 52                          (c) 40 : 177                   (d) 14 : 7

 

r       æ A

ö1 / 3

æ 27 ö^{1/ 3}     8     6

 

Solution : (a)       By using r µ A^{1 / 3}

Þ   1 = ç   1 ÷        = ç

÷       =      =

 

r2     è A2 ø

è 125 ø

5    10

 

Example: 3           If Avogadro’s number is 6 ´ 10²³ then the number of protons, neutrons and electrons in 14 g of ₆C¹⁴ are respectively

(a) 36 ´ 10²³, 48 ´ 10²³, 36 ´ 10²³                                   (b) 36 ´ 10²³, 36 ´ 10²³, 36 ´ 10²¹

 

 

(c) 48 ´ 10²³, 36 ´ 10²³, 48 ´ 10²¹                                   (d) 48 ´ 10²³, 48 ´ 10²³, 36 ´ 10²¹

 

Solution : (a)       Since the number of protons, neutrons and electrons in an atom of 6 C¹⁴

are 6, 8 and 6 respectively. As 14 gm

 

of 6 C¹⁴ contains 6 ´ 10²³ atoms, therefore the numbers of protons, neutrons and electrons in 14 gm of 6 C¹⁴

 

are

6 ´ 6 ´ 10²³ = 36 ´ 10²³ ,

8 ´ 6 ´ 10²³ = 48 ´ 10²³ ,

6 ´ 6 ´ 10²³ = 36 ´ 10²³ .

 

Example: 4           Two Cu⁶⁴ nuclei touch each other. The electrostatics repulsive energy of the system will be

(a) 0.788 MeV                        (b) 7.88 MeV                              (c) 126.15 MeV                  (d) 788 MeV

Solution : (c)        Radius of each nucleus R = R0(A)^{1 / 3} = 1.2 (64)^{1 / 3} = 4.8 fm

Distance between two nuclei (r) = 2R

 

So potential energy U = k × q 2

r

= 9 ´ 10⁹ ´ (1.6 ´ 10 ^–19 ´ 29)²

2 ´ 4.8 ´ 10 ^–15 ´ 1.6 ´ 10 ^–19

= 126.15 MeV.

 

Example: 5           When

₉₂ U ²³⁵ undergoes fission. 0.1% of its original mass is changed into energy. How much energy is

 

released if 1 kg of 92 U 235  undergoes fission                    [MP PET 1994; MP PMT/PET 1998; BHU 2001; BVP 2003]

(a) 9 ´ 10¹⁰ J                           (b) 9 ´ 10¹¹ J                               (c) 9 ´ 10¹² J                       (d) 9 ´ 10¹³ J

 

 

Solution : (d)       By using

E = Dm × c²

Þ E = æ 0.1 ´

ç 100

è

ö

1÷ (3

ø

´ 10⁸)² = 9 ´ 10¹³ J

 

Example: 6           1 g of hydrogen is converted into 0.993 g of helium in a thermonuclear reaction. The energy released is

[EAMCET (Med.) 1995; CPMT 1999]

(a) 63 ´ 10⁷ J                          (b) 63 ´ 10¹⁰ J                             (c) 63 ´ 10¹⁴ J                     (d) 63 ´ 10²⁰ J

Solution : (b)       Dm = 1 – 0.993 = 0.007 gm

2

\ E = Dmc² = 0.007 ´ 10^–3 ´ (3 ´ 10⁸)² = 63 ´ 10¹⁰ J

 

1

Example: 7           The binding energy per nucleon of deuteron

(² H)

and helium nucleus

(⁴ He)

is 1.1 MeV and 7 MeV

 

 

 

 

 

Solution : (c)

respectively. If two deuteron nuclei react to form a single helium nucleus, then the energy released is

[MP PMT 1992; Roorkee 1994; IIT-JEE 1996; AIIMS 1997; Haryana PMT 2000; Pb PMT 2001; CPMT 2001; AIEEE 2004]

(a) 13.9 MeV                           (b) 26.9 MeV                              (c) 23.6 MeV                      (d) 19.2 MeV

