Question

Define half-life of a radioactive substance.

Answer 1

Hint: Radioactive substances are elements with unstable nuclei which emits radiation to become stable. Express the number of elements after radioactive emission for time t. Describe the half-life of the radioactive element. Find the mathematical formula by considering that the number of radioactive elements will be reduced to half after the half-life of the radioactive element.

Complete step by step answer: Radioactive substance or radioactive elements are made of atoms which have unstable nuclei. To make these atoms stable these substances emit radiation which is called the radioactive emission.

All radioactive substances emit radiation. After emission the radioactive element transforms into another element. Half-life of a radioactive substance can be defined as the time taken by the substance to reduce to half due to radioactive decay.

Let the original no of atoms in a radioactive substance is ${{N}_{0}}$. After being decayed for a certain time t, the no of atoms remains is $N$. We can define a relation between these two quantities given as,

$N={{N}_{0}}{{e}^{-\lambda t}}$
Where, $\lambda $ is the decay constant of the element.

Now, half life is defined as the time at which the original number of atoms reduced to half because of radioactive emission. So, let the half-life is ${{t}_{\dfrac{1}{2}}}$, when the no of atoms becomes $N/2$ from $N$ .
So, we can write,
$\dfrac{N}{2}=N{{e}^{-\lambda {{t}_{\dfrac{1}{2}}}}}$
Taking $\log $ on both side of the equation, we get,

$\begin{align}
  & \ln \left( \dfrac{N}{2} \right)=\ln \left( N{{e}^{-\lambda {{t}_{\dfrac{1}{2}}}}} \right) \\
 & \ln N-\ln 2=\ln N+\ln {{e}^{-\lambda {{t}_{\dfrac{1}{2}}}}} \\
 & -\ln 2=-\lambda {{t}_{\dfrac{1}{2}}} \\
 & \lambda =\dfrac{\ln 2}{{{t}_{\dfrac{1}{2}}}} \\
\end{align}$

Putting the value of $\ln 2\approx 0.693$ in the above equation, we get,
$\lambda =\dfrac{0.693}{{{t}_{\dfrac{1}{2}}}}$

So, we can write,
${{t}_{\dfrac{1}{2}}}=\dfrac{0.693}{\lambda }$
This is the formula for finding the half-life of a radioactive element.

Note: The decay constant will be different for different elements. So, the half-life of the elements will also be different depending on the value of the decay constant. If the decay constant is large for an element then the half-life of the element will be small and if the decay constant is small then the half-life of the element will be large.