Initially there are $8$ grams of a radioactive material in a container.The half-life of the material is $2$ days.
How much of the radioactive substance will remain after $5$ days ?
By exponential decay I know that after $2$ days I have $4$ grams of the radiactive material and that $2$ days later I will have $\cfrac{2}{2}=1$ gram of the substance.
Now I don't know how to find how much radioactive I have after $5$ days ,i.e. in the "beginning",roughly speaking, of the $6^{th}$ day.
My book provides as a solution the following equation $\cfrac{2-x}{2}=\cfrac{x-1}{x}$ ($x$ is the amount of material after $5$ days) from which it follows that $x=\sqrt{2}$ but I don't know how this equation have been set up,what is meaning behind this.
If someone can help me understand how that equation is derived I am really grateful.
Best Answer
The remaining quantities at the beginning of the days are known to be
$$0\to8g\\2\to4g\\4\to2g\\5\to xg\\6\to1g.$$
Then you express that the relative quantity that disappears in a single day (the daily decay) is a constant. With the disintegrations from day $4$ to day $5$ and day $5$ to day $6$,
$$\frac{2-x}2=\text{Cst}=\frac{x-1}{x}.$$
It is simpler to express the fraction that remains, giving $$\frac x2=\text{Cst}=\frac 1x$$ and you immediately get $x=\sqrt2g$.