₁ H ² + 1H ² ® 2He⁴ + Q

Total binding energy of helium nucleus = 4 ´ 7 = 28 MeV Total binding energy of each deutron = 2 ´ 1.1 = 2.2 MeV Hence energy released = 28 – 2 ´ 2.2 = 23.6 MeV

 

Example: 8  The masses of neutron and proton are 1.0087 amu and 1.0073 amu respectively. If the neutrons and protons combine to form a helium nucleus (alpha particles) of mass 4.0015 amu. The binding energy of the helium nucleus will be [1 amu= 931 MeV]                                                                       [CPMT 1986; MP PMT 1995; CBSE 2003]

(a) 28.4 MeV                           (b) 20.8 MeV                              (c) 27.3 MeV                      (d) 14.2 MeV

Solution : (a)   Helium nucleus consist of two neutrons and two protons.

So binding energy E = Dm amu = Dm ´ 931 MeV

Þ E = (2 ´ m_p + 2m_n – M) ´ 931 MeV = (2 ´ 1.0073 + 2 ´ 1.0087 – 4.0015) ´ 931 = 28.4 MeV

Example: 9 A atomic power reactor furnace can deliver 300 MW. The energy released due to fission of each of uranium atom U ²³⁸ is 170 MeV. The number of uranium atoms fissioned per hour will be                                            [UPSEAT 2000] (a) 5 ´ 10¹⁵                       (b) 10 ´ 10²⁰                        (c) 40 ´ 10²¹                  (d) 30 ´ 10²⁵

 

 

 

 

Solution : (c)        By using

P = W = n ´ E

where n = Number of uranium atom fissioned and E = Energy released due to

 

t

each fission so

t

300 ´ 106 = n ´ 170 ´ 10⁶ ´ 1.6 ´ 10^–19

3600

Þ n = 40 ´ 10²¹

 

Example: 10        The binding energy per nucleon of O¹⁶ is 7.97 MeV and that of O¹⁷ is 7.75 MeV. The energy (in MeV) required to remove a neutron from O¹⁷ is                                                                                                 [IIT-JEE 1995]

(a) 3.52                           (b) 3.64                              (c) 4.23                        (d) 7.86

 

Solution : (c)

O17 ® O16 + 0n1

\ Energy required = Binding of O¹⁷ – binding energy of O¹⁶ = 17 ´ 7.75 – 16 ´ 7.97 = 4.23 MeV

 

Example: 11 A gamma ray photon creates an electron-positron pair. If the rest mass energy of an electron is 0.5 MeV and the total kinetic energy of the electron-positron pair is 0.78 MeV, then the energy of the gamma ray photon must be                                                                                                                                        [MP PMT  1991]

(a) 0.78 MeV                           (b) 1.78 MeV                              (c) 1.28 MeV                      (d) 0.28 MeV

Solution : (b)      Energy of g-rays photon = 0.5 + 0.5 +0.78 = 1.78 MeV

Example: 12      What is the mass of one Curie of U²³⁴                                                                                           [MNR 1985]

(a) 3.7 ´ 10¹⁰ gm                    (b) 2.348 ´ 10²³ gm                   (c)  1.48 ´ 10^–11 gm        (d) 6.25 ´ 10^–34 gm

Solution : (c)    1 curie = 3.71 ´ 10¹⁰ disintegration/sec and mass of 6.02 ´ 10²³ atoms of U ²³⁴ = 234 gm

\ Mass of 3.71 ´ 10¹⁰ atoms = 234 ´ 3.71´ 1010 = 1.48 ´ 10^–11 gm

6.02 ´ 10²³

 

Example: 13        In the nuclear fusion reaction ² H +³ H ®⁴ He + n,

given that the repulsive potential energy between the two

 

1        1           2

 

nuclei is

• 7 ´ 10^–14 J , the temperature at which the gases must be heated to initiate the reaction is nearly

 

[Boltzmann’s constant k = 1.38 ´ 10^–23 J/K ]                                                                                  [AIEEE 2003]

 

(a)

10⁹ K

(b)

10⁷ K

(c)

10⁵ K

(d)

10³ K

 

Solution : (a)       Kinetic energy of molecules of a gas at a temperature T is 3/2 kT

 

 

\ To initiate the reaction

3 kT = 7.7 ´ 10^–14 J

2

Þ T = 3.7 ´ 10⁹ K.

