Nuclear Chemistry : Radioactivity

Radioactivity

“Radioactivity is a process in which nuclei of certain elements undergo spontaneous disintegration without excitation by any external means.’’

Henry Becquerel (1891) observed the spontaneous emission of invisible, penetrating rays from potassium uranyl sulphate K2UO2(SO4)2, which influenced photographic plate in dark and were able to produce luminosity in substances like ZnS.

Later on, Madam Marie Curie and her husband P. Curie named this phenomenon of spontaneous emission of penetrating rays as, Radioactivity. They also pointed out that radioactivity is a characteristic property of an unstable or excited nucleus, i.e., a nuclear property is independent of all the external conditions such as pressure, temperature, nature of other atoms associated with unstable atom but depends upon the amount of unstable atom.

Curies also discovered a new radioactive element Radium from pitchblende (an ore of U i.e. U3O3) which is about 3 million times more radioactive than uranium. Now a days about 42 radioactive elements are known.

The elements whose atoms disintegrate and emit radiations are called radioactive elements.

Radioactivity can be detected and measured by a number of devices like ionisation chamber, Geiger Muller counter, proportional counter, flow counter, end window counter, scintillation counter, Wilson cloud chamber, electroscope, etc. The proper device depends upon the nature of the radioactive substance and the type of radiation emitted. GM counter and proportional counter are suitable for solids and liquids, ionisation chamber is most suitable for gases.

Lightest radioactive isotope is tritium (1H3); other lighter radioactive nuclides are 14C, 40K and 99Tc.

(1)     Nature of radioactive emissions : The nature of the radiations emitted from a radioactive substance was investigated by Rutherford (1904) by applying electric and magnetic fields to the radiation as shown in figure. 

It is observed that on applying the field, the rays emitted from the radioactive substances are separated into three types, called α, β, and γ-rays. The a-rays are deflected in a direction which shows that they carry positive charge; the β-rays are deflected in the opposite direction showing that they carry negative charge and the γ-rays are not deflected at all showing that they carry no charge.

(2)     Characteristics of radioactive rays : Radioactive rays are characterised by the following properties :

          (i)      They blacken photographic plates.

          (ii)     They pass through thin metal foils.

          (iii)    They produce ionization in gases through which they passes.

          (iv)    They produce luminescence in zinc sulphide, barium platinocyanide, calcium tungstate, etc.

Radioactive radiations are composed of three important rays, namely α, β and γ−rays which differ very much in their nature and properties, e.g. penetrating power, ionising power and effect on photographic plates. Remember that γ-rays are not produced simultaneously with α  and β-rays but are produced subsequently.

 

Comparison of a, β and γ-rays

a-Particle or a-Ray β-Particle or b-Ray γ-Ray
(1) Charge and mass : It carries 2 units positive charge and 4 unit mass. It carries 1 unit negative charge and no mass. These  are electromagnetic rays with very short wavelength (≈ 0.05 Å)
(2) Nature: It is represented as helium nucleus or helium ions 2He4 or He++. It is represented as electron −1e0. It is represented as 0γ0
(3) Action of magnetic field: These are deflected towards the cathode. These are deflected to anode. These are not deflected.
(4) Velocity:

2 × 109 cm/s or

2 × 107 m/sec (1/10th to that of light)

 2.36 to 2.83 × 1010 cm/s (2.36 to 2.83 × 108 m/s) Same as that of light

3 × 1010 cm/s

(3 × 108 m/s)
(5) Ionizing power : Very high nearly 100 times to that of β-rays. Low nearly 100 times to that of γ-rays. Very low.
(6) Effect on ZnS plate : They cause luminescence. Very little effect. Very little effect.
(7) Penetrating power : Low 100 times that of a-particles. 10 times that of β-particles.
(8) Range : Very small (8-12 cm.) More that of a-particles. More
(9) Nature of product : Product obtained by the loss of 1 a-particle has atomic number less by 2 units and mass number less by 4 units. Product obtained by the loss of 1 β-particle has atomic number more by 1 unit, without any change in mass number. There is no change in the atomic number as well as in mass number.

Note :

β-particles originates in the nucleus; they are not orbital electrons.

β-particles having their velocity almost equal to velocity of light are known as hard b-particles and the others having their velocity ≈ 1 × 1010 cm sec−1 are called soft b-particles.

γ-radiation always accompany alpha or beta emissions and thus are emitted after a– and β-decay.

Only one kind of emission at a time is noticed. No radioactive substance emits both a– and β-particles simultaneously.

 

 Rate of radioactive decay

“According to the law of radioactive decay, the quantity of a radioelement which disappears in unit time (rate of disintegration) is directly proportional to the amount present.”

