## Laws of Radioactive Disintegration and Half Life

Subject: Physics

#### Overview

The spontaneous breaking of a nucleus is known as radioactive disintegration. This note provides us an information on laws of radioactive disintegration and half life.
##### Laws of Radioactive Disintegration and Half Life

The spontaneous breaking of a nucleus is known as radioactive disintegration. Rutherford and Soddy made experimental study of the radioactive decay of various radioactive materials and gave the following laws:

Radioactivity decay is a random and spontaneous process. It is not influenced by external conditions such as temperature, pressure, electric field etc. Each decay is an independent event occurs by a chance to take first.

In any radioactive decay, either a α-particle or β-particle is emitted by the atom. Emission of both types is impossible at a time. Moreover, an atom does not emit more than one α-particle or more than one β-particle at a time.

When a nuclide emits α-alpha particle, its mass number is reduced by four and atomic number by two.

\begin{align*} _ZX^A \rightarrow _{Z-2}Y^{A-4} + _2He^4 \\ \text {Example}: _{92} U^{238} \rightarrow _{90} Th^{234} + _2He^4 \end{align*}

When a nuclide emits a β-particle, its mass number remains unchanged but atomic number increases by 1.

\begin{align*} _ZX^A \rightarrow _{Z+1}Y^{A} + _{-1}e^0 \\ \text {Example}: _{6} C^{14} \rightarrow _{7} N^{14} + _2He^4 + _{-1}e^0 \end{align*}

When a nuclide emits a gamma ray, neither the mass number nor the atomic number changes.

$$_ZX^A \rightarrow _{Z}Y^{A} +\gamma$$

The gamma radiation is emitted by the excited nucleus. These laws are called the displacement laws.

The rate of disintegration of a radioactive substance is directly proportional to the number of atoms present at that instant. This is called decay law.

Let N0 be the number of atoms present in the radioactive sample at t =0 and N be the number of atoms left after time t. then, the rate of disintegration, dN/dt is proportional to N. so,

\begin{align*} \frac {dN}{dt} &\propto N \\ \text {or,} \: \frac {dN}{dt} &= -\lambda N \\ \end{align*}

Where λ is a constant of proportionality called as disintegration constant or decay constant. The – sign indicates that N decreases as time increases. The number of disintegration per second, dN/dt is called the activity of radioactive sample.

The above equation can be written as

\begin{align*} \frac {dN}{N} &= -\lambda dt \\ \text {integrating on both sides, we get} \\ \int _{N_0}^N \frac {dN}{N} &= -\lambda \int _0^t dt\\ \text {or,} \: [\log _e N]_{N_0}^N &= -\lambda [t]_0^t \\ \text {or,} \: \log _e N - \log_e N_0 &= -\lambda t \\ \text {or,} \log _e \frac {N}{N_0} &= -\lambda t \\ \text {or,} N &= N_0 e^{-\lambda t} \\ \end{align*}

This equation is known as decay constant. It shows that number of active nuclei in a radioactive sample decreases exponentially with time as shown in the figure.

#### Decay Constant

From the law of radioactive disintegration, we have

\begin{align*} \frac {dN}{dt} &= -\lambda N \\ \text {or,} \: \lambda &= \frac {-\frac {dN}{dt}}{N} \\ \end{align*} Hence the decay constant is defined as the ratio of rate of decay per unit atom present. \begin{align*} \\ \text {If we put t} = \frac {1}{\lambda } \text {in decay equation }\\ N &= N_0 e^{-\lambda t} we get, \\ N &= N_0 e^{-1} = \frac {N_0} {e} \\ &= \frac {N_0}{2.718} = 0.37 N_0 = 37\%\: of \:N_0 \\ \end{align*}

Since decay constant may also be defined as the reciprocal of time during which the number of radioactive atoms of a radioactive substance falls to 37% of its original value.

#### Half Life

The time is taken by a radioactive substance to disintegrate half of its atoms is called the half-life of that substance. It is denoted by T1/2 or simply T. Its value is different for a different substance.

