Table of Contents

Half-Life Calculator

Q: How many half-lives does it take to reach 1% remaining?

It takes approximately 6.64 half-lives for a substance to decay to 1% of its initial amount. This is calculated by n = log2(1/0.01) = log2(100) ≈ 6.64$ .

Q: Why is ln(2) used in the decay constant formula?

The ln(2) (or 0.693 ) factor arises because the half-life is the time at which the quantity is 1/2 of the initial amount. The natural logarithm ln is used because the decay process is a continuous, natural exponential decay process, e¯λt .

Q: What is the elimination rate constant k in pharmacokinetics?

In pharmacokinetics, the elimination rate constant k describes how quickly a drug is removed from the body. It is mathematically equivalent to the decay constant λ in physics, and they are related by the formula k = ln(2) / T1/2(elimination) .

Q: How do I convert the decay constant $\lambda$ to half-life T1/2?

To convert the decay constant λ (units of 1/time ) into the half-life T1/2 (units of time} ), use the formula T1/2 = ln(2)/λ .

Last updated: October 24, 2025

TL;DR: Use our Half-Life Calculator to compute the remaining amount, time, half-life ( $T_{1/2}$ ), or decay constant (λ) for first-order decay. Enter any known values—initial amount/activity, remaining amount/activity, time, or $T_{1/2}$ and get step-by-step results using the core exponential decay formulas.

Half-life ( $T_{1/2}$ ) is the time required for a decaying quantity, such as a radioactive isotope or a drug concentration, to fall to half its initial value. For first-order processes, the remaining quantity $N(t)$ is calculated using , and the decay constant λ is directly related by

Half‑Life Calculator

Compute radioactive decay, time to thresholds and more

Inputs

ModeChoose what you want to calculate.

Decay constant (λ)

Provide λ if known (units inverse of time) 1/time.

Initial quantity/activity (N₀ or A₀)

Enter a positive number; units are optional.

Remaining quantity/activity (N(t) or A(t))

Enter the remaining quantity or activity.

Half‑life (T_1/2)

Enter a positive half‑life duration.

Time elapsed (t)

Time since start of decay.

Target quantity/activity

Enter the target amount you wish to reach.

Results

The most comprehensive Half-Life Calculator for solving radioactive decay, pharmacokinetics, and first-order chemical reactions. Use the tool below to find the remaining amount/activity, the time elapsed, the half-life ( $T_{1/2}$ ), or the decay constant λ.

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Understanding the Core Half-Life Formulas

The half-life ( $T_{1/2}$ ) describes the exponential decay of a quantity, such as the mass of a radionuclide or the concentration of a drug in the body. This process is typically governed by first-order kinetics. The calculator employs two equivalent mathematical models.

Half-Life Formula (Base 1/2)

This form is the most intuitive, directly relating the elapsed time to the number of half-lives that have occurred, $n = t/T_{1/2}$ .

Where:

$N(t)$ is the Remaining quantity (or activity) after time $t$ .
$N_0$ is the Initial quantity (or activity).
$T_{1/2}$ is the Half-Life.
$t$ is the Time elapsed.

Exponential Decay Formula (Base $e$)

This form uses the natural exponential function and the decay constant ( $\lambda$ ), which is the fractional rate of decay. This form is common in differential equations and advanced chemistry/physics.

The Relationship between Half-Life and Decay Constant

The two formulas are fundamentally linked. Setting the remaining amount $N(t)$ to $0.5 N_0$ when $t = T_{1/2}$ yields the essential relationship:

This conversion is crucial for moving between half-life-based calculations (common in basic labs and nuclear medicine) and the continuous exponential model (common in reactor physics and pharmacokinetics).

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How the Calculator Works: Solving for the Inverse Problems

Our calculator’s multi-mode solver is specifically designed to handle the inverse problems—solving for time ( $t$ ), half-life ( $T_{1/2}$ ), or the decay constant λ. This goes beyond the basic “remaining amount” problem.

