Question

The half-life of radium is 1600 years. After how much time $\dfrac{1}{16}$th part of radium will remain disintegrated in a sample?

Answer 1

Hint: Radium is an element with atomic number 88. It is having the symbol Ra. It is a pure white alkaline earth metal but when exposed to oxygen changes to black color. Let’s see definition of some terms -
-Half-life- it is the time required for a quantity to reduce to half of its initial value. It is used in nuclear chemistry to describe radioactive decay. It is given by the equation- $t\dfrac{1}{2}=\dfrac{0.693}{\lambda }$
-Radioactive decay- It is a process that shows how long stable atoms survive or how an unstable atomic nucleus loses energy by radiation.

Complete step by step solution:
-To solve the question, we have to use the formula-
$\lambda =\dfrac{2.303}{t}\times \log \dfrac{{{N}_{\circ }}}{N}$

 Where, $\lambda $-Decay constant. It is a fraction of the total number of atoms that disintegrate in time.
- ${{N}_{\circ }}$is the initial quantity of substance

- N is the quantity still remaining and that was not decayed.


-The values given in question are-$t\dfrac{1}{2}$=1600years , $\dfrac{N}{{{N}_{\circ }}}$=1/16
We have to find t=?
So, putting all the given values in above equation, we get-
\[\lambda =\dfrac{2.303}{t}\times \log 16\]
-There is relation between $\lambda $and $t\dfrac{1}{2}$,
\[\lambda =\dfrac{0.693}{t\dfrac{1}{2}}\]
Now putting values, we get,
\[\begin{align}
  & \lambda =\dfrac{0.693}{1600} \\
 & \lambda =\text{0}\text{.000433125} \\
\end{align}\]
-Now to calculate t,
\[\begin{align}
  & t=\dfrac{2.303}{\lambda }\log 16 \\
 & t=\dfrac{2.303}{0.000433125}\log 16 \\
 & t=6402years \\
\end{align}\]
Hence, we can say that after 6402 years the $\dfrac{1}{16}$th part of radium will remain disintegrated .
- This question can be also solved by following way-
Other formula can be also used that is -$\dfrac{N}{{{N}_{\circ }}}={{\left( \dfrac{1}{2} \right)}^{n}}$$\dfrac{N}{{{N}_{\circ }}}={{\left( \dfrac{1}{2} \right)}^{n}}$
     \[\dfrac{1}{16}={{\left( \dfrac{1}{2} \right)}^{n}}\]
\[\dfrac{1}{16}={{\left( \dfrac{1}{2} \right)}^{n}}\]
Now, we have to put a value of n so that the can become.
So, n = 4 . As $\left( 2\times 2\times 2\times 2 \right)$,
Formula of n is, $n=\dfrac{t}{t\dfrac{1}{2}}$
\[\begin{align}
  & 4=\dfrac{t}{1600} \\
 & \therefore t=4\times 1600 \\
 & t=6400years \\
\end{align}\]

Hence, we can solve the given question by any one of these methods. Answers by solving any method comes almost the same.


Additional information:
-Radium Element- Radium is silvery, lustrous, soft, radioactive element. It readily oxidizes on exposure to air, turning from pure white to black. It is also to be noted that it is luminescent, and also corrodes in water to form radium hydroxide. Although is the heaviest member of the alkaline-earth group and is the most volatile.
-Types of radioactive decay- There are three of the most common types of decay are alpha decay, beta decay, and gamma decay, all of which involve the emission of one or more particles or photons.
-Effects of Radiation- Radiation can either kill cells or damage the DNA within them, which damages their ability to reproduce and as we know that it can eventually lead to disease like cancer. When radiation is present, very high energy particles pass through your body. These can collide with atoms in your body and can disrupt atomic structure.

Note:
-While solving the question one should remember that the units of time given in question should not be converted to other units of time. Means, for an example-if time given in question is in years, then it should not be converted to seconds or minutes and the answer must be solved in units of years only.