Relation between half life and decay constant:

Let N0 be the initial number of atoms in a radioactive substance of decay constant λ . Then after time T, the number of atoms left behind N0/2. So,

\begin{align*} t = T\: \text {and} \: N = \frac {N_0}{2} \\ \text {Substituting these values in the equation,} \: N =N_0 e^{-\lambda t}, \\\text {we get} \\ \frac {N_0}{2} &= N_0 e^{-\lambda T} \\ \text {or,} \: \frac 12 &= e^{-\lambda T} \\ e^{\lambda T} &= 2 \\ \text {or,} \: \lambda T&= \log _e 2 = 0.693 \\ T &= \frac {0.693}{\lambda } \\ \end{align*}

This is the relation between the half-life and decay constant. Thus the half life of the radioactive substance is inversely proportional to its decay constant.

Average Life or Mean Life

The average life or mean life of a radioactive substance is equal to the sum of total life of the atoms divided by the total number of atoms of element.

\begin{align*} \text {Mean life} &= \frac {\text {sum of life of all the atoms}}{\text {total number of atoms}} \\ \end{align*} It can be shown that the mean life of a radioactive substance is equal to the reciprocal of the decay constant. \begin{align*} T_{mean} &= \frac {1}{\lambda } \\ \text {But} \lambda = \frac {0.693}{T} \text {where T is the half life of the substance.} \\ \therefore T_{mean} &= \frac {T}{0.693} = 1.443 T \\ \end{align*}

Thus the mean life of a radioactive substance is longer than its half life.

The rate of decay of a radioactive substance is called the activity (R) of the substance.

\begin{align*} R &= \frac {dN}{dt} = -\lambda N\\ \text {or,} \: |R| &= \lambda N = \frac {0.693} {T} N \\ \text {So,} R \propto N. \text {If} \: R_0 \text {is the activity of a substance at time}\: t = 0, \text {then} \\ R_0 &= \lambda N_0 \\ \text {or,} \frac {R}{R_0} &= \frac {N}{N_0} = \frac {N_0 e^{-\lambda t} }{N_0} = e^{-\lambda t} \\ \text {or,} \: R &= R_0 e^{-\lambda t} \\ \end{align*}

Number of atoms left behind after n half lives

Let N0 be the total number of atoms of a radioactivity substance present at time t = 0. Then, the number of atoms present after one half life, $T = N = \frac {N_0}{2}$

After two half life, the number present $= \frac 12 \frac {N_0}{2} = \left ( \frac 12 \right ) ^2 N_0$

After n-half life time, the number present $= \left ( \frac 12 \right ) ^n N_0$

In general after time t = nT, the number of atoms left is given by $\left ( \frac 12 \right ) ^{1/T} N_0$

Also, after time t = nT, the activity of a radioactive substance is $\left ( \frac 12 \right ) ^{1/T} R_0$

Further, if R be the activity after n half life times, and R0 be the initial activity, then

\begin{align*} R &= \left ( \frac 12 \right ) ^{1/T} R_0 \\ \text {or,} \: R_0 &= 2^n R \\ \end{align*}

The activity of a radioactive substance is measured in terms of disintegration per second. Following are units of radioactivity.

1. Curie (Ci)
It is defined as the activity of a radioactive substance which gives 3.7×1010 integration per second. It is equal to the activity of 1 g of pure radium.
$$1 \text {Ci} = 3.7 \times 10^{10} \text {disintegration}/\text {second}$$
2. Rutherford (rd)
It is defined as the activity of radioactive substance which gives rise to 106 disintegration per second.
$$1 \text{rd} = 10^6 \text {disintegration}/\text {second}$$
3. Becquerel (Bq)
\begin{align*} 1\text {Bq} = 1 \text {disintegration}/\text {second} \\ \text {Ci} = 3.7 \times 10^{10} \text {disintegration}/\text {second} = 3.7 \times 10^{10} \text {Bq} = 3.7 \times 10^4 \text {rd} \\ \end{align*}

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.

##### Things to remember

The spontaneous breaking of a nucleus is known as radioactive disintegration.

The rate of disintegration of a radioactive substance is directly proportional to the number of atoms present at that instant. This is called decay law.

The time is taken by a radioactive substance to disintegrate half of its atoms is called the half-life of that substance.

The average life or mean life of a radioactive substance is equal to the sum of total life of the atoms divided by the total number of atoms of element.

The activity of a radioactive substance is measured in terms of disintegration per second.

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