Half-life
Section: Nuclear Physics | Syllabus: Cambridge AS Level Physics 9702
What is Half-life? Half-life The time taken for half the nuclei of that isotope in any sample to decay. Exam Tip Learn this exact definition: "the time taken for half the nuclei of that isotope in any sample to decay." This is the precise wording used in the syllabus and mark schemes.
Equivalent Ways to Express Half-life Half-life can also be described as: The time taken for the count rate (activity) to fall to half its original value The time taken for the mass of radioactive material to halve Important Half-life is constant for a particular isotope and cannot be changed by temperature, pressure, or chemical reactions.
Half-life Values Different isotopes have vastly different half-lives: Isotope Half-life Use/Notes Polonium-214 0.00016 seconds Very unstable Iodine-131 8 days Medical diagnosis Cobalt-60 5.3 years Cancer treatment Carbon-14 5,730 years Archaeological dating Uranium-238 4.5 billion years Nuclear fuel, geological dating The Half-life Pattern After each half-life period, half of the remaining radioactive nuclei decay.
FIG 5.21: Half-life decay pattern Visual representation showing decay over time: Start with 16 nuclei (all radioactive, filled circles), After 1 half-life: 8 nuclei, After 2 half-lives: 4 nuclei, After 3 half-lives: 2 nuclei, After 4 half-lives: 1 nucleus.
Decayed nuclei shown as faded. The Half-life Table Number of Half-lives Fraction Remaining Percentage Remaining Fraction Decayed 0 1 100% 0 1 1/2 50% 1/2 2 1/4 25% 3/4 3 1/8 12.5% 7/8 4 1/16 6.25% 15/16 5 1/32 3.125% 31/32 Pattern After n half-lives, the fraction remaining = (1/2) n Half-life Calculations You must be able to use the half-life definition in simple calculations, which might involve information in tables or decay curves.
Note Core calculations will not include background radiation. Supplement candidates must also be able to calculate half-life from data where background radiation has not been subtracted. Example: Basic Half-life Calculation Question: A radioactive sample has an initial count rate of 800 counts/s.
The half-life is 10 minutes. What is the count rate after 30 minutes? Solution Number of half-lives = 30 ÷ 10 = 3 half-lives After 1 half-life (10 min): 800 ÷ 2 = 400 counts/s After 2 half-lives (20 min): 400 ÷ 2 = 200 counts/s After 3 half-lives (30 min): 200 ÷ 2 = 100 counts/s Example: Finding the Half-life Question: A sample has an initial activity of 1200 counts/min.
After 24 hours, the activity is 150 counts/min. What is the half-life? Solution Start: 1200 counts/min After 1 half-life: 1200 ÷ 2 = 600 counts/min After 2 half-lives: 600 ÷ 2 = 300 counts/min After 3 half-lives: 300 ÷ 2 = 150 counts/min ✓ 3 half-lives = 24 hours, so 1 half-life = 24 ÷ 3 = 8 hours Example: Mass Remaining After Decay Question: A sample contains 80 g of a radioactive isotope with a half-life of 6 days.
How much remains after 18 days? Solution Number of half-lives = 18 ÷ 6 = 3 half-lives After 1 half-life (6 days): 80 ÷ 2 = 40 g After 2 half-lives (12 days): 40 ÷ 2 = 20 g After 3 half-lives (18 days): 20 ÷ 2 = 10 g Example: Fraction Remaining After Half-lives Question: After 4 half-lives, what fraction of the original sample remains?
Solution Fraction remaining = (1/2)⁴ = 1/16 Or: 1 × 1/2 × 1/2 × 1/2 × 1/2 = 1/16 Decay Curves When we plot activity (count rate) against time, we get a characteristic decay curve. FIG 5.22: Radioactive decay curve Exponential decay curve with X-axis: Time (half-life intervals marked), Y-axis: Count rate/Activity.
Curve starts at maximum, decreases exponentially. Horizontal dashed lines at 1/2, 1/4, 1/8 of initial value. Vertical dashed lines showing half-life intervals. Features of Decay Curves Exponential decay: The curve never reaches zero Decreasing gradient: The rate of decay slows over time Constant half-life: Takes the same time to halve regardless of starting point No background radiation: Pure decay curves assume background has been subtracted Key Feature The curve gets closer and closer to zero but never quite reaches it - there's always a small amount of radioactive material remaining.
Reading Half-life from Decay Curves Method 1: Using Any Starting Point Choose any point on the curve and note the count rate Calculate half of this count rate Find where the curve crosses this half-value Measure the time difference between these two points This time difference is the half-life FIG 5.23: Reading half-life from a decay curve Decay curve showing method: Mark point A at count rate 600 at time t₁, Mark point B at count rate 300 (half of 600) at time t₂.
Arrow showing time difference = half-life. Can repeat with different starting point to verify. Example: Reading Half-life from a Graph Question: From a decay curve: At time = 0 hours, count rate = 800 counts/s.
At time = 5 hours, count rate = 400 counts/s. What is the half-life? Solution Count rate halved from 800 to 400 Time taken = 5 - 0 = 5 hours = Half-life Example: Multiple Measurements Verification Question: From the same graph: At tim…
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