Time Elapsed (t) from Half-Life

To find the time it takes for a substance to decay from an initial amount $N_0$ to a remaining amount $N(t)$ , we rearrange the Base $1/2$ formula using logarithms:

Start:
Isolate the Decay Factor:
Take $\ln$ of both sides:
Simplify using log rules:

Half-Life (T 1/2) from Two Measurements

Similarly, if you know the initial amount $N_0$ , the remaining amount $N(t)$ , and the time elapsed $t$ , you can calculate the half-life:

Decay Constant (λ) from Half-Life

The decay constant is solved directly:

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Unit Conversion, Sig Figs, and Activity (Bq/Ci) in Half-Life Calculations

Robust Unit Conversions

Our calculator features robust unit toggles for time (seconds, minutes, hours, days, years) and activity ( $\text{Bq} \leftrightarrow \text{Ci}$ ).

Time: All time inputs are internally converted to a common base unit (e.g., seconds) before calculation.
Activity: $1 \text{ Ci} = 3.7 \times 10^{10} \text{ Bq}$ . The calculator allows input/output in either unit, automatically handling the conversion.
Mass/Moles: The calculator can optionally convert between mass (grams) and moles using the substance’s molar mass, which is critical for chemistry applications.

Significant Figures and Scientific Notation

The results are displayed with user-controlled significant figures to ensure academic precision. Large numbers (common in initial amounts, e.g., $10^{23}$ atoms) or tiny numbers (common in decay constants, e.g., $10^{-9} \text{ 1/y}$ ) are automatically rendered in scientific notation for clarity.

The Role of Activity A(t)

In nuclear physics and medicine, it is often more important to track the activity $A(t)$ —the number of decays per unit of time—rather than the number of atoms $N(t)$ . Activity is related to the number of atoms by:

A(t) = \lambda N(t)

Since

N(t)

decays exponentially,

A(t)

also decays exponentially:

. The half-life for activity is the same as the half-life for the number of atoms.

Effective vs Physical vs Biological Half-Life

For applications in nuclear medicine and pharmacokinetics (PK), you must consider three types of half-life:

1. Physical Half-Life

Definition: The time required for a radioactive substance (radionuclide) to decay to half its initial activity due to nuclear instability. This is the radioactive decay half-life used in basic physics.

2.Biological Half-Life

Definition: The time required for a substance (radioactive or not, e.g., a drug or toxin) to reduce to half its amount in the body via natural elimination processes (metabolism, excretion, etc.).

3.Effective Half-Life

Definition: The time required for a radionuclide’s activity in the body to be reduced by half, taking both physical decay and biological elimination into account. This is the value used for calculating radiation dose in a patient.
Formula (Inverse Addition): The rates of decay and elimination are additive, leading to this inverse relationship for their half-lives:

Worked Examples: Calculate Time, Decay Constant, and Remaining Amount

Example 1: Finding Remaining Quantity (Simple Decay)

Scenario: Iodine-131 ( $T_{1/2} = 8.02 \text{ days}$ ) is used in a lab with an initial activity of $1500 \text{ Ci}$ . What is the remaining activity after $10 \text{ minutes}$ ?

Example 2: Finding Half-Life (from Two Measurements)

Scenario: A sample of an unknown isotope decays from $1500 \text{ Bq}$ to $250 \text{ Bq}$ over a period of $10 \text{ minutes}$ . What is its half-life?

Interactive Half-Life Graph and Decay Tables

Interactive Decay Graph

The interactive graph below plots the decay of $N(t)$ over time. Use the options to switch the Y-axis between linear (for visualization of the initial rapid drop) and semi-log (where the exponential decay plots as a straight line, useful for confirming first-order kinetics).

Markers: The plot includes vertical and horizontal lines marking the values at $1 T_{1/2}, 2 T_{1/2}, 3 T_{1/2}$ , etc.
Export: Data can be exported as a PNG image or a CSV file for further analysis.

