The half-life of a radioactive substance is 3200 years. Find the quantity of the substance left at time if ?
The general exponential decay function is defined as:
is the initial quantity, is the “proportionality constant”, is the initial time, and would be any time duration.
For radioactive decay problems, is treated as the “decay constant”
Since , or the “half-life,” is the amount of time at which a radioactive material’s quantity is reduced to half, we can turn into,
where and .
We then solve for by canceling like terms and taking the natural logarithm of the equation:
((Recall that and ))
With these in mind, for Exercise 4.1.1, only algebra would be needed.
Given that and ,
And since , solving for would yield
Therefore, the quantity over time of a 20 gram substance with a 3200 years half-life can be found using,
We don’t simply use or as the solution because the resulting equation will NOT give us different values of at , only at . Remember that is just a value of .
Solution by Brian and Jian HuiPrint this page