The law of radioactive decay may also be expressed mathematically.

Suppose the number of atoms of the radioactive element present at the commencement of observation, i.e. when t = 0 is N0, and after time t, the number of atoms remaining unchanged is Nt, then the rate of decay of atoms is  \[-\frac{d{{N}_{t}}}{dt}  (the word ‘d’ indicates a very-very small fraction; the negative sign shows that the number of atoms Nt decreases as time t increases)

Now since the change in number of atoms is proportional to the total number of atoms Nt, the relation becomes \[-\frac{d{{N}_{t}}}{dt}=\lambda {{N}_{t}}, where λ is a radioactive constant or decay constant.

  • Rate of decay of nuclide is independent of temperature, so its energy of activation is zero.
  • Since the rate of decay is directly proportional to the amount of the radioactive nuclide present and as the number of undecomposed atom decreases with increase in time, the rate of decay also decreases with the increase in time. Various forms of equation for radioactive decay are,

\[{{N}_{t}}={{N}_{0}}{{e}^{-\lambda t}}\,\,;\,\,\log {{N}_{0}}-\log {{N}_{t}}=0.4343\,\,\lambda t\,\,\,;\,\,\log \frac{{{N}_{0}}}{{{N}_{t}}}=\frac{\lambda t}{2.303}      

\[\lambda =\frac{2.303}{t}\log \frac{{{N}_{0}}}{{{N}_{t}}}                                               

where N0 = Initial number of atoms of the given nuclide, i.e. at time 0

Nt = Number of atoms of that nuclide present after time t.

λ = Decay constant

Note :   This equation is similar to that of first order reaction, hence we can say that radioactive disintegration are examples of first order reactions. However, unlike first order rate constant (K), the decay constant (λ) is independent of temperature.

Decay constant (l) : The ratio between the number of atoms disintegrating in unit time to the total number of atoms present at that time is called the decay constant of that nuclide.

Characteristics of decay constant (λ) :

  • It is characteristic of a nuclide (not for an element).
  • Its units are time−1.
  • Its value is always less than one.

 

Half life and Average life period

(1)     Half-life period (T1/2 or t1/2) : Rutherford in 1904 introduced a constant known as half-life period of the radioelement for evaluating its radioactivity or for comparing its radioactivity with the activities of other radioelements. The half-life period of a radioelement is defined, as the time required by a given amount of the element to decay to one-half of its initial value.

                                Mathematically,  \[{{t}_{1/2}}=\frac{0.693}{\lambda }

Now since λ is a constant, we can conclude that half-life period of a particular radioelement is independent of the amount of the radioelement. In other words, whatever might be the amount of the radioactive element present at a time, it will always decompose to its half at the end of one half-life period.

Half-life period is a measure of the radioactivity of the element since shorter the half-life period of an element, greater is the number of the disintegrating atoms and hence greater is its radioactivity. The half-life periods or the half-lives of different radioelements vary widely, ranging form a fraction of a second to million of years.

Fraction and Percent of radioactive nuclides left after n-Half-Lives

No. of half-lives passed (n)

Fraction of mass

Percent of mass
Decayed Left Decayed Left
0 0 1.0 0 100
 \[\frac{1}{2}  \[\frac{\sqrt{2}-1}{\sqrt{2}}=0.293  \[\frac{1}{\sqrt{2}}=0.707 29.3 79.7
1  \[\frac{1}{2}=0.50  \[\frac{1}{2}=0.50 50 50
2  \[\frac{3}{4}=0.75  \[\frac{1}{4}=0.25 75 25
3  \[\frac{7}{8}=0.875  \[\frac{1}{8}=0.125 87.5 12.5
4  \[\frac{15}{16}=0.9375  \[\frac{1}{16}=0.0625 93.75 6.25
5  \[\frac{31}{32}=0.96875  \[\frac{1}{32}=0.03125 96.75 3.125
 ∞ Total 0 100 0

 

Let the initial amount of a radioactive substance be N0

After one half-life period (t1/2) it becomes = N0/2

After two half-life periods (2t1/2) it becomes = N0/4                 

After three half-life periods (3t1/2) it becomes = N0/8

After n half life periods (2t1/2) it shall becomes  \[={{\left( \frac{1}{2} \right)}^{n}}{{N}_{0}}

 

Half life periods of some isotopes

Radio isotope
Half life
 Radio isotope Half life
 \[_{92}^{238}U 4.5 × 109 years  \[_{15}^{32}P 14.3 days
\[_{90}^{230}Th  8.3 × 104 years  \[_{53}^{131}I 8.0 days
 \[_{88}^{226}Ra 1.58 × 103 years  \[_{84}^{214}Po 1.5 × 10−4 seconds
 \[_{90}^{234}Th 24 days  \[_{6}^{14}C 5 × 103 years
 \[_{26}^{59}Fe 44.3 days  \[_{86}^{222}Rn 3.82 days

 

Thus, for the total disintegration of a radioactive substance an infinite time will be required.

