(NCM) Half-Life Calculations Lesson

Half-Life Calculations

Half-lives have a very wide range, from billions of years to fractions of a second. Listed in the table below are the half-lives of some common and important radioisotopes.

Half Life Rates Table
Nuclide	Half-life (t_1/2)
Carbon-14	5730 years
Cobalt-60	5.27 years
Francium-220	27.5 seconds
Hydrongen-3	12.26 years
Cobalt-60	5.27 years
Iodine-131	8.07 days
Nitrogen-16	7.2 seconds
Phosphourus-32 Plutonium-239	14.3 days 24,100 years
Potassium-40 Radium-226	1.28 x 10⁹ years 1600 years
Radon-222	3.82 days
Strontium-90	28.1 days
Technetium-99 Thorium-234 Uranium-238 Uranium-238	2.13 x 10⁵ years 24.1 days 7.04 x 10⁸ years 4.47 x 10⁹ years

Before you start making half-life calculations, there are some terms you should be familiar with. In every problem, you will be using or looking for the following variables:

Original amount: This is the amount of the isotope you start with. The amount is usually given in grams or milligrams of the substance.
Remaining amount: This is the amount of radioactive substance remaining after a certain amount of time. The amount is usually given in grams or milligrams of the substance.

*TIP: Remember that we are talking about a radioactive isotope breaking down. Therefore, the original amount will always be greater than the remaining amount. This is a good check to make sure that you have calculated the problem correctly.

Total Time: This is the amount of time elapsed or gone by. Time can be given in seconds, minutes, days, years, etc.
Time of half-life: This is the given amount of time for a specific isotope in which half of the original substances decays. In other words, this is the time of one half-life for a particular isotope.

*TIP: Total time is always going to be greater than the time of one half-life. This is a good check to make sure that you have calculated the problem correctly.

Number of half-lives: You will always want to determine how many half-lives have occurred. This can be found by 2 methods depending on what is given in the problem.
- Calculated when given both original amount and remaining amount.
  - For example: If you are given that one half-life for an isotope is 11 days and the total time that has elapsed is 44 days, how many half-lives have occurred?

$LaTeX: \frac{44\: days}{11\: days}$ $\frac{44\: days}{11\: days}$ = 4 half-lives

By knowing 4 half-lives have gone by, you can know that you have $LaTeX: \frac{1}{16}$ $\frac{1}{16}$ of the original amount remaining (look at the chart below).

Radioactive Isotope Breakdown
Number of Half-lives Passed	Percentage of Radioactive Isotope Remaining	Fraction of Radioactive Isotope Remaining
1	50%	$LaTeX: \frac{1}{2}$ $\frac{1}{2}$
2	25%	$LaTeX: \frac{1}{4}$ $\frac{1}{4}$
3	12.5%	$LaTeX: \frac{1}{8}$ $\frac{1}{8}$
4	6.25%	$LaTeX: \frac{1}{16}$ $\frac{1}{16}$
5	3.125%	$LaTeX: \frac{1}{32}$ $\frac{1}{32}$

Calculated when given total time and time of half-life.
- For example: If you are given that you start with an amount of 80 grams of a substance and the remaining amount after a certain period of time unknown to you is 2.5 grams, how many half-lives have occurred?

$LaTeX: \frac{2.5g}{80g}$ $\frac{2.5g}{80g}$ = 0.03125, changing to a percent you have 3.125% remaining

By knowing 3.125% (or 1/32), you can know that 5 half-lives have occurred (look at the chart above).

Knowing the vocabulary of what will be needed to calculate half-lives, look at the example problem below. You will see how to use the half-life of a sample to determine the amount of radioisotope that remains after a certain period of time has passed without using a formula.

Problem: Strontium-90 has a half-life of 28.1 days. If you start with a 5.00 mg sample of the isotope, how much remains after 140.5 days have passed?

Step 1: List the known values and plan the problem.

Known Variables

Original amount = 5.00 mg

t _½ = 28.1 days
Total time = 140.5 days

Unknown Variable

Remaining amount of Sr-90 = ?

Step 2: Determine how many half-lives have passed.

One half-life for Sr-90 is 28.1 days and the total time that has elapsed is 140.5 days, how many half-lives have occurred?

$LaTeX: \frac{140.5}{28.1}$ $\frac{140.5}{28.1}$ = 5 half-lives

By knowing 5 half-lives have gone by, you can know that you have $LaTeX: \frac{1}{32}$ $\frac{1}{32}$ of the original amount remaining (look at the chart above).

Step 3: Solve for how much of the substance is remaining.

Remaining amount of Sr-90 = 5.00 mg × $LaTeX: \frac{1}{32}$ $\frac{1}{32}$ = 0.156 mg

Step 4: Think about your result.

The remaining amount is less than the starting amount. Is this correct? Yes, since the nucleus is breaking down you should expect less.

Practice Problems

Radioactive Dating

Radioactive dating is a process by which the approximate age of an object is determined through the use of certain radioactive nuclides. For example, carbon-14 has a half-life of 5,730 years and is used to measure the age of organic material. The ratio of carbon-14 to carbon-12 in living things remains constant while the organism is alive because fresh carbon-14 is entering the organism whenever it consumes nutrients. When the organism dies, this consumption stops, and no new carbon-14 is added to the organism. As time goes by, the ratio of carbon-14 to carbon-12 in the organism gradually declines, because carbon-14 radioactively decays while carbon-12 is stable. Analysis of this ratio allows archaeologists to estimate the age of organisms that were alive many thousands of years ago. The ages of many rocks and minerals are far greater than the ages of fossils. Uranium-containing minerals that have been analyzed in a similar way have allowed scientists to determine that the Earth is over 4 billion years old.

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