Question

(A) Half-life of a first order reaction is independent of the initial concentration of the reactant (R) ${{t}_{{1}/{2}\;}}\left( first\text{ order} \right)=\dfrac{\tau }{1.44}$ , where $\tau $ is the average life.
[A] Both (R) and (A) are true and reason is the correct explanation of the assertion.
[B] Both (R) and (A) are true and reason is not the correct explanation of the assertion.
[C] Assertion (A) is true but reason (R) is false.
[D] Assertion (A) and reason (R) both are false.
[E] Assertion (A) is false but reason (R) is true.

Answer 1

Hint: Half-life of the element is the time required for the substance to reduce to its half. We can explain this on the basis of radioactive disintegration law. The average life period is the reciprocal of the decay constant.

Complete Step by Step Solution: We know that the spontaneous disintegration of certain nuclei to form new elements gives us an active radiation which affects photographic plates and affects electric and magnetic fields. This is known as radioactivity.
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 3000 years 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.
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-
\[n={{n}_{\circ }}{{e}^{-\lambda t}}\]
From here, we can find the half-life of an element which comes out to be-
\[{{t}_{{1}/{2}\;}}=\dfrac{0.693}{\lambda }\]
Where, lambda $\left( \lambda \right)$ is the decay constant. We can see from the above equation that there is no term of number of atoms present. It means that half-life of the element does not depend on the concentration of the reactant initially.
Now, let us discuss the average life period.
We define the average life period as the reciprocal of the decay constant of a radioactive element. We denote average life by the symbol, $\tau $, tau.
So, we can write that- $\tau =\dfrac{1}{\lambda }$
The half-life is related to the average life period as-
${{t}_{{1}/{2}\;}}=\dfrac{\tau }{1.44}$
We can understand from the above discussion that both the statements given to us are correct but the reason is not necessarily correct for the explanation of the assertion.

Therefore, the correct answer is option [B] Both (R) and (A) are true but the reason is not the correct explanation of the assertion.

Note: Higher the half-life of a radioactive element is, higher is its stability. However, chemical changes depend on factors like temperature, pressure, concentration and we can speed up or slow down the chemical changes by this factor but the half-life remains unchanged. The half-life of a radioactive element is independent of these factors.