Question # Lodine - 131, a radioactive substance that is effective in localing brain tumors, a half-life of only eight days. A hospital purchased 22 grams of the substance but had to wail five days before it could be used. How much of the substance was left after five days?

Decimals
ANSWERED Lodine - 131, a radioactive substance that is effective in localing brain tumors, a half-life of only eight days. A hospital purchased 22 grams of the substance but had to wail five days before it could be used. How much of the substance was left after five days? 2021-03-08
Data analysis
Given data,
Initial amount of substance $$(P_{0}) = 22 gms$$
Half-life $$(t_{1/2})=8\ days$$
Time period $$(t) = 5\ days$$
Amount of substance left after 't' = ?
Solution
The decay of an radioactive substance is exponential.
And the amount of substance (P) left fter 't' days is given by,
$$P=P_{0}e^{-\lambda\ t}$$
Where lambda is decay constant.
Given $$t_{1/2} = 8\ days$$
That is by 8 days, amount left will become half.
$$\Rightarrow\ (P_{0}/2)=P_{0}e^{-\lambda(8)}$$
$$\Rightarrow\ 8\lambda=\ln(2)$$
$$\Rightarrow\ \lambda=(1/8)\ln(2)$$
$$\Rightarrow\ \lambda = 0.0866434$$ (rounded to 7 decimals)
Now after 5 days, amount left,
$$\Rightarrow\ P=22(e^{0.0866434(5)})$$
$$= 22 (0.6484198)$$
$$\Rightarrow\ P = 14.2652356$$ NAK Hemce amount left after 5 days is 14.265 grams (rounded to 3 decimals)