Half-life & decay tool • 2026 standards
\( N(t) = N_0 \cdot e^{-\lambda t} \)
Where:
Alternative Forms:
Common Isotopes:
This formula describes the exponential decay of radioactive substances over time.
| Time | Remaining (g) | Decayed (g) | Activity (Bq) | % Remaining |
|---|
Radioactive decay is the spontaneous transformation of unstable atomic nuclei into more stable configurations, releasing energy in the form of radiation. It follows exponential decay laws described by the decay constant.
\( N(t) = N_0 \cdot e^{-\lambda t} \)
Where N(t) is the quantity at time t, N₀ is the initial quantity, λ is the decay constant, and t is time.
Carbon dating, medical imaging (PET scans), nuclear power generation, cancer treatment (radiation therapy), geological dating, smoke detectors, and nuclear weapons applications utilize radioactive decay principles.
What does the half-life of a radioactive substance represent?
The answer is B) The time for half of the nuclei to decay. The half-life is the time required for exactly half of the radioactive nuclei in a sample to undergo radioactive decay. After one half-life, 50% of the original nuclei remain; after two half-lives, 25% remain; after three half-lives, 12.5% remain, and so on.
Half-life is a fundamental concept in nuclear physics that describes the characteristic time scale of radioactive decay. It's a probabilistic measure - each nucleus has a 50% chance of decaying within one half-life period. The exponential nature of decay means that even after many half-lives, some radioactive nuclei will still remain, though in diminishing quantities.
Half-life (t₁/₂): Time for half of the radioactive nuclei to decay
Radioactive Decay: Spontaneous transformation of unstable nuclei
Activity: Rate of radioactive decay (decays per second)
• Half-life is constant for each isotope
• Decay follows exponential law: N(t) = N₀e^(-λt)
• After n half-lives, fraction remaining = 1/2ⁿ
• Remember: half-life is the time for 50% decay
• Each half-life reduces quantity by half
• Use the formula t₁/₂ = ln(2)/λ
• Thinking half-life means complete decay
• Confusing half-life with mean lifetime
• Forgetting exponential nature of decay
If Carbon-14 has a half-life of 5,730 years, what fraction of a sample remains after 17,190 years?
Step 1: Calculate number of half-lives = Time elapsed ÷ Half-life
Number of half-lives = 17,190 ÷ 5,730 = 3
Step 2: Calculate fraction remaining = (1/2)ⁿ where n = number of half-lives
Fraction remaining = (1/2)³ = 1/8 = 0.125 = 12.5%
Alternatively, using the decay formula: N(t) = N₀e^(-λt)
Where λ = ln(2)/t₁/₂ = 0.693/5730 = 1.21×10⁻⁴ per year
N(17190) = N₀e^(-(1.21×10⁻⁴)(17190)) = N₀e^(-2.08) = 0.125N₀
Therefore, 12.5% of the original sample remains.
This problem demonstrates the exponential nature of radioactive decay. After each half-life period, the remaining quantity is halved. After 3 half-lives, the remaining fraction is (1/2)³ = 1/8. This exponential relationship is fundamental to understanding radiometric dating and nuclear medicine applications.
Exponential Decay: Decrease by a constant proportion per time period
Decay Constant (λ): Probability of decay per unit time
Carbon Dating: Using C-14 decay to date organic materials
• Fraction remaining after n half-lives = (1/2)ⁿ
• λ = ln(2)/t₁/₂
• N(t) = N₀e^(-λt)
• Count the number of half-lives
• Use (1/2)ⁿ for quick calculations
• Remember: exponential decay never reaches zero
• Linear thinking instead of exponential
• Forgetting to divide by half-life
• Confusing decay constant with half-life
Iodine-131 has a half-life of 8.02 days and is used in medical treatments. If a patient receives 100 μg of I-131, how much remains after 24.06 days? What percentage has decayed?
Step 1: Calculate number of half-lives = 24.06 ÷ 8.02 = 3
Step 2: Calculate remaining fraction = (1/2)³ = 1/8 = 0.125
Step 3: Calculate remaining amount = 100 μg × 0.125 = 12.5 μg
Step 4: Calculate decayed amount = 100 μg - 12.5 μg = 87.5 μg
Step 5: Calculate percentage decayed = (87.5/100) × 100% = 87.5%
Therefore, 12.5 μg remains (12.5%) and 87.5% has decayed.
This problem demonstrates practical applications of radioactive decay in medicine. Iodine-131 is commonly used for thyroid treatments and imaging. The relatively short half-life makes it useful for medical applications while ensuring the radioactivity decreases to safe levels within days after treatment.
