24.110 CP - Chemistry: Molecular Nature Of Matter And Change

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.

Step-by-Step Solution

Answer

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

Step 1: 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. 

Step 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:

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

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