# How long will it take for 3/4 of the sample of 131 iodine that has half-life of 8.1 days?

Feb 14, 2015

Since you didn't specify whether 3/4 of the sample remains or undergoes decay, I'll show you both cases.

Here's the equation for exponential decay used in nuclear half-life calculations

$A \left(t\right) = {A}_{0} \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\text{1/2}\right)}$, where

$A \left(t\right)$ - the amount left after t years;
${A}_{0}$ - the initial quantity of the substance that will undergo decay;
${t}_{\text{1/2}}$ - the half-life of the decaying quantity.

First case - 3/4 of the sample undergoes radioactive decay.

If 3/4 of the sample undergoes radioactive decay, you wil be left with 1/4 of the original sample. This means that $A \left(t\right)$ will be equal to ${A}_{0} \left(t\right) \cdot \frac{1}{4}$. Plug this into the above equation and you'll get

${A}_{0} \left(t\right) \cdot \frac{1}{4} = {A}_{0} \left(t\right) \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\text{1/2}\right)}$, or $\frac{1}{4} = {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\text{1/2}\right)}$

This implies that $\frac{t}{t} _ \left(\text{1/2}\right) = 2$, since $\frac{1}{4} = {\left(\frac{1}{2}\right)}^{2}$.

Therefore, t = 2 * t_("1/2") = 2 * 8.1 = "16.2 days"

It will take 16.2 days for 3/4 of your sample to undergo radioactive decay.

Second case - 3/4 of the sample remains, i.e. does not undergo radioactive decay.

The same principle applies in this case as well, only this time 1/4 of the sample will decay and 3/4 will remain. This means that $A \left(t\right) = {A}_{0} \left(t\right) \cdot \frac{3}{4}$. So,

${A}_{0} \left(t\right) \cdot \frac{3}{4} = {A}_{0} \left(t\right) \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\text{1/2}\right)}$, or $\frac{3}{4} = {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\text{1/2}\right)}$

As a result,

t/t_("1/2") = log_(("1/2"))(3/4) = 0.415

$t = 0.415 \cdot {t}_{\text{1/2}}$

$t = 0.415 \cdot 8.1 = \text{3.36 days}$

It will take 3.36 days for 1/4 of the sample to undergo radioactive decay.