Half-Life Calculator

What half-life tells you

Half-life is the time needed for a quantity to fall to half its starting amount. It is most familiar from radioactive decay, where unstable atomic nuclei emit particles or energy as they move toward a more stable state, but the same exponential pattern appears in pharmacology, chemical kinetics, and other natural processes.

The important detail is that half-life removes half of what is currently present, not half of the original amount every time. After one half-life, 50% remains; after two, 25% remains; after three, 12.5% remains. The quantity gets closer to zero without reaching zero in the ideal mathematical model.

This calculator lets you choose Half-Life, Elapsed Time, Remaining Quantity, or Initial Quantity from the Calculate dropdown. It also reports the decay constant and mean lifetime. Use the same type of quantity for the initial and remaining values, and select matching time units for elapsed time and half-life.

How to Calculate Half-Life

Step One: Identify the initial and remaining quantities

Record the amount before decay and the amount measured after a known period.

initial quantity = Nâ‚€ remaining quantity = N

Using the calculator defaults: Nâ‚€ = 100 and N = 5.

Step Two: Find the quantity ratio

Divide the initial quantity by the remaining quantity.

quantity ratio = initial quantity ÷ remaining quantity

For the defaults: 100 ÷ 5 = 20.

Step Three: Apply the half-life equation

Multiply elapsed time by ln(2), then divide by ln(Nâ‚€/N).

half-life = elapsed time × ln(2) ÷ ln(initial quantity ÷ remaining quantity)

For 15 seconds, 100 initial units, and 5 remaining units: 15 × 0.693147 ÷ ln(20) = 3.470673 seconds.

Half-Life Formula

The standard decay equation calculates the remaining amount after a known time:

remaining quantity = initial quantity × (1/2)^(elapsed time ÷ half-life)

In symbols:

N = Nâ‚€ × (1/2)^(t/t½)

N is the remaining quantity, N₀ is the initial quantity, t is elapsed time, and t½ is the half-life. The time values must be expressed in compatible units.

The half-life is related to the decay constant λ:

decay constant = ln(2) ÷ half-life

The mean lifetime Ï„ is related to both values:

mean lifetime = half-life ÷ ln(2) = 1 ÷ decay constant

How to Calculate Time Elapsed Using Radioactive Decay

Elapsed time can be estimated by comparing the quantity remaining in a sample with the amount that was present at the start. This is the basic idea behind radiometric dating. For example, carbon-14 in living material is continually replenished, but after an organism dies, the carbon-14 amount decreases while stable carbon-12 remains comparatively unchanged.

Carbon-14 has a half-life of approximately 5,730 years. If a sample retains 15% of its original carbon-14, the elapsed time can be estimated from the remaining fraction.

Half-Life Formula for Time Elapsed

Rearrange the decay equation to isolate time:

elapsed time = half-life × ln(remaining quantity ÷ initial quantity) ÷ -ln(2)

For 15% remaining and a 5,730-year half-life:

t = 5,730 × ln(15 ÷ 100) ÷ -ln(2)

t = 5,730 × ln(0.15) ÷ -0.693147

t ≈ 15,680 years

This result is an estimate based on the decay model and assumes the original quantity and the half-life are known accurately.

How to Calculate Remaining Quantity

Step One: Calculate how many half-lives have passed

Divide elapsed time by half-life.

number of half-lives = elapsed time ÷ half-life

For 15 seconds and a 5-second half-life: 15 ÷ 5 = 3 half-lives.

Step Two: Raise one-half to that power

remaining fraction = (1/2)^(number of half-lives)

For three half-lives: (1/2)^3 = 0.125.

Step Three: Multiply by the initial quantity

remaining quantity = initial quantity × remaining fraction

For 100 units: 100 × 0.125 = 12.5 units.

Decay Constant and Mean Lifetime

The decay constant describes how quickly a substance decays as a fraction of its current amount. It is not a fixed number of units lost per second; it is a rate applied to what remains.

decay constant = ln(2) ÷ half-life

Mean lifetime is the average lifetime of particles in an exponential-decay process:

mean lifetime = half-life ÷ ln(2)

For a 5-second half-life, the decay constant is 0.1386294361 per second and the mean lifetime is 7.213475204 seconds.

Half-Life Decay Table

The table below shows the fraction and percentage remaining after successive half-lives.

Important assumptions

The formulas assume a first-order exponential-decay process with a constant half-life. Radioactive half-life is characteristic of a particular isotope, while environmental conditions can affect some chemical, biological, or physical processes. For radiometric dating, the sample must also be interpreted with appropriate assumptions about its original composition and contamination.

References

• U.S. Nuclear Regulatory Commission, Half-life glossary: https://www.nrc.gov/reading-rm/basic-ref/glossary/half-life-radiological.html
• U.S. Nuclear Regulatory Commission, Radiation basics: https://www.nrc.gov/about-nrc/radiation/health-effects/radiation-basics.html
• NCBI Bookshelf, pharmacokinetic elimination half-life: https://www.ncbi.nlm.nih.gov/books/NBK554498/