Radio active decay

**av8or91** · December 15th 2009, 06:34 AM

Radioactive substance has a half life of 18 years. Decay rate remains constant how long will it before 95 percent of the sample has decayed?

Thanks,!

**Soroban** · December 15th 2009, 07:14 AM

Hello, av8or91!

A radioactive substance has a half life of 18 years. Decay rate remains constant.
How long will it be before 95 percent of the sample has decayed?

The half-life function is: . $A \;=\;A_o\,e^{-kt}$
. . where $A_o$ is the intial amount and . $k \:=\:\frac{\ln(2)}{\text{half-life}}$

Our function is: . $A \;=\;A_o\,e^{-\frac{\ln 2}{18}t}$

If 95% of the sample has decayed, then: . $A \,=\,0.05A_o$

We have: . $A_oe^{-\frac{\ln2}{18}t} \:=\:0.05A_o \quad\Rightarrow\quad e^{-\frac{\ln2}{18}t} \:=\:0.05$

Take logs: . $\ln\left(e^{-\frac{\ln2}{18}t}\right) \:=\:\ln(0.05) \quad\Rightarrow\quad -\tfrac{\ln2}{18}t\ln(e) \:=\:\ln(0.05)$

. . $t \:=\:\frac{18\ln(0.06)}{-\ln2} \;=\; 77.79470571 \;\approx\;77.8\text{ years}$

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