Question

$50\% $ of the amount of a radioactive substance decomposes in $5$ years. The time required for the decomposition of $99.9\% $ of the substance will be
A. 10 years
B. between 10 and 40 years
C. less than 10 years
D. between 49 and 50 years

Answer 1

Hint: The formula for decomposed amount should be known and proper information about original amount and amount left after some time can be calculated by the given question.

Complete step by step answer:
$kt$ $ = 2.303\log N \circ /N$
Here, half life \[{t_{1/2}}\] $ = 5$years
Decomposed amount is equal to $99.9\% $ as given.
Let the actual amount, $N \circ $be $100$
Amount left after time $t$ $ = 100 - 99.9$
$ \Rightarrow $$ = 0.1$
- After solving, we get amount left after $t$ time
$ \Rightarrow $$N$ $ = 0.1$
- According to the formula,
$\because $$kt$$ = 2.303\log N \circ /N$
By putting the values,
$ \Rightarrow $ \[0.693/{t_{_{1/2}}} \times t\]= $2.303\log 100/0.1$
$ \Rightarrow $\[0.693/5 \times t\] \[ = {\text{ }}2.303log{10^3}\]
After evaluating,
$ \Rightarrow $ \[0.693 \times t\]\[ = 2.303 \times 3 \times 5\]
By solving,
$ \Rightarrow $\[t = 49.85\] years
The correct option is option “D” .

Additional Information .
- Radioactive substances are the atoms which decay with their own naturally. They are capable of giving off alpha particles, beta particles and gamma radiation.
- Long-lived radioactive elements which are uranium, thorium and potassium and any of their products which decay, just like radium and radon are examples of NORM. These elements have always been there in the Earth's crust and atmosphere, and are fixed in some places, such as uranium bodies which may be mined.
- Decomposition process can be stated as the dead organic substances which are broken down into smaller and simpler organic or inorganic matter just like carbon dioxide, water, simple sugars and mineral salts.
- A decomposition reaction takes place when 1 reactant breaks down into 2 or more products. This can be shown by the general equation: AB → A + B. Instances of decomposition reactions involve the breakdown of hydrogen peroxide to water and oxygen, and the breakdown of water to hydrogen and oxygen.

Note: Check the formula carefully, find the amount left after time t firstly then based on this, figure out other steps turn by turn.