 

Example: 14        A nucleus with mass number 220 initially at rest emits an a-particle. If the Q value of the reaction is 5.5 MeV. Calculate the kinetic energy of the a-particle                                                                 [IIT-JEE (Screening) 2003]

(a) 4.4 MeV                             (b) 5.4 MeV                                (c) 5.6 MeV                        (d) 6.5 MeV

 

Solution : (b)

 

M = 220                               m1 = 216

k₂ p₂

m₂ = 4

 

 

 

Q-value of the reaction is 5.5 eV i.e. k₁ + k ₂ = 5.5 MeV

……(i)

 

By conservation of linear momentum

p1 = p2 Þ

=                    Þ k₂ = 54 k₁          ……(ii)

 

Ne

On solving equation (i) and (ii) we get k₂ = 5.4 MeV.

 

Example: 15         Let mp

be the mass of a proton, mn

the mass of a neutron, M1 the mass of a 20

nucleus and

M ₂ the mass of a

 

20

10

40 Ca nucleus. Then                                                                                                  [IIT 1998; DPMT 2000]

 

(a)

M2 = 2M1

(b)

M2 > 2M1

(c)

M2 < 2M1

(d)

M1 < 10(mn + mp )

 

 

Ne

Solution : (c, d)    Due to mass defect (which is finally responsible for the binding energy of the nucleus), mass of a nucleus is

 

10

always less then the sum of masses of it’s constituent particles ²⁰

is made up of 10 protons plus 10

 

10

neutrons. Therefore, mass of ²⁰ Ne nucleus M1 < 10 (mp + mn )

Also heavier the nucleus, more is he mass defect thus 20 (mn + mp ) – M2 > 10(mp + mn) – M1

 

or          10 (mp + mn ) > M 2 – M1

Þ             M2 < M1 + 10 (mp + mn)

Þ M 2 < M1 + M1   Þ

M 2 < 2M1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Radioactivity.

The phenomenon of spontaneous emission of radiatons by heavy elements is called radioactivity. The elements which shows this phenomenon are called radioactive elements.

• Radioactivity was discovered by Henery Becquerel in uranium salt in the year
• After the discovery of radioactivity in uranium, Piere Curie and Madame Curie discovered a new radioactive element called radium (which is 10⁶ times more radioactive than uranium)
• Some examples of radio active substances are : Uranium, Radium, Thorium, Polonium, Neptunium
• Radioactivity of a sample cannot be controlled by any physical (pressure, temperature, electric or magnetic field) or chemical
• All the elements with atomic number (Z ) > 82 are naturally
• The conversion of lighter elements into radioactive elements by the bombardment of fast moving particles is called artificial or induced
• Radioactivity is a nuclear event and not atomic. Hence electronic configuration of atom don’t have any relationship with

Nuclear radiatons

 

 

According to Rutherford’s experiment when a sample of radioactive substance is put in a lead box and allow the emission of radiation through a small hole only. When the radiation enters into the external electric field, they splits into three parts

 

 

 

 

 

 

• Radiations which deflects towards negative plate are called a-rays (stream of positively charged particles)
• Radiations which deflects towards positive plate are called b particles (stream of negatively charged particles)
• Radiations which are undeflected called g-rays. (E.M. waves or photons)

Note : @Exactly same results were obtained when these radiations were subjected to magnetic field.

@    No radioactive substance emits both a and b particles simultaneously. Also g–rays are emitted after the emission of a or b-particles.

@    b-particles are not orbital electrons they come from nucleus. The neutron in the nucleus decays into proton and an electron. This electron is emitted out of the nucleus in the form of b-rays.

 

Properties of a, b and g-rays

 

Features	a– particles	b – particles	g – rays
1. Identity	Helium      nucleus      or	Fast       moving       electron	Photons (E.M. waves)
	doubly ionised helium atom (2He4)	(-b 0 or b – )
2. Charge	+ 2e	– e	Zero
3. Mass 4 mp (mp = mass of proton = 1.87 ´ 10–27	4 mp	me	Massless
4. Speed	» 107 m/s	1% to 99% of speed of light	Speed of light
5. Range of kinetic energy	4 MeV to 9 MeV	All possible values between a	Between     a     minimum
		minimum certain value to	value to 2.23 MeV
		1.2 MeV
6. Penetration power (g, b, a)	1	100	10,000
	(Stopped by a paper)	(100 times of a)	(100 times of b upto 30
			cm of iron (or Pb) sheet
7. Ionisation power (a > b > g)	10,000	100	1
8.    Effect    of    electric    or	Deflected	Deflected	Not deflected
magnetic field
9. Energy spectrum	Line and discrete	Continuous	Line and discrete

 

 

 

 

Radioactive Disintegration.

