Half life formula - what is half life equation? examples

Answer

Half-life of a radioactive nuclide can be calculated using measurements on change in the quantity (mass or number of atoms) of the nuclide and the total time it takes for the decay to occur. We can derive the half-life equation from the conventional equation that connects initial mass, mass remaining and the total time it takes the nuclide to decay

                  $N_t$ = ${\left(\frac{1}{2}\right)}^{\frac{t}{t_{1/2}}}$$N_o$,                                                                                   1

Where $N_t$ is the mass or quantity remaining after time t, $N_o$ is the initial mass/ quantity. By rearanging the above equation, we obtain the half-life formula/equation interms of the above parameters.

Half life formula is:

                Half-life ($t_{1/2}$) = $\frac{t \log 2}{\log \left({\displaystyle \frac{N_o}{N_t}}\right)}$                                                                      2

Where

$N_o$= Initial mass of the substance 

$N_t$ = Quantity os the substance remaining

$t$ = Time elapsed 

$t_{\frac{1}{2}}$= Half life of the substance

Example 1

Calculate the half-life of Gold-198 given that 325.7 mg of this radioactive isotope decayed to 10.2 mg in 13.5 days.

                      Solution

           $t_{1/2}$ = $\frac{13.5\log 2}{\log {\displaystyle \frac{325.7}{10.2}}}$

                 = 2.70 days

Half-life can also be expressed interms of the number of half-lives, n, and total time t as in the equation below

            number of half-lives (n) = $\frac{t}{t_{{\displaystyle \frac{1}{2}}}}$, 

                           Half-life($t_{1/2}$) = $\frac{t}{n}$

Example 2

A certain radioactive isotope took 720 hours to decay, given that the half-life of this isotope is 10 days, Calculate the number of half-lives.

                 720 hours= $\frac{720}{24}$ days

                               = 30 days

 number of half-lives (n) = $\frac{30}{10}$

                               = 3

Example 3

Given that Ra-226 took 4800 years to decay in 3 half-lives, calculate the half-life of Ra-226.

                      $t_{1/2}$= $\frac{t}{n}$, but we have n= 3, t=4800 years

                            = $\frac{4800}{3}$

                            = 1600 years

We can use equation 2 to determine half-life of a radioisotope if we have percentage remaining or percentage decayed.

Example 4

It takes 689.28 of a radioactive isotope to decay to 92 % of its original mass, calculate the half-life of the isotope.

Solution

The percentage/ fraction remaining is $\frac{92}{100}$, therefore, $N_t$ =  92, $N_o$= 100, substituting these values in equation 2, we have

            $t_{1/2}$ = $\frac{689.28\log 2}{\log {\displaystyle \frac{92}{100}}}$

                   = 5730 years

• If we have the percentage decayed, then we less that value from 100 so that we can obtain the percentage remaining and calculate half-life as illustrated in example 4 above.
• Equation 1 can be expressed in terms of n, as;           $N_t$ = ${\frac{1}{2}}^n$$N_o$

Yes · 0 No · 0 Add Comment · 0

Add ·

Report