Calculate half-life from decay constant or find the decay constant from half-life using seconds, minutes, hours, days, or years.
Halving Time Formula
The halving time—more commonly called half-life—is the time required for an exponentially decaying quantity to decrease to one-half of its starting value. This calculator uses the standard relationship between halving time and the decay constant.
When the halving time is known and you want to solve for the decay constant, rearrange the equation:
Because the natural logarithm of 2 is a constant, the formula is often approximated as:
| Quantity | Description | Unit Guidance |
|---|---|---|
| Halving Time | The amount of time needed for the quantity to fall to 50% of its original value. | Seconds, minutes, hours, days, years, or any other consistent time unit. |
| Decay Constant | The continuous rate at which the quantity decays. | Inverse time, such as 1/s, 1/min, 1/hr, 1/day, or 1/year. |
| ln(2) | A fixed mathematical constant used to convert between decay constant and half-life. | Dimensionless. |
How to Use the Calculator
- Select half-life or decay constant, then enter the known positive value.
- Select the input and result units independently. The calculator converts reciprocal-time units before calculating. A year is defined here as 365 days.
- Use a positive half-life or decay constant. Zero decay constant has no finite half-life, and a negative decay constant describes growth.
- Interpret the result as the time required for the quantity to be cut in half once.
What the Formula Means
Halving time applies to processes that follow exponential decay, meaning the quantity decreases by the same percentage over equal time intervals, not by the same absolute amount. The underlying model is:
At the halving time, the remaining amount is one-half of the initial amount:
Substituting that condition into the decay model produces the halving time formula used by the calculator.
Example
If a substance has a decay constant of 0.05 per day, the halving time is:
This means the amount will be reduced by half about every 13.86 days. If the starting amount is 80 grams, then after one halving time 40 grams remain, after two halving times 20 grams remain, and after three halving times 10 grams remain.
Amount Remaining After Multiple Halving Times
Once the halving time is known, the amount remaining after multiple half-lives can be estimated with:
Here, n is the number of halving times that have passed.
| Halving Times Elapsed | Remaining Percentage |
|---|---|
| 1 | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 4 | 6.25% |
| 5 | 3.125% |
Common Uses of Halving Time
- Radioactive decay: estimating how quickly unstable isotopes lose activity.
- Chemistry: modeling the breakdown of reactants or contaminants.
- Pharmacokinetics: approximating how quickly a drug concentration declines in the body.
- Biology and environmental science: tracking degradation, die-off, or elimination processes.
- General exponential decay models: any system where loss is proportional to the current amount.
Common Mistakes
- Mixing units: a decay constant in per hour produces a halving time in hours, not days or minutes unless converted.
- Confusing linear and exponential decay: halving time only fits processes that decay by a constant fraction over time.
- Using percentage values directly: convert rates into a proper decay constant before applying the formula.
- Assuming the same amount disappears each interval: exponential decay removes the same proportion, not the same fixed quantity.
Quick Interpretation Guide
- A larger decay constant means a shorter halving time.
- A smaller decay constant means a longer halving time.
- If the halving time is known, you can compare how quickly different substances or systems decay on the same time basis.