(1)  Law of radioactive disintegration

According to Rutherford and Soddy law for radioactive decay is as follows.

“At any instant the rate of decay of radioactive atoms is proportional to the number of atoms present at that

 

instant” i.e.

• dNµ N dt

Þ dN = –lN . It can be proved that N = N₀e^–lt

dt

 

This equation can also be written in terms of mass i.e. M = M₀e^–lt

where N = Number of atoms remains undecayed after time t, N₀ = Number of atoms present initially (i.e. at t = 0), M = Mass of radioactive nuclei at time t, M₀ = Mass of radioactive nuclei at time t = 0, N₀ – N = Number of disintegrated nucleus in time t

dN = rate of decay, l = Decay constant or disintegration constant or radioactivity constant or Rutherford

dt

Soddy’s constant or the probability of decay per unit time of a nucleus.

Note : @l depends only on the nature of substance. It is independent of time and any physical or chemical changes.

 

N

 

t

(2)  Activity

It is defined as the rate of disintegration (or count rate) of the substance (or the number of atoms of any

 

material decaying per second) i.e.

A = – dN = lN = lN

dt

0 e –lt

= A0 e –lt

 

where A₀ = Activity of t = 0, A = Activity after time t

 

 

Units of activity (Radioactivity)

It’s units are Becqueral (Bq), Curie (Ci) and Rutherford (Rd)

 

 

1 Becquerel = 1 disintegration/sec,    1 Rutherford = 10⁶ dis/sec,     1 Curie = 3.7 ´ 10¹¹ dis/sec

Note : @Activity per gm of a substance is known as specific activity. The specific activity of 1 gm of radium – 226 is 1 Curie.

@    1 millicurie = 37 Rutherford

@    The activity of a radioactive substance decreases as the number of undecayed nuclei decreases with time.

 

 

@     Activity µ

1

 

Half life

 

 

(3)  Half life (T_1/2)

Time interval in which the mass of a radioactive substance or the number of it’s atom reduces to half of it’s

N0

initial value is called the half life of the substance.                                              N

 

i.e.               if

N = N0

2

then t = T1 / 2

 

Hence from

N = N0 e ^–lt

N₀/2 N₀/4

 

N                  –l (T   )

log 2

0.693

N₀/8

 

l

     0 = N0 e

2

1 / 2

Þ T1 / 2 =           e        =     l

0      1      2T        3T                      t

 

 

Time (t)	Number of undecayed atoms (N) (N0 = Number of initial atoms)	Remaining fraction of active atoms (N/N0) probability of survival		Fraction of atoms decayed (N0 – N) /N0 probability of decay
t = 0	N0	1 1 2 1 4 1 8   æ 1 ö10 ç 2 ÷ è   ø æ 1 ön ç 2 ÷ è   ø	(100%) (50%)   (25%)   (12.5%)     » 0.1%	0 1          (50%) 2 3         (75%) 4 7          (87.5%) 8 » 99.9%     ìï      æ 1 ön üï í1 – ç ÷ ý ïî       è 2 ø   ïþ
t = T1/2	N 0
	2
t = 2(T1/2)	1 ´ N 0 = N 0
	2     2      (2)2
t = 3(T1/2)	1 ´ N0 = N0
	2    (2)     (2)3
t = 10 (T1/2)	N 0
	(2)10
t = n (N1/2)	N
	(2)2

 

 

Useful relation

 

After n half-lives, number of undecayed atoms

æ 1 öⁿ

N = N₀ ç 2 ÷

æ 1 öt / T1 / 2

= N₀ ç 2 ÷

 

 

(4)  Mean (or average) life (t)

è   ø              è   ø

 

The time for which a radioactive material remains active is defined as mean (average) life of that material.

