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
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)