Question

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

Answer 1

Answer:

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

Step-by-step 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%

$\frac{N}{N_0}=\frac{75\%}{100\%}=\frac{75}{100}=\frac34$, t= 10

$N=N_0e^{-\lambda t}$

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

$\Rightarrow \frac34 =e^{-\lambda .10}$

Taking ln both sides

$\Rightarrow ln|\frac34| =ln|e^{-\lambda .10}|$

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

$\Rightarrow \lambda=\frac{ ln|\frac34|}{-10}$

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

$N=N_0e^{-\lambda t}$

$\therefore 121.5=512e^{-\frac{ln|\frac34|}{-10}.t}$

$\Rightarrow 121.5=512e^{\frac{ln|\frac34|}{10}.t}$

$\Rightarrow \frac{121.5}{512}=e^{\frac{ln|\frac34|}{10}.t}$

Taking ln both sides

$\Rightarrow ln|\frac{121.5}{512}|=ln|e^{\frac{ln|\frac34|}{10}.t}|$

$\Rightarrow ln|\frac{121.5}{512}|={\frac{ln|\frac34|}{10}.t}$

$\Rightarrow t=\frac{ln|\frac{121.5}{512}|}{\frac{ln|\frac34|}{10}}$

$\Rightarrow t=\frac{10.ln|\frac{121.5}{512}|}{{ln|\frac34|}}$

⇒t=50 years

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