What is the half-life of (Na^24) if a research assistant made 160 mg of radioactive sodium (Na^24) and found that there was only 20 mg left 45 h later?
We need to find an equation of the form:
We are given:
We need to solve for
Divide by 160:
Taking natural logarithms of both sides:
Dividing by 45:
Since by definition the half life is the time period when we have half of the starting amount:
So we need to solve for t in:
Taking natural logarithms:
The half life is 15 hours.
- Quick Way
As the amount of a decaying substance halves over each half-life (hence the name), halving the amount in steps requires 3 steps to get from 160 to 20:
#160 to 80 to 40 to 20#
So the half-life is 15 years.
- More formal way
Plugging the values that are given into