Other definitions

 

 

• It is defined as the sum of lives of all atoms divided by the total number of atoms

i.e. t = Sum of the lives of all the atoms = 1

 

Total number of atoms           l

 

 

• From

N = N

0 e –lt Þ

ln N N0

t

= –l

slope of the line shown in the graph

 

i.e. the magnitude of inverse of slope of ln N

N0

vs t

curve is known as mean life (t).

N

N

ln

 

• From

N = N

0 e –lt                                                                                                                                                                                                                0

 

If t = 1 = t Þ

l

N = N

₀e ^-1 = N

æ 1 ö = 0.37 N

0 ç  ÷

e                      0

= 37%

of N₀.

t

 

è ø

 

i.e. mean life is the time interval in which number of undecayed atoms (N) becomes 37% of original number of atoms.                                                                     or

1 times or 0.37 times or

e

 

It is the time in which number of decayed atoms (N

– N) becomes

æ1 – 1 ö

times or 0.63 times or 63% of

 

 

original number of atoms.

₀                    ç         ÷

e

è        ø

 

• From T1/ 2

= 0.693

l

Þ 1 = t =

l

1

 

0.693

.(t

 

1 / 2

) = 1.44 (T1 / 2 )

 

i.e. mean life is about 44% more than that of half life. Which gives us t > T_(1/2)

Note : @ Half life and mean life of a substance doesn’t change with time or with pressure, temperature etc.

Radioactive Series.

If the isotope that results from a radioactive decay is itself radioactive then it will also decay and so on.

The sequence of decays is known as radioactive decay series. Most of the radio-nuclides found in nature are members of four radioactive series. These are as follows

Mass number	Series (Nature)	Parent	Stable and product	Integer	n	Number of lost particles
4n	Thorium (natural)	90 Th 232	82 Pb 208	52	a = 6, b = 4
4n + 1	Neptunium	93 Np 237	83 Bi 209	52	a = 8, b = 5
	(Artificial)
4n + 2	Uranium (Natural)	92 U 238	82 Pb 206	51	a = 8, b = 6
4n + 3	Actinium (Natural)	89 Ac 227	82 Pb 207	51	a = 7, b = 4

 

Note : @The 4n + 1 series starts from

94 PU 241

but commonly known as neptunium series because

 

neptunium is the longest lived member of the series.

@    The 4n + 3 series actually starts from ₉₂ U ²³⁵ .

Successive Disintegration and Radioactive Equilibrium.

 

 

Suppose a radioactive element A disintegrates to form another radioactive element B which intern disintegrates to still another element C; such decays are called successive disintegration.

 

Rate of disintegration of

A = dN1

dt

= –l₁ N1

(which is also the rate of formation of B)

 

Rate of disintegration of

B = dN 2

dt

= –l₂ N 2

 

\ Net rate of formation of B = Rate of disintegration of A – Rate of disintegration of B

= l₁N₁ – l₂N₂

Equilibrium

In radioactive equilibrium, the rate of decay of any radioactive product is just equal to it’s rate of production from the previous member.

 

i.e.                                               l N

= l N

Þ         l₁

= N 2

= t 2 =

 

(T1 / 2 )

 

 

1   1         2 2

l       N         t       (T     )

 

2           2          1           1 / 2 1

Note : @ In successive disintegration if N₀ is the initial number of nuclei of A at t = 0 then number of nuclei of

 

product B at time t is given by N   =

l1 N0

(e ^{–l t} – e ^{–l t} ) where l l

– decay constant of A and B.

 

 

Uses of radioactive isotopes

• In medicine

²    (l

1                    2

2

1 2

– l₁)

 

• For testing blood-chromium – 51 (ii) For testing blood circulation – Na – 24

• For detecting brain tumor- Radio mercury – 203
• For detecting fault in thyroid gland – Radio iodine – 131
• For cancer – cobalt – 60
• For blood – Gold – 189
• For skin diseases – Phospohorous – 31

(2)  In Archaeology

• For determining age of archaeological sample (carbon dating) C¹⁴
• For determining age of meteorites – K ⁴⁰
• For determining age of earth-Lead isotopes

(3)  In agriculture

 

• For protecting potato crop from earthworm- CO⁶⁰

• For artificial rains – AgI (iii) As fertilizers –

P 32

 

• As tracers – (Tracer) : Very small quantity of radioisotopes present in a mixture is known as tracer
  • Tracer technique is used for studying biochemical reaction in tracer and animals.

