Class 12 > Unit # 27: Nuclear Physics > Law of Radioactive Decay

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Law of Radioactive Decay & Half Life

Video Lecture: Law of Radioactive Decay & Half Life

Watch the complete lecture explaining the concept of radioactive decay, exponential equations, sample activity, and half-life derivations:

Watch directly on YouTube (https://youtu.be/lvInA6NO3NU)

Law of Radioactive Decay

Statement: The rate of decay in a radioactive process is directly proportional to the number of parent nuclides present in the unstable nuclides of the given species at that instant.

Figure: Negative exponential decay curve showing the decrease of parent nuclides over time.

Mathematical Form

If $\Delta N$ be the number of nuclides disintegrated in time $\Delta t$ and $N$ be the number of nuclides at time $t$, then the rate of decay is expressed as:

$-\frac{\Delta N}{\Delta t} \propto N \quad \text{--- (i)}$

Removing the proportionality sign introduces a constant $\lambda$ (decay constant):

$-\frac{\Delta N}{\Delta t} = \lambda N \quad \text{--- (ii)}$

Expressed in differential form:

$\frac{dN}{dt} = -\lambda N \quad \text{--- (iii)}$

Where $\lambda$ is the decay constant, and the negative sign shows that the number of atoms decreases with respect to time.

The Exponential Decay Equation

Integrating equation (iii) on both sides:

$\int_{N_0}^{N} \frac{dN}{N} = -\lambda \int_{0}^{t} dt \quad \text{--- (iv)}$

Applying initial conditions $N = N_0$ at $t = 0$:

$\ln N - \ln N_0 = -\lambda t$

Using logarithm properties ($\ln \left(\frac{N}{N_0}\right) = -\lambda t$):

$\frac{N}{N_0} = e^{-\lambda t}$

Yielding the final exponential decay equation:

$N = N_0 e^{-\lambda t}$

Activity of the Sample

Definition: The activity ($A$) of a sample is the number of disintegrations (decays) occurring per unit time.

Mathematically, activity is written as:

$A = -\frac{dN}{dt} = \lambda N$

In terms of initial activity ($A_0 = \lambda N_0$):

$A = A_0 e^{-\lambda t}$

Half Life of an Element

Definition: It is the time in which half of the radioactive elements decay from the parent element to the daughter element. It is denoted by $T_{1/2}$.

Example

Suppose we have 10,000 radioactive atoms. If in a specific time interval, exactly half of them decay to 5,000 atoms, that duration is defined as the half-life of that radioactive element.

Mathematical Derivation

Substituting $N = \frac{N_0}{2}$ and $t = T_{1/2}$ into the exponential decay formula:

$\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}} \implies \frac{1}{2} = e^{-\lambda T_{1/2}}$

Taking reciprocal and natural logarithms on both sides:

$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$

Where $\lambda$ is the Decay Constant.

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