Half-Life (DP IB Physics: HL)

• It is impossible to know when a particular unstable nucleus will decay
• But the rate at which the activity of a sample decreases can be known
  • This is known as the half-life

• Half-life is defined as:

The time taken for half the undecayed nuclei to decay or the activity of a source to decay by half

• In other words, the time it takes for the activity of a sample to fall to half its original level
• Different isotopes have different half-lives and half-lives can vary from a fraction of a second to billions of years in length

Using Half-life

• Scientists can measure the half-lives of different isotopes accurately:
• Uranium-235 has a half-life of 704 million years
  • This means it would take 704 million years for the activity of a uranium-235 sample to decrease to half its original amount

• Carbon-14 has a half-life of 5700 years
  • So after 5700 years, there would be 50% of the original amount of carbon-14 remaining
  • After two half-lives, or 11 400 years, there would be just 25% of the carbon-14 remaining

• With each half-life, the amount remaining decreases by half

Graph showing how the activity of a radioactive sample changes over time. Each time the original activity halves, another half-life has passed

• The time it takes for the activity of the sample to decrease from 100 % to 50 % is the half-life
• It is the same length of time as it would take to decrease from 50 % activity to 25 % activity
• The half-life is constant for a particular isotope

• To calculate the half-life of a sample, the procedure is:
  • Measure the initial activity, A₀, of the sample
  • Determine the half-life of this original activity
  • Measure how the activity changes with time

• The time taken for the activity to decrease to half its original value is the half-life

Step 1: Draw lines on the graph to determine the time it takes for technetium to drop to half of its original activity

Step 2: Read the half-life from the graph

• • In the diagram above the initial activity, A₀, is 8 × 10⁷ Bq
  • The time taken to decrease to 4 × 10⁷ Bq, or ½A₀, is 6 hours
  • The time taken to decrease to 2 × 10⁷ Bq is 6 more hours
  • The time taken to decrease to 1 × 10⁷ Bq is 6 more hours
  • Therefore, the half-life of this isotope is 6 hours

Step 1: Calculate how many times the number of un-decayed atoms has halved

• • There were 2 000 000 atoms to start with
  • 1 000 000 atoms would remain after 1 half-life
  • 500 000 atoms would remain after 2 half-lives
  • Therefore, the sample has undergone 2 half-lives

Step 2: Divide the time period by the number of half-lives

• • The time period is a year
  • The number of half-lives is 2
  • 1 year divided by 2 is half a year or 6 months
  • Therefore, the half-life is 6 months

Step 1: Calculate the number of half-lives

• • The time period is 15 years
  • The half-life is 3 years

15 ÷ 3 = 5

• • There have been 5 half-lives

Step 2: Raise 1/2 to the number of half-lives

(1/2)⁵ = 1/32

• • So 1/32 of the original nuclei are remaining

Step 3: Write the ratio correctly

• • If 1/32 of the original nuclei are remaining, then 31/32 must have decayed
  • Therefore, the ratio is 31 decayed : 32 remaining, or 31:32