Question

Determine the amount of time for polonium-210 to decay to one fourth its original quantity. The half-life of polonium-210 is 138 days.

Answer 1

Answer: 276 days

Explanation:

This problem can be solved using the Radioactive Half Life Formula:  

$A=A_{o}.2^{\frac{-t}{h}}$ (1)  

Where:  

$A=\frac{1}{4}A_{o}$ is the final amount of the material

$A_{o}$ is the initial amount of the material  

$t$ is the time elapsed  

$h=138 days$ is the half life of polonium-210

Knowing this, let's substitute the values and find $t$ from (1):

$\frac{1}{4}A_{o}=A_{o}2^{\frac{-t}{138 days}}$ (2)  

$\frac{A_{o}}{4A_{o}}=2^{\frac{-t}{138 days}}$ (3)  

$\frac{1}{4}=2^{\frac{-t}{138 days}}$ (4)  

Applying natural logarithm in both sides:

$ln(\frac{1}{4})=ln(2^{\frac{-t}{138 days}})$ (5)  

$-1.386=-\frac{t}{138days}ln(2)$ (6)  

Clearing $t$:

$t=276days$ (7)