# Half-Life Calculator

Exponential decay for radioactive isotopes and drugs: remaining amount, derived half-life, time to target, or steady-state accumulation. Uses N(t) = N0·e^(-λt).

> Canonical page: https://elysiatools.com/en/tools/half-life-calculator

- **Category:** Math & Numbers

- **Keywords:** half-life, radioactive decay, isotope, pharmacokinetics, drug accumulation, exponential decay, chemistry, physics

## Overview

The Half-Life Calculator computes exponential decay for radioactive isotopes and pharmaceuticals using the standard decay formula. It allows you to determine the remaining quantity of a substance, calculate a derived half-life from measurements, find the time required to reach a target amount, or model steady-state drug accumulation over multiple doses.

## Inputs

- **Calculation Mode** (select)
- **Half-life** (number): Half-life in the selected time unit. Required for remaining/time/dose modes.
- **Initial Amount N0** (number): Initial quantity (mass, activity, count, dose…). Required for remaining/half-life modes.
- **Remaining Amount N(t)** (number): Measured remaining quantity. Required for half-life/time modes.
- **Elapsed Time t** (number): Elapsed time in the selected unit. Required for remaining/half-life modes.
- **Percent Remaining (%)** (number): Alternative to N(t) in time mode: target percentage of N0 still present.
- **Time Unit** (select)
- **Single Dose (mg)** (number): Amount of drug administered at each dose.
- **Dosing Interval** (number): Time between consecutive doses, in the selected time unit.
- **Decimal Places** (number)

## When to use

- When calculating the remaining activity or mass of a radioactive isotope after a specific elapsed time.
- When determining the elimination rate and steady-state concentration of a drug based on its dosing interval and half-life.
- When estimating the age of organic materials or calculating the time needed for a substance to decay to a safe threshold.

## How it works

- Select the calculation mode: remaining amount, derived half-life, time to target, or drug accumulation.
- Input the known parameters such as the initial amount, elapsed time, half-life, or dosing interval.
- Choose the appropriate time unit (seconds, minutes, hours, days, or years) and set the desired decimal precision.
- Click calculate to compute the exponential decay constant, mean lifetime, and the final target value.

## Use cases

- Radiocarbon dating calibration by calculating the decay of Carbon-14 over thousands of years.
- Pharmacokinetic modeling to determine how much of a medication remains in a patient's system before the next dose.
- Nuclear medicine planning to calculate the remaining activity of short-lived isotopes like Technetium-99m.

## Frequently asked questions

### What formula does this calculator use for decay?

It uses the exponential decay formula N(t) = N0 * e^(-λt), where λ (decay constant) is equal to ln(2) divided by the half-life.

### Can I calculate drug accumulation over multiple doses?

Yes, select the 'Drug accumulation (steady state)' mode and input the single dose amount, dosing interval, and drug half-life.

### How do I find the half-life of an unknown substance?

Choose the 'Derive half-life from measurements' mode, then enter the initial amount, remaining amount, and elapsed time.

### What is the difference between N(t) and percent remaining?

N(t) represents the absolute remaining quantity (e.g., in grams or mg), while percent remaining is the proportion left relative to the initial amount (N0).

### Can I use different time units for the calculation?

Yes, you can select seconds, minutes, hours, days, or years, but ensure all input time values match the selected unit.

