Question

Exactly 0.1 of the radioactive nuclei in a sample decay per hour. Thus, after n hours, the fraction of nuclei remaining is(0.900)ⁿ. Find the value of n equal to one half-life.

Short answer

The value of ' n ' that is equal to a half-life is 6.58 h .

Step-by-step solution

Half-life

Half-life of a substance is the time taken by the substance to decay to half of its initial concentration. The half-life of a substance is its characteristic property and is independent of the concentration of the sample at any point of time. Half-life is denoted by t_1/2.

Find the value of n equal to one half-life

Consider the given information:

Fraction radioactive nuclei that gets decayed in 1hr =0.1

Remaining fraction of radioactive nuclei after n hrs=(0.900)ⁿ

The total fraction of sample is taken as 1. So, the fraction of sample at one half-life is 0.500 .

As a result,

(0.900)ⁿ =0.500 .

The value of 'n' that corresponds to a half-life can be calculated as follows:

$$\begin{gathered} (0.900{)}^n=0.500 \\ n\ln (0.900)=\ln (0.500) \\ n=\frac{\ln (0.500)}{\ln (0.900)} \\ =6.58hrs \end{gathered}$$

Therefore, the required value of ' n ' that is equal to a half-life is6.58 h .