Question

The half-life of $^{99}{\text{Tc}}$ is 6 hr, hence average life is:
A.$4.17{\text{ h}}$
B.${\text{3}}{\text{.0 h}}$
C.${\text{8}}{\text{.66 h}}$
D.${\text{8}}{\text{.00 h}}$

Answer 1

To calculate the average life we need to know the relation between half life and the average life for a reaction. The radioactive decay follows the first order reaction.

Formula used:
${{\text{t}}_{{\text{av}}}} = \dfrac{1}{{\text{k}}}$ here ${{\text{t}}_{{\text{av}}}}$ is the average life and ${\text{k}}$ is the rate constant.
For a first order reaction: ${\text{k}} = \dfrac{{0.693}}{{{{\text{t}}_{1/2}}}}$ here ${{\text{t}}_{1/2}}$ is the half life of a first order reaction.

Complete step by step answer:
The reaction follows a first order reaction because it is a radioactive substance.
The half life of a radioactive substance is given to us, so we will first calculate the rate constant for the reaction. Rate constant will be:
${\text{k}} = \dfrac{{0.693}}{{6{\text{ h}}}} = 0.1155{\text{ }}{{\text{h}}^{ - 1}}$
In the formula for average life we will add the value of rate constant and hence we will get:
${{\text{t}}_{{\text{av}}}} = \dfrac{1}{{0.1155{\text{ }}{{\text{h}}^{ - 1}}}} = 8.66{\text{ h}}$

Hence the correct option is C.

Additional information:
A first order reaction is that reaction in which the rate of reaction is directly proportional to the concentration of reaction. As we have also seen in the formula that the half life for a first order reaction does not depend on the concentration of reaction. The above used formula for half life is valid only as a first order reaction. If we change the order of reaction, then the formula of the half life will change. Average life is the time when 63 percent of the reactant has been reacted. Half life is the time when half of the reactant has been reacted.

Note:
The k is called the rate constant. The rate constant only depends on the nature of reaction and temperature. This means if temperature remains constant then rate constant will be constant for a particular reaction.