Exponential decay – Half life formula

Decay calculator in Matlab 

The formulas below are used in calculations involving the exponential decay of, for example, radioactive materials   

   

Let’s code a Matlab function to calculate the final amount of a substance, given the elapsed time, half-life and initial amount. This is just a code of the ‘final amount’ formula above
 

function end_amt = exp_decay(beg_amt, hl, time)
% beg_amt = beginning amount
% hl = half life
% time = elapsed time
n = time / hl;
end_amt = beg_amt/2^n;
 

This is a possible code to calculate the half-life of a substance, if we’re given the initial and final amounts, and the elapsed time between those masses:
 

function hl = half_life(beg_amt, fin_amt, time)
% beg_amt = beginning amount
% fin_amt = final amount
% time = elapsed time
hl = log10(2) * time / log10(beg_amt/fin_amt);
 

And this code calculates the time needed for a substance undergoing exponential decay to go from an initial mass to a final mass. We need to know the half-life, naturally.

function t = elapsed_time(beg_amt, fin_amt, hl)
% beg_amt = beginning amount
% fin_amt = final amount
% hl = half life
t = hl * log10(beg_amt / fin_amt) / log10(2); 

Now, let’s test our decay calculators to discover ending amounts: if we start with 100 grams of radium, and we know its half-life is 1620 years, how much would be left after 3240 years (two half-lives)?
 

We can type this from the command window or from any other script. It’s important to follow the order of the parameters:
 

end_amt = exp_decay(100, 1620, 3240) 

 
Matlab shows that the ending amount would be 25, as expected.

 
If 10 grams are present now, how much will be present in 50 years? 

end_amt = exp_decay(10, 1690, 50) 

The result on screen is end_amt = 9.7970 

 
Example:

We need to understand what we’re being asked. We can make a drawing to understand better...
 

We could create an m-file named ‘test_exp_decay’, for example, and we’d type this:

clear, clc, format compact 

% Initial amount of salt is 25 kg
% after 10 hours, 15 kg of salt is left
beg_amt = 25;
fin_amt = 15;
time = 10;
hl = half_life(beg_amt, fin_amt, time) 

% How much salt is left after 1 day?
time = 24;
end_amt = exp_decay(beg_amt, hl, time)
 
% How long does it take until 1/2 kg of salt is left?
fin_amt = 0.5;
t = elapsed_time(beg_amt, fin_amt, hl) 

The code represents the given problem and its solution. First, we solve for the half-life; second, we solve for the ending amount; finally, we solve for the elapsed time considered in the formulas.

Matlab answer is:

hl = 13.5692
end_amt = 7.3367
t =  76.5824