How can you model half life decay?

Question

How can you model half life decay?

1 Answer

Impact of this question

3572 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-07-03T02:20:57

The equation would be:

$[A] = \frac{1}{2_{1/2}^{t / t_{1/2}}} {[A]}_{0}$ $[A] = 1/(2^(t"/"t_"1/2"))[A]_0$

Read on to know what it means.

Just focus on the main principle:

The upcoming concentration of reactant $A$ $A$ after half-life time $t_{1/2}$ $t_"1/2"$ becomes half of the current concentration.

So, if we define the current concentration as ${[A]}_{n}$ $[A]_n$ and the upcoming concentration as ${[A]}_{n + 1}$ $[A]_(n+1)$ , then...

${[A]}_{n + 1} = \frac{1}{2} {[A]}_{n}$ $[A]_(n+1) = 1/2[A]_n$ $(1)$ $" "\mathbf((1))$

We call the (1) the recursive half-life decay equation for one half-life occurrence, i.e. when $t_{1/2}$ $t_"1/2"$ has passed by only once. This isn't very useful though, because half-lives can range from very slow (thousands of years) to very fast (milliseconds!).

Let's go through another half-life, until we've gone through $n$ $\mathbf(n)$ half-lives. For this, we rewrite ${[A]}_{n}$ $[A]_n$ as ${[A]}_{0}$ $[A]_0$ (the initial concentration), and ${[A]}_{n + 1}$ $[A]_(n+1)$ as $[A]$ $[A]$ (the upcoming concentration).

Notice how ${[A]}_{0}$ $[A]_0$ will always be the same, but $[A]$ $[A]$ will keep changing over time.

$[A] = (\frac{1}{2}) (\frac{1}{2}) \dots (\frac{1}{2}) {[A]}_{0}$ $[A] = (1/2)(1/2)cdots(1/2)[A]_0$

$= {(\frac{1}{2})}^{n} {[A]}_{0}$ $= (1/2)^n[A]_0$

$\Rightarrow [A] = \frac{1}{2^{n}} {[A]}_{0}$ $=> [A] = 1/(2^n)[A]_0$ $(2)$ $" "\mathbf((2))$

Now we have (2), the equation for any number of half-life decays... once we know how many half-lives passed by.

However, (2) can be made more convenient since we know that each half-life takes $t_{1/2}$ $t_"1/2"$ time to occur. When $n$ $n$ half-lives occur, each one taking $t_{1/2}$ $t_"1/2"$ to occur, it must occur over a set amount of time $t$ $t$ . So:

$n t_{1/2} = t$ $nt_"1/2" = t$ $(3)$ $" "\mathbf((3))$

That means $n = \frac{t}{t_{1/2}}$ $n = t/t_"1/2"$ , which is saying that we can divide the total time passed during the process by the time it takes to lose half of $A$ $A$ again to get the number of half-lives that passed by.

Therefore:

$[A] = \frac{1}{2_{1/2}^{t / t_{1/2}}} {[A]}_{0}$ $color(blue)([A] = 1/(2^(t"/"t_"1/2"))[A]_0)$ $(4)$ $" "\mathbf((4))$

So, we can use (4) to determine half-lives of any typical radioactive element for which we know $t$ $t$ , the time passed during the half-life decay(s) AND:

${[A]}_{0}$ $[A]_0$ , the initial concentration, and $[A]$ $[A]$ , the upcoming concentration, OR
$\frac{[A]}{{[A]}_{0}}$ $([A])/[A]_0$ , the fraction of the element left after time $t$ $t$ passes.

Science

Math

Humanities

... and beyond

How can you model half life decay?

1 Answer

Related questions

Impact of this question