 

Time (T) Amount of radioactive substance (N) Amount of radioactive substance decomposed (N0 – N)
0 (N0) 0
t1/2  \[\frac{1}{2}{{N}_{0}}={{\left( \frac{1}{2} \right)}^{1}}{{N}_{0}}  \[\frac{1}{2}{{N}_{0}}=\left[ 1-\frac{1}{2} \right]{{N}_{0}}
2t1/2  \[\frac{1}{4}{{N}_{0}}={{\left( \frac{1}{2} \right)}^{2}}{{N}_{0}}  \[\frac{3}{4}{{N}_{0}}=\left[ 1-\frac{1}{4} \right]{{N}_{0}}
3t1/2  \[\frac{1}{8}{{N}_{0}}={{\left( \frac{1}{2} \right)}^{3}}{{N}_{0}}  \[\frac{7}{8}{{N}_{0}}=\left[ 1-\frac{1}{8} \right]{{N}_{0}}
4t1/2  \[\frac{1}{16}{{N}_{0}}={{\left( \frac{1}{2} \right)}^{4}}{{N}_{0}}  \[\frac{15}{16}{{N}_{0}}=\left[ 1-\frac{1}{16} \right]{{N}_{0}}
nt/2  \[{{\left( \frac{1}{2} \right)}^{n}}{{N}_{0}  \[\left[ 1-{{\left( \frac{1}{2} \right)}^{n}} \right]{{N}_{0}}

 

Amount of radioactive substance left after n half-life periods

\[N={{\left( \frac{1}{2} \right)}^{n}}{{N}_{0}}                               

and Total time T = n × t1/2

where n is a whole number.

(2)     Average-life period (T) : Since total decay period of any element is infinity, it is meaningless to use the term total decay period (total life period) for radioelements. Thus the term average life is used which the following relation determines.

             Average life (T)  \[=\frac{\text{Sum}\,\,\text{of}\,\,\text{lives}\,\,\text{of}\,\,\text{the}\,\,\text{nuclei}}{\text{Total}\,\,\text{number}\,\,\text{of}\,\,\text{nuclei}}

Relation between average life and half-life : Average life (T) of an element is the inverse of its decay constant, i.e., \[T=\frac{1}{\lambda }, Substituting the value of λ in the above equation, \[T=\frac{{{t}_{1/2}}}{0.693}=1.44\,\,\,{{t}_{1/2}}

Thus, Average life (T) = 1.44 × Half life (T1/2) =  \[\sqrt{2}\times {{t}_{1/2}}

Thus, the average life period of a radioisotope is approximately under-root two times of its half life period.

Note :    This is because greater the value of λ, i.e., faster is the disintegration, the smaller is the average life (T).

 

Radioactive disintegration series

The phenomenon of natural radioactivity continues till stable nuclei are formed. All the nuclei from the initial element to the final stable element constitute a series known as disintegration series. Further we know that mass numbers change only when a-particles are emitted (and not when b-particles are emitted) causing the change in mass of 4 units at each step. Hence the mass numbers of all elements in a series will fit into one of the formulae.

4n, 4n + 1, 4n + 2 and 4n + 3

Hence there can be only four disintegration series

 

Series 4n 4n + 1 4n + 2 4n + 3
n 58 59 59 58
Parent element 90Th232 94Pu241 92U238 92U235
Half life 1.39 × 1010 years 10 years 4.5 × 109

years

 7.07 × 108 years
Prominent element 90Th232 93Np237 92U238 89Ac227
Half life 1.39 × 1010 years 2.2 × 106

years

 4.5 × 109 years 13.5 years
Name of series Thorium (Natural) Neptunium (Artificial) Uranium (Natural) Actinium (Natural)
End product 82Pb208 83Bi209 82Pb206 82Pb207
n 52 52 51 51
Number of lost particles α = 6

β = 4

 α = 8

β = 5

 α = 8

β = 6

 α = 7

β = 4

The numbers indicate that in a particular series the mass numbers of all the members are either divisible by 4 (in case of 4n) or divisible by 4 with remainder of 1, 2 or 3 (in the rest three series), n being an integer. In other words, the mass numbers of the members of 4n, 4n + 1, 4n +2 and 4n + 3 series are exactly divisible by 4, 4 + 1, 4 + 2 and 4 + 3 respectively.

