Half-life and population doubling

Learning Objectives Define half-life and population doubling time. Calculate the remaining amount of a substance after a given number of half-lives. Determine the population size after a given number of doubling periods. Calculate the number of half-lives or doubling periods that have occurred over a total time. Apply proportional reasoning to solve real-world problems involving half-life and population doubling. Identify patterns of decay and growth in simple scenarios. Ever wonder how scientists figure out how old ancient artifacts are, or how quickly a group of animals can grow? 🕰️📈 In this lesson, you'll explore the fascinating concepts of half-life, which describes how things decay, and population doubling, which shows how things grow. These ideas help us underst...

TermDefinitionExample Half-lifeThe amount of time it takes for half of a substance to decay or disappear.If a substance has a half-life of 10 years, then after 10 years, half of the original amount will be left. Population Doubling TimeThe amount of time it takes for a population (like bacteria or animals) to double in size.If a bacterial population doubles every 20 minutes, then after 20 minutes, there will be twice as many bacteria as before. Initial Amount/PopulationThe starting quantity of a substance or the starting number of individuals in a population.You start with 100 grams of a radioactive material, so 100 grams is the initial amount. Remaining Amount/PopulationThe quantity of a substance or the number of individuals in a population left after a certain period of time.After one...

Calculating Remaining Amount (Half-life) $$A_f = A_0 \times \left(\frac{1}{2}\right)^n$$ To find the final amount ($A_f$) of a substance after decay, multiply the initial amount ($A_0$) by one-half for each half-life ($n$) that has passed. This is the same as dividing by 2, 'n' times. Calculating Final Population (Doubling) $$P_f = P_0 \times 2^n$$ To find the final population ($P_f$) after growth, multiply the initial population ($P_0$) by 2 for each doubling period ($n$) that has passed. Calculating Number of Periods $$n = \frac{T_{total}}{T_{period}}$$ To find the number of half-lives or doubling periods ($n$), divide the total time ($T_{total}$) by the time for one period ($T_{period}$, which is either the half-life or the doubling time).