Medical Isotope: Radioactive material used in medicine
Thyroid Treatment: Using I-131 to treat hyperthyroidism
Microgram (μg): One millionth of a gram
• Shorter half-life = faster decay
• Medical isotopes balance effectiveness with safety
• Activity decreases with time
• Count half-lives for quick estimates
• Use exponential formula for precision
• Consider safety implications
• Forgetting to calculate percentage decayed
• Confusing remaining with decayed amounts
• Not considering units properly
Uranium-238 has a half-life of 4.468 billion years. How long does it take for 75% of a U-238 sample to decay? Express your answer in billions of years.
Step 1: If 75% decays, 25% remains, so N(t)/N₀ = 0.25
Step 2: Use the decay formula: N(t) = N₀e^(-λt)
Step 3: 0.25 = e^(-λt), so ln(0.25) = -λt
Step 4: Calculate decay constant: λ = ln(2)/t₁/₂ = 0.693/(4.468×10⁹) = 1.55×10⁻¹⁰ per year
Step 5: Solve for t: t = -ln(0.25)/λ = -ln(0.25)/(1.55×10⁻¹⁰)
Step 6: t = 8.99×10⁹ years = 8.99 billion years
Alternatively: 0.25 = (1/2)ⁿ, so 2⁻ⁿ = 2⁻², therefore n = 2 half-lives
t = 2 × 4.468 = 8.936 billion years (same result)
Therefore, it takes approximately 8.94 billion years for 75% of U-238 to decay.
This problem shows how to work backwards from a remaining fraction to find time. For 75% decay, 25% remains, which is equivalent to 2 half-lives (since (1/2)² = 1/4 = 0.25). This demonstrates the relationship between remaining fraction and number of half-lives.
Uranium Series: Chain of radioactive decays starting with U-238
Geological Time Scale: Measuring Earth's historyRadioactive Dating: Using decay to determine age
• N(t)/N₀ = e^(-λt) for any fraction
• λ = ln(2)/t₁/₂
• For fraction F remaining: F = (1/2)ⁿ
• Convert percentage to decimal for calculations
• Use logarithms to solve for time
• Verify with half-life multiples
• Forgetting to convert percentage to decimal
• Incorrectly setting up the exponential equation
• Arithmetic errors with large numbers
Which of the following correctly expresses the relationship between half-life (t₁/₂) and decay constant (λ)?
The answer is B) λ = ln(2) / t₁/₂. Starting from the decay equation N(t) = N₀e^(-λt), at t = t₁/₂, N(t) = N₀/2. Substituting: N₀/2 = N₀e^(-λt₁/₂). Dividing by N₀: 1/2 = e^(-λt₁/₂). Taking natural log: ln(1/2) = -λt₁/₂. Since ln(1/2) = -ln(2): -ln(2) = -λt₁/₂. Therefore: λ = ln(2) / t₁/₂.
The decay constant λ represents the probability per unit time that a nucleus will decay. The relationship λ = ln(2)/t₁/₂ connects the microscopic decay probability to the macroscopic half-life. This fundamental relationship allows conversion between these two important parameters in radioactive decay calculations.
Decay Constant (λ): Probability of decay per unit time
Half-life (t₁/₂): Time for half of nuclei to decay
ln(2): Natural logarithm of 2 ≈ 0.693
• λ = ln(2)/t₁/₂
• t₁/₂ = ln(2)/λ
• Larger λ means shorter t₁/₂
• Remember ln(2) ≈ 0.693
• λ and t₁/₂ are inversely related
• Use dimensional analysis to verify
• Confusing the relationship between λ and t₁/₂
• Forgetting the ln(2) factor
• Reversing the fraction
Q: How do I calculate the activity of a radioactive sample?
A: The activity A of a radioactive sample is given by A = λN, where λ is the decay constant and N is the number of radioactive nuclei.
Using the decay formula N(t) = N₀e^(-λt), the activity at time t is:
A(t) = λN(t) = λN₀e^(-λt) = A₀e^(-λt)
Where A₀ is the initial activity. The SI unit of activity is the Becquerel (Bq), where 1 Bq = 1 decay per second.
For example, if λ = 1.21×10⁻⁴ per year and N = 10²⁰ nuclei, then A = 1.21×10⁻⁴ × 10²⁰ = 1.21×10¹⁶ Bq.
Q: What's the difference between half-life and mean lifetime?
A: The half-life (t₁/₂) is the time for half of the nuclei to decay, while the mean lifetime (τ) is the average time a nucleus exists before decaying.
Relationships:
• Half-life: t₁/₂ = ln(2)/λ ≈ 0.693/λ
• Mean lifetime: τ = 1/λ
• Therefore: τ = t₁/₂/ln(2) ≈ 1.44 × t₁/₂
The mean lifetime is always longer than the half-life by a factor of 1/ln(2) ≈ 1.44. This is because the exponential decay curve has a long tail.