Question

How do you find the half -life of the uranium – 235?

Answer 1

Hint: Half-life of any radioactive substance can be defined as the time needed by the radioactive substance or we can say time needed by one half of the atoms to disintegrate or convert into a different substance.

Formula used:
$N\left( t \right)={{N}_{o}}{{e}^{-\lambda t}}$

Complete step by step solution:
As we know to find the half-life of radioactive substance (in the case ${{U}^{_{235}}}$ ) you need to begin with the initial quantity of the substance, N (o), then after a time interval, t, you have to measure the remaining quantity of the substance N (t)

Consider Formula to find the radioactive decay.
$N\left( t \right)={{N}_{o}}{{e}^{-\lambda t}}....\left( 1 \right)$
Where, $N\left( t \right)$=is the quantity that left after decay.
$N\left( o \right)$ = the initial quantity of the substance.
$\lambda $ = decay constant.
t = time.

Now let’s discover half – life formula, now from definition when half substance decays and the time taken for that is ${{t}_{\dfrac{1}{2}}}$ then,
t = $\Rightarrow t={{t}_{\dfrac{1}{2}}}$
$\Rightarrow N\left( {{t}_{\dfrac{1}{2}}} \right)=\dfrac{N\left( o \right)}{2}$

Substitute both the values in the equation (1)
$\begin{align}
  & \Rightarrow \dfrac{N\left( o \right)}{2}=N\left( o \right){{e}^{\lambda {{t}_{\dfrac{1}{2}}}}} \\
 & \therefore \dfrac{1}{2}={{e}^{\lambda {{t}_{\dfrac{1}{2}}}}} \\
\end{align}$

Taking log to remove exponential,
$\begin{align}
  & \Rightarrow {{l}_{n}}=\left( \dfrac{1}{2} \right)=\lambda {{t}_{\dfrac{1}{2}}} \\
 & \Rightarrow -{{l}_{n}}\left( 2 \right)=\lambda {{t}_{\dfrac{1}{2}}} \\
 & \therefore {{t}_{\dfrac{1}{2}}}=\dfrac{-{{l}_{n}}\left( 2 \right)}{\lambda } \\
\end{align}$

From the above formula we can say that to find half- life we need $\lambda $ or half-life to find which can be found by quantification that is decay in the time interval.

From the experiment half – life of ${{U}^{_{235}}}$is $7.04\times {{10}^{8}}$ years.

Note:
To find half – life of ${{U}^{_{235}}}$ we can’t divide the value of 235 by $\dfrac{1}{2}$ because 235 is atomic mass number to find half – life we need time interval in which ${{U}^{_{235}}}$ decaying it will give us value of the remaining ${{U}^{_{235}}}$ at the time interval t with the value we can find but the life of ${{U}^{_{235}}}$ using the equation (1).