## Related tools

- [Angular Velocity Converter (rad/s / rpm / deg/s / Hz)](https://elysiatools.com/en/tools/angular-velocity-converter): Convert angular velocity (angular speed) between radian per second (rad/s, SI base), revolutions per minute (rpm = 2π/60 rad/s), degrees per second (deg/s = π/180 rad/s), and hertz (Hz, used as revolution per second = 2π rad/s). Converts via rad/s to the target unit and lists all four equivalents. Note: 1 Hz in angular-frequency context means one full revolution per second, so 1 Hz = 2π rad/s ≈ 6.283185307 rad/s. Reference: vinyl LP 33⅓ rpm ≈ 3.49 rad/s, car engine idle ~800 rpm ≈ 83.8 rad/s.
- [Duct Size Calculator (Flow Rate × Velocity)](https://elysiatools.com/en/tools/duct-size-calculator): Size a duct cross-section from the air flow rate Q and the design mean velocity v: area A = Q/v. Circular duct diameter D = √(4A/π). Rectangular duct with aspect ratio r = a/b gives b = √(A/r) and a = r·b, plus the ASHRAE equivalent diameter D_eq = 1.30·(a·b)^0.625/(a+b)^0.25. Flow rate in m³/s/m³/h/CFM, velocity in m/s; dimensions reported in mm and inches.
- [Fatigue Limit Calculator (Goodman / Gerber / Soderberg)](https://elysiatools.com/en/tools/fatigue-limit-calculator): Mean-stress fatigue correction under cyclic loading. Given stress amplitude σ_a, mean stress σ_m, and material σ_uts / σ_-1 (endurance limit) / σ_y, compute safety factors from three classical criteria: Modified Goodman (linear, conservative), Gerber (parabolic, better for ductile metals), and Soderberg (uses σ_y, most conservative). Reports the governing (smallest) value and whether the operating point lies inside the Goodman line.
- [Margin of Error Calculator](https://elysiatools.com/en/tools/margin-of-error-calculator): Calculate the margin of error for a mean or proportion from sample size, confidence level, and variability
- [One Sample t Test Calculator](https://elysiatools.com/en/tools/one-sample-t-test-calculator): Run a one-sample t test from raw data or summary statistics, including p-value and confidence interval
- [Paired t Test Calculator](https://elysiatools.com/en/tools/paired-t-test-calculator): Run paired-sample t tests from before and after values or precomputed paired differences
- [Pipe Pressure Drop Calculator (Darcy-Weisbach)](https://elysiatools.com/en/tools/pipe-pressure-drop-darcy): Compute the Darcy-Weisbach major (friction) pressure drop in a straight pipe: ΔP = f·(L/D)·(ρ·v²/2) Pa and head loss h_f = f·(L/D)·v²/(2g) m. The user supplies the Darcy friction factor f (not the Fanning factor). Length in m/km/ft, diameter in m/cm/mm/inch, density in kg/m³/g/cm³/lb/ft³ — all normalised to SI internally. Returns ΔP in Pa, kPa and bar, and head loss in m and ft. Gravity defaults to 9.81 m/s² and may be overridden.
- [Slope Stability Calculator (Infinite Slope)](https://elysiatools.com/en/tools/slope-stability-calculator): Calculate the infinite-slope factor of safety: FS = c'/(γ·z·cosβ·sinβ) + tanφ'/tanβ. Supports dry slopes and seepage parallel to the slope. Returns FS and a stable/marginal/unstable classification.

## Samples

- [Web Image Processing Python Samples](https://elysiatools.com/en/samples/web-image-processing-python): Web Python image processing examples using PIL/Pillow including reading, saving, resizing, and format conversion
- [Android Image Processing Java Samples](https://elysiatools.com/en/samples/android-image-processing-java): Android Java image processing examples including reading/saving images, scaling, and format conversion
- [Android Image Processing Kotlin Samples](https://elysiatools.com/en/samples/android-image-processing-kotlin): Android Kotlin image processing examples including reading/saving images, scaling, and format conversion
- [Web Image Processing Rust Samples](https://elysiatools.com/en/samples/web-image-processing-rust): Web Rust image processing examples including image read/save, scaling, and format conversion

## Related content

- [Pharmacokinetic Teaching and Dose-Model Checks](https://elysiatools.com/en/hubs/pharmacokinetic-dose-planning-calculators): Explore pharmacokinetic formulas, model assumptions, and result consistency for teaching or QA without giving medication dosing or clinical advice.