Half Life EQ: How is the half-life of a radioactive element used to determine how much of a sample is left after a given period of time?

Half Life

EQ: How is the half-life of a radioactive element used to determine how much of a sample is left after a given period of time?

What is “half-life?”

• The time taken for the radioactivity of a specified isotope to fall to half of its original value

• A radioactive material will have some nuclei (plural for nucleus) that are stable and some that are unstable.– Stable nuclei – don’t change– Unstable nuclei – will change into stable nuclei at

different rates depending on the material.

The “speed” of half-life

• Some unstable nuclei will change quickly.– For example, Lithium-8 has a half-life of 0.85 seconds

• Some unstable nuclei will change slowly.– For example, uranium-238 has a half-life of 4.51

BILLION years.

• It’s important to remember that “half-life” is an amount of time.

An analogy

• You have 8 markers.• Every minute you give away HALF your markers.• After 1 MINUTE (or one half-life) you only have 4

markers left.• After 2 MINUTES (or two half-lives) you have 2 markers

left.• After 3 MINUTES (or three half-lives) you have 1 marker

left.• Each “half-life” period, your total markers cuts in half.• This is an example of how “half-life” works.

How half-life is used…just a few examples

• Radioactive dating– Scientists use Carbon-14 to date materials,

assisting them in a variety of areas.• The formation of the earth• The geologic timeline• Fossil Dating

• Pharmacy– How long a particular drug remains in your system

Sample Problem #1

• If 100.0 grams of carbon-14 decays until only 25.0 grams of carbon is left after 11,460 years, what is the half life of carbon-14?

Step 1: List the “Given” and “Unknown” Values

• Given:– Initial Mass of the Sample – 100g– Final Mass of the Sample – 25g– Total time of decay – 11,460 years

• Unknown:– Number of half lives = ? Half lives– Half life = ? years

Step 2: Write down the equation relating half life, the number of half lives, and the decay time.

• Step 2(B): Rearrange it to solve for half life.

Step 2Total time of decay = number of half lives x (number of years/half life)

Step 2(B)(Number of Years/Half life) = (total time of decay/number of half lives)

Step 3: Calculate how many half lives have passed during the decay

of the sample.

Fraction of the sample remaining = final mass/initial mass

In this example: 25/100 = ¼

Therefore: 2 half lives have passed because ½ + ½ = ¼

Step 4: Calculate the half life

Equation: Number of years/half life

In this example: 11,460 years/2 half lives

Therefore: Carbon-14’s half life is 5730 years

Sample Problem #2

• Thallium-208 has a half-life of 3.053 minutes. How long will it take for 120g to decay to 7.5g?

Step 1: List the “given” and “unknown” values

• Given– Half life – 3.053 minutes– Initial Mass – 120g– Final Mass – 7.5g

• Unknown– Number of half lives = ? Half lives– Total time of decay

Step 2: Write down the equation relating half life, the number of half lives, and the decay time, and then

rearrange it to solve for the total time of decay.

• Total time of decay = number of half lives X (number of minutes/half life)

Step 3: Calculate how many half lives have passed during the decay

of the given sample.

Fraction of the sample remaining = 7.5g/120g 0.0625 = 1/16

Therefore: 4 half lives have passed because ½ + ½ + ½ + ½ = 1/16

Step 4: Calculate the total time required for the radioactive decay.

Equation: Total time of decay = number of half lives x (number of minutes/half life)

In this example: Total time of decay = 4 half lives X (3.053 min/half life)

Therefore: Total time of decay = 12.21 minutes

Sample Problem #3

• Gold-198 has a half-life of 2.7 days. How much of a 96g sample of gold-198 will be left after 8.1 days?

Step 1: Identify the “given” and the “unknown”

• Given– Half-life – 2.7 days– Total time of decay – 8.1 days– Initial Mass – 96g

• Unknown– Number of half-lives = ? Half-lives– Final Mass = ? g

Step 2: Write down the equation relating half-life, the number of half-lives, and the decay time, and then

rearrange it to solve for the number of half-lives.

Total time of decay = number of half-lives X (number of days/half life)

Step 3: Calculate how many half-lives have passed during the decay

of the given sample.

Number of half lives = 8.1 days/2.7 days = 3 half-lives

Step 4: Calculate how much of the sample will remain after 3 half lives.

Final mass of the sample = initial mass of the sample X the fraction of the sample remaining

Fraction of the sample remaining after 3 half lives = ½ + ½ + ½ = 1/8

Final mass of the sample = 96g X 1/8 = 12g

Practice Page

• Complete the nine practice problems in your journal.

• You may use these notes.• You may work in groups.• Raise your hand when you have completed the

assignment to receive your stamp.• WE WILL GO OVER THE ANSWERS after you

make an attempt at completing them on your own.

Web Reference

• http://webcheck-test.eharcourtschool.com/hrwfw/INT-D-bck/http/sh/hk6_0030390966/teacher/osp/data/chap09/sec1/math_skills2.pdf