HalfLife
Radioactive elements goes through decay at different rates. The time it takes for the element to degrade by half is called a halflife. Learn more here.
HalfLife Learning Targets:
 Be able to define halflife
 Determine the amount of radioactive substance remaining after an amount of time (mathematically and visually with a table)
 Determine the amount of radioactive substance remaining after a number of half lives (mathematically and visually with a table)
Half Life of Common Radioactive Isotopes
A halflife is the time it takes for half of the radioactive sample to decay.
 Half lives range from a fraction of a second to billions of years.
 The length of a half life is shorter the less stable the radioactive isotope.
Radiometric Dating
Radiometric dating is a process of determining the age of an object by the amount of radioactive isotope that has degraded from an initial sample.
Carbon14 is a radioactive form of carbon used for radiometric dating of once living organisms.
Carbon14 has a halflife of 5,730 years. All life molecules have Carbon, testing the amount of radioactive carbon present in a sample can determine its age.
An archaeologist that tests a bone and finds it has half the normal amount of radioactive carbon14 just determined its age is 5,730 years old.
The halflife of Radium226 is 1600 years. This means a 1 kg radioactive sample today would decay to a 1/2 kg radioactive sample after 1600 years.
Analyze the table along with the animation to identify the pattern and relationship between number of halflives, kilograms of radioactive sample size, nonradioactive sample size, and percent decayed.
Half Lives  Radioactive  Nonradioactive  Percent Decayed 
0  1 kg  0 kg  0% 
1  1/2 kg  1/2 kg  50% 
2  1/4 kg  3/4 kg  75% 
3  1/8 kg  7/8 kg  87.5% 
Half Life Equations
We will work through a few sample problems on this page using a table form and the following equations.
y=1/(2^{n})
T_{1/2} = t/n
 y represents the fraction of radioactive material remaining
 n represents the number of half lives
 T_{1/2 }represents the length of one half life
Guided HalfLife Example
A. How much of the original radioactive material remains after four halflives?
Givens List:

 n = 4
 y=?
Equation:

 y=1/(2^{n})
Work:

 y=1/(2^{4})
 y= 1/16 or 0.0625
B. How much of a 56 gram radioactive sample radioactive remains after four halflives?
First: Find the fraction left
Givens List:

 n = 4
 y=?
Equation:

 y=1/(2^{n})
Work:

 y=1/(2^{4})
 y=0.0625
Second: Multiply the fraction by the original sample size
0.0625 x 56 grams = 3.5 grams of a radioactive sample remains
Using a table to answer example A and B above
#1 Make a table as seen below starting with 0 half lives and columns for any necessary information. Fill in one in this column if asked about a fraction, fill in 100% if asked about a percent.
#2 Fill in the table with an extra row for every half life until you get to the number of halflives in the question
#3 The completed table here could be used to answer questions about the fraction or sample remaining for up to four halflives of a 56 gram sample.
C. How long is a halflife for a substance that has 2.5 half lives every 50,000 years?
Givens:

 n = 2.5
 t = 50,000 years
 T_{1/2} = ?
Equation:

 T_{1/2} = t/n
Work:

 T_{1/2} = 50000/2.5
 T_{1/2} = 20,000 years
D. A 100 gram unknown radioactive material has a halflife of 4000 years. How many grams of the radioactive sample will remain after 20,000 years?
Step 1: Determine the number of halflives
Givens:

 T_{1/2} = 4000 years
 t = 20,000 years
Starting Equation:

 T_{1/2} = t/n rearranges to n = t/T_{1/2}
Work:

 n = 20000/4000
 n = 5
Step 2: find the fraction left after five halflives
Givens:

 y = ?
 n = 5
Equation:

 y =1/(2^{n})
Work:

 y =1/(2^{5})
 y = 0.03125
Step 3: Multiply the fraction by the original sample mass
0.03125 x 100 g = 3.125 grams
Example Problems
Note: You can use a table or math for any of the following examples
1. What percentage of radioactive substance remains after ten halflives?
2. If a substance starts out with 400g, how much remains after five halflives?
3. How many complete halflives have occurred if 12.5% of the original radioactive material remains?
4. How many complete halflives have occurred is 18.75 grams of an originally 600 gram radioactive substance remains?
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