If a substance decays at a rate of 25% every 10 years, how long will it take 512 grams of the substance to decay to 121.5 grams

Question

asked Nov 3, 2021 182k views

1 Answer

Matt Adams · Answer 1 · 2021-11-09T06:58:18+0000

Answer:

It will take 50 years to decay from 512 grams to 121.5 grams.

Explanation:

The decay formula :

$N=N_0e^(-\lambda t)$

where

N= amount of substance after t time

N₀= initial of substance

t= time.

A substance decays at a rate 25% every 10 years.

So, remaining amount of the substance is = (100%-25%)= 75%

$(N)/(N_0)=(75\%)/(100\%)=(75)/(100)=\frac34$ , t= 10

$N=N_0e^(-\lambda t)$

$\Rightarrow \frac {N}{N_0}=e^(-\lambda t)$

$\Rightarrow \frac34 =e^(-\lambda .10)$

Taking ln both sides

$\Rightarrow ln|\frac34| =ln|e^(-\lambda .10)|$

$\Rightarrow ln|\frac34|=-10\lambda$

$\Rightarrow \lambda=( ln|\frac34|)/(-10)$

Now , N₀= 512 grams, N= 121.5 grams, t=?

$N=N_0e^(-\lambda t)$

$\therefore 121.5=512e^\frac34$

$\Rightarrow 121.5=512e^(ln$

$\Rightarrow (121.5)/(512)=e^(ln$

Taking ln both sides

$\Rightarrow ln|(121.5)/(512)|=ln|e^\frac34|$

$\Rightarrow ln|(121.5)/(512)|=\frac34$

$\Rightarrow t=(ln|(121.5)/(512)|)/((ln|\frac34|)/(10))$

$\Rightarrow t=\frac10.ln{ln}$

⇒t=50 years

It will take 50 years to decay from 512 grams to 121.5 grams.