Exponential Decay Calculator

Calculator

Solve for

Choose the unknown you need.

Time unit

Times are interpreted in this unit.

Display precision (2–12)

Controls rounding in results and exports.

Initial amount (N0)

Particles, charge, temperature offset, or concentration.

Remaining amount (N)

Use with N0 to solve for time or λ.

Time (t)

Elapsed time in the chosen unit.

Decay constant (λ)

If blank, half-life can define it.

Half-life (t1/2)

If λ is empty, t1/2 is used.

Tip: Provide either λ or t1/2. When both are provided, λ is used.

Example data table

These sample values demonstrate typical decay calculations.

N0	Half-life (s)	Time (s)	λ (1/s)	Computed N(t)
1	5	12	0.138629	189.464571

Formula used

N(t) = N0 · e^(−λt)
t1/2 = ln(2) / λ
τ = 1 / λ (mean life)
λ = (1/t) · ln(N0/N)
t = (1/λ) · ln(N0/N)

The calculator converts your time unit to seconds internally, then converts results back to your chosen unit.

How to use this calculator

Select what you want to solve for.
Pick a time unit for all time inputs.
Enter known values, leaving the unknown blank.
Click Calculate to view results above the form.
Use the export buttons to download CSV or PDF.

FAQs

1) What does the decay constant represent?

It measures how fast a quantity decreases. Larger λ means faster decay. It has units of inverse time, like 1/s or 1/day.

2) When should I use half-life instead of λ?

Use half-life when it is reported in lab data or material tables. The calculator converts it to λ using ln(2)/t1/2 automatically.

3) Can this model describe cooling or discharging?

Yes, when the process follows a first-order law. For cooling, the quantity is the temperature difference from ambient.

4) Why can't remaining amount exceed initial amount?

For pure exponential decay with positive λ, the curve is always decreasing. If N is larger than N0, the inputs imply growth, not decay.

5) What if my data shows multiple decay phases?

A single λ may not fit. Consider fitting separate segments or using a sum of exponentials. This calculator targets one dominant decay rate.

6) How accurate are the results?

Accuracy depends on your input quality and whether the decay assumption holds. Numeric calculations use double precision and adjustable rounding for display.

7) How do the export files work?

CSV is generated from the result table for spreadsheets. PDF is built in your browser from the same values, suitable for printing or sharing.