Note :   

  • 4n + 1 series is an artificial series while the rest three are natural.
  • The end product in the 4n + 1 series is bismuth, while in the rest three, a stable isotope of lead is the end product.
  • The 4n + 1 series starts from plutonium 94Pu241 but commonly known as neptunium series because neptunium is the longest-lived member of the series.
  • The 4n + 3 series actually starts from 92U235.

 

Activity of population, Radioactive equilibrium and Units of radioactivity

(1)     Activity of population or specific activity : It is the measure of radioactivity of a radioactive substance. It is defined as ‘ the number of radioactive nuclei, which decay per second per gram of radioactive isotope.’ Mathematically, if ‘m‘ is the mass of radioactive isotope, then

\[\text{Specific}\,\,\text{activity}=\frac{\text{Rate}\,\,\text{of}\,\,\text{decay}}{m}=\frac{\lambda N}{m}=\lambda \times \frac{\text{Avogadro}\,\,\text{number}}{\text{Atomic}\,\,\text{mass}\,\,\text{in}\,\,g}

where N is the number of radioactive nuclei which undergoes disintegration.

(2)     Radioactive equilibrium : Suppose a radioactive element A disintegrates to form another radioactive element B which in turn disintegrates to still another element C.

A → B → C

In the starting, the amount of A (in term of atoms) is large while that of B is very small. Hence the rate of disintegration of A into B is high while that of B into C is low. With the passage of time, A go on disintegrating while more and more of B is formed. As a result, the rate of disintegration of A to B goes on decreasing while that of B to C goes on increasing. Ultimately, a stage is reached when the rate of disintegration of A to B is equal to that of B to C with the result the amount of B remains constant. Under these conditions B is said to be in equilibrium with A. For a radioactive equilibrium to be established half-life of the parent must be much more than half-life of the daughter.

It is important to note that the term equilibrium is used for reversible reactions but the radioactive reactions are irreversible, hence it is preferred to say that B is in a steady state rather than in equilibrium state.

At a steady state,

\[\frac{{{N}_{A}}}{{{N}_{B}}}=\frac{{{\lambda }_{B}}}{{{\lambda }_{A}}}=\frac{{{T}_{A}}}{{{T}_{B}}}\,\,\,\left( \because \stackrel{\scriptscriptstyle\to}{\leftarrow}T=\frac{1}{\lambda } \right)

Where λA and λB are the radioactive constants for the processes A → B and B → C respectively. Where TA and TB are the average life periods of A and B respectively.

In terms of half-life periods, \[\frac{{{N}_{A}}}{{{N}_{B}}}=\frac{{{({{t}_{1/2}})}_{A}}}{{{({{t}_{1/2}})}_{B}}}

Thus at a steady state (at radioactive equilibrium), the amounts (number of atoms) of the different radioelements present in the reaction series are inversely proportional to their radioactive constants or directly proportional to their half-life and also average life periods.

It is important to note that the radioactive equilibrium differs from ordinary chemical equilibrium because in the former the amounts of the different substances involved are not constant and the changes are not reversible.

(3)     Units of radioactivity : The standard unit in radioactivity is curie (c) which is defined as that amount of any radioactive material which gives 3.7 × 1010 disintegration’s per second (dps), i.e.,

            1c = Activity of 1g of Ra226 = 3.7 × 1010 dps

The millicurie (mc) and microcurie (mc) are equal to 10−3 and 10−6 curies i.e. 3.7 × 107 and 3.7 × 104 dps respectively.

1c = 103 mc = 106 μc ; 1c = 3.7 × 1010 dps ; 1mc = 3.7 × 107 dps ;  1μc = 3.7 × 104 dps    

But now a day, the unit curie is replaced by rutherford (rd) which is defined as the amount of a radioactive substance which undergoes 106 dps i.e., 1 rd = 106 dps. The millicurie and microcurie correspondingly rutherford units are millirutherford (mrd) and microrutherford (mrd) respectively.

1c = 3.7 × 1010 dps = 37 × 103 rd ; 1 mc = 3.7 × 107 dps = 37 rd ;

1μc = 3.7 × 104 dps = 37 mrd

However, is SI system the unit of radioactivity is Becquerel (Bq)

1 Bq = 1 disintegration per second = 1 dps = 1μrd

106 Bq = 1 rd              

3.7 × 1010 Bq = 1c

(4)     The Geiger-Nuttal relationship : It gives the relationship between decay constant of an a– radioactive substance and the range of the a-particle emitted.

log λ = A + BlogR

Where R is the range or the distance which an a-particle travels from source before it ceases to have ionizing power. A is a constant which varies from one series to another and B is a constant for all series. It is obvious that the greater the value of λ the greater the range of the a-particle.