(5)  In industries

• For detecting leakage in oil or water pipe lines (ii) For determining the age of

 

 

21

1      l₁, T₁, t₁

l

 

 

 

 

 

 

 

Example: 16      When 90 Th²²⁸

transforms to 83 Bi²¹² , then the number of the emitted a–and b–particles is, respectively

[MP PET 2002]

 

 

Solution : (d)

(a) 8a, 7b                        (b) 4a, 7b                           (c) 4a, 4b                     (d) 4a, 1b

Z =90 ThA= 228 ® Z‘=83 Bi A‘= 212

Number of a-particles emitted na = A – A‘ = 228 – 212 = 4

 

4              4

Number of b–particles emitted nb = 2na – Z + Z‘ = 2 ´ 4 – 90 + 83 = 1.

Example: 17      A radioactive substance decays to 1/16^th of its initial activity in 40 days. The half-life of the radioactive substance expressed in days is                                                                                                   [AIEEE 2003]

(a) 2.5                             (b) 5                                  (c)  10                           (d) 20

 

æ 1 ö t / T1 / 2

N         1     æ 1 ö ^{40 / T}1 / 2

 

Solution : (c)        By using

N = N ₀ ç 2 ÷              Þ N     = 16 = ç 2 ÷

Þ T1 / 2 = 10

days.

 

è   ø                        ₀             è ø

Example: 18      A sample of radioactive element has a mass of 10 gm at an instant t = 0. The approximate mass of this element in the sample after two mean lives is                                                                            [CBSE PMT 2003]

(a) 2.50 gm                             (b) 3.70 gm                                 (c) 6.30 gm                         (d) 1.35 gm

 

 

–lt

–l(2t )

æ 2 ö

–l

ç l ÷

æ 1 ö²

 

Solution : (d)       By using M = M₀e

Þ M = 10e            = 10e     ^{è   ø} = 10 ç   ÷   = 1.359 gm

 

 

Example: 19      The half-life of ²¹⁵ At

è e ø

is 100 ms. The time taken for the radioactivity of a sample of ²¹⁵ At

to decay to 1/16^th of

 

its initial value is                                                                                                    [IIT-JEE (Screening) 2002]

(a) 400 ms                                (b) 6.3 ms                                     (c) 40 ms                              (d) 300 ms

 

æ 1 öⁿ

N        æ 1 ö ^{t / T}1 / 2

1     æ 1 ö t / 100

 

Solution : (a)       By using

N = N ₀ ç 2 ÷      Þ N     = ç 2 ÷            Þ   16 = ç 2 ÷

Þ t = 400 m sec.

 

è   ø                 ₀     è   ø                              è ø

Example: 20      The mean lives of a radioactive substance for a and b emissions are 1620 years and 405 years respectively. After how much time will the activity be reduced to one fourth                                                              [RPET 1999]

• 405 year (b) 1620 year                      (c) 449 year                  (d) None of these

 

 

Solution : (c)

l_a =    1    per year and l_b

=   1  

per year and it is given that the fraction of the remained activity A = 1

 

1620

405

A0       4

 

Total decay constant l = l_a + l_b

=    1   +   1   =   1 per year

 

1620     405     324

 

 

 

We know that A = A e ^–lt Þ t = 1 log         A0 Þ t = 1 log

 

4 = 2 log

2 = 324 ´ 2 ´ 0.693 = 449 years.

