Question

For a first order reaction, the half-life is 5 minutes. If the initial concentration is 128 mol/lit, what will be the concentration of the reactant after 30 minutes?
[A] 2 mole/lit
[B] 16 mole/lit
[C] 4 mole/lit
[D] 32 mole/lit

Answer 1

Hint: Half-life of the element is the time required for the substance to reduce to its half. To solve this, use the relation between the half-life of an element and the initial amount of the element. The relation is \[{{N}_{\circ }}{{\left( \dfrac{1}{2} \right)}^{n}}=N\].

Complete answer:
We know that the half-life period of a radioactive element is the time required for the substance to reduce to half of its initial value.
The general equation that we can use here to find out the amount of substance left after 30 minutes is-
     \[{{N}_{\circ }}{{\left( \dfrac{1}{2} \right)}^{n}}=N\]
Where, \[{{N}_{\circ }}\] is the initial amount of the element and N is the amount of the element left after n half-life time. ‘n’ is the number of half-life.
Here, the half-life of the element is given to us which is 5 minutes. After 30 minutes, its half-life will become 30 times the actual half-life.
 To understand this, let us take an example. Let us say we have 1g of an element with a half-life of 2 years. So, after two years the element will be half of its actual amount and after 4 years, it will be one-fourth.
Similarly, here the half-life is 5 minutes, so after 30 minutes, it will become 30 times of $\dfrac{1}{5}$ .
Therefore, n = $30\times \dfrac{1}{5}$ = 6.
We can write the equation as \[\dfrac{{{N}_{\circ }}}{N}={{\left( \dfrac{1}{2} \right)}^{6}}=\dfrac{1}{64}\]
As we can see that after 30 minutes, $\dfrac{1}{64}$ amount of the radioactive element would be left.
Now, the initial concentration is given to us as 128mol/lit.
So, the concentration after 30 minutes will be 2 mol/lit.

Therefore, the correct answer is option [A] 2 mole/lit.

Note:
According to the radioactive disintegration law, the rate of disintegration at any time is proportional to the number of atoms present at that time. This law gives us a relation of the number of atoms present at a particular time. The relation is-
     \[n={{n}_{\circ }}{{e}^{-\lambda t}}\]