Half-Life Calculator
Enter any three values to calculate the fourth, related to exponential decay and half-life.
Results:
| Time | Remaining Quantity | Fraction Remaining |
|---|---|---|
| Enter values to see decay table. | ||
What is a Half-Life Calculator?
A Half-Life Calculator is a tool used to determine various parameters related to exponential decay, most notably the half-life of a substance. Half-life (T½) is defined as the time it takes for half of the initial amount of a substance to undergo decay or be eliminated. The concept is widely used in physics (for radioactive decay), pharmacology (for drug metabolism), environmental science, and more. Our Half-Life Calculator allows you to find the half-life, initial quantity, final quantity, or time elapsed given the other three values.
Anyone studying radioactive decay, pharmacokinetics, or other processes involving exponential decay can benefit from using a Half-Life Calculator. It's useful for students, researchers, and professionals in fields like nuclear physics, medicine, and chemistry.
A common misconception is that half-life means half the substance disappears in the first half-life, and the other half disappears in the second half-life, making it all gone after two half-lives. However, it means half of the *remaining* substance decays in each subsequent half-life period, so it approaches zero but theoretically never reaches it in a finite time.
Half-Life Calculator Formula and Mathematical Explanation
The decay of a substance following first-order kinetics is described by the formula:
N(t) = N₀ * (1/2)^(t / T½)
Where:
- N(t) is the quantity of the substance remaining after time t.
- N₀ is the initial quantity of the substance at time t=0.
- t is the time elapsed.
- T½ is the half-life of the substance.
From this formula, if we know N₀, N(t), and t, we can solve for T½:
N(t) / N₀ = (1/2)^(t / T½)
ln(N(t) / N₀) = (t / T½) * ln(1/2)
ln(N(t) / N₀) = -(t / T½) * ln(2)
T½ = -t * ln(2) / ln(N(t) / N₀) = t * ln(2) / ln(N₀ / N(t))
Similarly, we can rearrange to solve for t, N(t), or N₀ if the other values are known. The Half-Life Calculator uses these rearrangements.
Another related concept is the decay constant (λ), which is related to half-life by:
λ = ln(2) / T½
And the decay formula can also be written as N(t) = N₀ * e^(-λt).
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| N₀ | Initial quantity | grams, moles, number of atoms, concentration units | > 0 |
| N(t) | Remaining quantity at time t | Same as N₀ | 0 < N(t) ≤ N₀ |
| t | Time elapsed | seconds, minutes, hours, days, years | ≥ 0 |
| T½ | Half-life | Same as t | > 0 |
| λ | Decay constant | 1/time units (e.g., 1/s, 1/year) | > 0 |
Practical Examples (Real-World Use Cases)
Example 1: Carbon-14 Dating
An archaeologist finds a wooden artifact. They measure the carbon-14 activity and find it has 75% of the carbon-14 activity of living wood. The half-life of Carbon-14 is approximately 5730 years. How old is the artifact?
- N₀ = 100% (or 1)
- N(t) = 75% (or 0.75)
- T½ = 5730 years
- We need to find t. Using the formula: t = T½ * ln(N(t)/N₀) / ln(1/2) = 5730 * ln(0.75) / (-ln(2)) ≈ 2378 years.
Our Half-Life Calculator can find 't' if you input N₀, N(t), and T½.
Example 2: Drug Elimination
A patient is given a drug with a half-life of 6 hours. If the initial plasma concentration is 200 mg/L, what will the concentration be after 12 hours?
- N₀ = 200 mg/L
- T½ = 6 hours
- t = 12 hours
- We need to find N(t). After 12 hours, two half-lives have passed (12/6 = 2). After 1 half-life (6 hrs): 200 / 2 = 100 mg/L After 2 half-lives (12 hrs): 100 / 2 = 50 mg/L. Using the formula: N(t) = 200 * (1/2)^(12/6) = 200 * (1/2)^2 = 200 * 0.25 = 50 mg/L.
The Half-Life Calculator can easily find N(t).
How to Use This Half-Life Calculator
- Enter Known Values: Input three of the four values: Initial Quantity (N₀), Remaining Quantity (N(t)), Time Elapsed (t), or Half-Life (T½). Ensure Remaining Quantity is not greater than Initial Quantity.
- Calculate: Click the "Calculate" button or simply change input values. The calculator will determine the missing value and update the results automatically. If you enter Half-Life, it will prioritize using that to calculate other values if N0 and t are also given to find N(t), or N0 and N(t) to find t, etc.
- View Results: The primary result (the calculated value) will be highlighted. Intermediate values like the decay constant and number of half-lives elapsed will also be shown.
- Examine Table and Chart: The table and chart will update to show the decay over time based on the calculated or entered half-life and initial quantity.
- Reset: Use the "Reset" button to clear inputs and results to their default values.
- Copy: Use the "Copy Results" button to copy the main results and assumptions to your clipboard.
When making decisions, remember that the Half-Life Calculator assumes first-order decay, which is common but not universal.
Key Factors That Affect Half-Life Results
- Initial Quantity (N₀): The starting amount. While it doesn't affect the half-life itself (which is an intrinsic property), it determines the actual amount remaining after a certain time.
- Remaining Quantity (N(t)): The amount left after time t. This, along with N₀ and t, is used to calculate T½.
- Time Elapsed (t): The duration over which decay is observed. It's directly proportional to the number of half-lives if T½ is known.
- The Substance Itself: Different substances have vastly different half-lives, from fractions of a second to billions of years for radioactive isotopes, or minutes to days for drugs. The half-life is an intrinsic property.
- Decay Process: The calculator assumes exponential decay (first-order kinetics). If the decay process is different (e.g., zero-order), the formula and results will differ.
- Measurement Accuracy: The accuracy of the calculated half-life or other parameters depends on the accuracy of the input measurements (N₀, N(t), t).
Using an accurate Half-Life Calculator like this one is essential for reliable results.