 

⁰                 l      ^e   A

l     e             l     e

 

Example: 21     At any instant the ratio of the amount of radioactive substances is 2 : 1. If their half lives be respectively 12 and 16 hours, then after two days, what will be the ratio of the substances                                         [RPMT 1996] (a) 1 : 1                           (b) 2 : 1                              (c) 1 : 2                        (d) 1 : 4

2´24

æ 1 ö 12

n

æ 1 öⁿ             N         (N )     (1 / 2)ⁿ1             2    ç 2 ÷              1

 

Solution : (a)       By using

N = N ₀ ç   ÷        Þ

1 =       0 1 ´

=     ´ è   ø           =

 

è 2 ø

N2      (N0 )2

(1 / 2) 2

1          2´24      1

æ 1 ö

 

ç ÷ 16

è 2 ø

Example: 22      From a newly formed radioactive substance (Half-life 2 hours), the intensity of radiation is 64 times the permissible safe level. The minimum time after which work can be done safely from this source is

[IIT 1983; SCRA 1996]

• 6 hours (b) 12 hours                                (c) 24 hours                        (d) 128 hours

 

æ 1 öⁿ

A        1     æ 1 ö⁰

æ 1 öⁿ

 

Solution : (b)       By using

A = A₀ ç 2 ÷        Þ     A   = 64 = ç 2 ÷   = ç 2 ÷

Þ n = 6

 

è   ø                   ₀             è   ø        è ø

 

Þ       t     = 6

T1 / 2

Þ t = 6 ´ 2 = 12 hours.

 

Example: 23      nucleus of mass number A, originally at rest, emits an a-particle with speed v. The daughter nucleus recoils

with a speed						[DCE 2000; AIIMS 2004]
(a) 2v /(A + 4)	(b)	4v /(A + 4)	(c)	4v /(A – 4)	(d)	2v /(A – 4)

Solution : (c)                                                                                   m

v

+

 

 

Rest

 

According to conservation of momentum 4v = (A – 4)v‘ Þ

v‘ =   4v    .

A – 4

 

Example: 24      The counting rate observed from a radioactive source at t = 0 second was 1600 counts per second and at t = 8 seconds it was 100 counts per second. The counting rate observed as counts per second at t = 6 seconds will be                                                                                                           [MP PET 1996; UPSEAT 2000]

(a) 400                            (b) 300                               (c)  200                         (d) 150

 

æ 1 öⁿ

æ 1 ö8 / T1 / 2

1     æ 1 ö^{8 / T}1 / 2

 

Solution : (c)        By using

A = A₀ ç 2 ÷   Þ 100 = 1600 ç 2 ÷

Þ   16 = ç 2 ÷               Þ T_1 / 2 = 2 sec

 

è   ø                                  è   ø

è   ø

æ 1 ö6 / 2

 

Again by using the same relation the count rate at t = 6 sec will be

A = 1600 ç 2 ÷

= 200 .

 

è ø

Example: 25      The kinetic energy of a neutron beam is 0.0837 eV. The half-life of neutrons is 693s and the mass of neutrons is 1.675 ´ 10^–27 kg. The fraction of decay in travelling a distance of 40m will be

 

(a)

10-3

10-4

 

3

10-5

10-6

 

Solution : (c)        v =              =

= 4 ´ 10

m/sec

 

 

 

 

\ Time taken by neutrons to travel a distance of 40 m

Dt‘ =

40

4 ´ 10³

= 10 ^–2 sec

 

\     dN = l N dt

Þ dN = l dt

N

 

 

 

 

\ Fraction of neutrons decayed in Dt sec in

DN = l Dt = 0.693 Dt = 0.693 ´ 10 ^–2 = 10 ^–5

 

 

N                         T                  693

Example: 26      The fraction of atoms of radioactive element that decays in 6 days is 7/8. The fraction that decays in 10 days will be (a) 77/80                                       (b) 71/80                             (c) 31/32                      (d) 15/16

 

æ N ₀ ö

 

æ log

 N₀ ö

 

æ 1 ö t / T1 / 2

T_{1/ 2} log _e ç N ÷

N            t        ç       e     N ÷

 

Solution : (c)        By using

N = N ₀ ç   ÷

Þ   t =                    è        ø

Þ t µ log e    0 Þ   ¹ = è                     ø1

 

è 2 ø

log e (2)

N          t 2       æ log

 N ₀ ö

 

 

Hence 6 =

10

log e (8 / 1) log e (N ₀ / N)

Þ log N 0

e     N

= 10 log e

6

 

• = log e

 

32 Þ

ç

è

N 0 = 32 .

N

_e          ÷

N ø 2

 

So fraction that decays = 1 – 1

32

= 31 .

32