Percent Remaining vs. Half-Lives Table

This table provides quick reference values for common decay thresholds.

Number of Half-Lives (n)	Fraction Remaining	Percent Remaining (%)
1	1/2	50.0%
2	1/4	25.0%
3	1/8	12.5%
4	1/16	6.25%
5	1/32	3.125%
10	1/1024	0.0977%

Common Half-Life Calculation Pitfalls & QA

To ensure accurate results, be aware of these common errors:

Mixed Units: Always ensure that $t$ and $T_{1/2}$ are in the same time unit (seconds, minutes, years, etc.) before calculation, or use our unit toggle to let the calculator handle the conversion.
$\ln$ vs. $\log_{10}$ : The decay constant λ is derived using the natural logarithm ( $\ln$ ), not the base-10 logarithm ( $\log_{10}$ ). Using the wrong log function will result in a factor of $\ln(10)/\ln(2) \approx 3.32$ error.
Non-First-Order Processes: The formulas on this page only apply to first-order kinetics (where the decay rate is proportional to the concentration/amount). Zero-order or second-order reactions require different kinetic equations. In pharmacokinetics, the elimination half-life is usually only constant for first-order elimination.

Practical Use Cases: Radiometric Dating, PK Elimination, and Decay

Radiometric Dating

Carbon-14 Dating: Carbon-14 ( $T_{1/2} \approx 5,730 \text{ years}$ ) dating uses the remaining concentration of C-14 in an organic artifact compared to its initial concentration (in a living organism) to determine its age. This requires solving for the time elapsed ( $t$ ).

Nuclear Medicine and Radiation Safety

Technetium-99m ( $\text{Tc-99m}$ ): $\text{Tc-99m}$ is a short-lived isotope ( $T_{1/2} = 6.01 \text{ hours}$ ) used in imaging. Dose planning requires calculating the effective half-life to determine how long the radio-pharmaceutical remains in the patient’s system.

Pharmacokinetics (PK)

Elimination Half-Life: The half-life of a drug is its elimination half-life and is often denoted as $T_{1/2}(\text{elimination})$ . It is used to determine drug dosing frequency, the time to reach steady-state concentration, and the time for the drug to be virtually eliminated (about $5$ to $7$ half-lives). The elimination rate constant $k$ is equivalent to the decay constant λ in physics: $k = \lambda$ .

Step-by-Step Guide: Using the Half Life Calculator

Choose What to Solve For: Use the Mode dropdown menu (e.g., “Remaining quantity/activity,” “Half-life,” or “Time to target quantity”) to set the goal of your calculation.
Enter Known Values & Units: Input the initial amount ( $N_0$ ), remaining amount ( $N(t)$ ), half-life ( $T_{1/2}$ ), or time elapsed ( $t$ ). Use the unit toggles (s, min, h, d, y, Bq, Ci) to match your input data.
Get Results with Steps: Click Calculate. The results will include the final answer, the intermediate number of half-lives ( $n=t/T_{1/2}$ ), and the underlying algebraic formula with your numbers substituted.
Inspect Graph: Review the interactive decay curve to visualize the process.
Export or Adjust: Use the Copy options for your data, or hit Reset to start a new calculation.

Frequently Asked Questions (FAQs)

What is the half-life formula?

How many half-lives does it take to reach 1% remaining?

It takes approximately $6.64$ half-lives for a substance to decay to $1%$ of its initial amount. This is calculated by $n = log2(1/0.01) = log2(100) \approx 6.64$$ .

How do you calculate the time from half-life?

Why is ln(2) used in the decay constant formula?

The $ln(2)$ (or $0.693$ ) factor arises because the half-life is the time at which the quantity is $1/2$ of the initial amount. The natural logarithm $ln$ is used because the decay process is a continuous, natural exponential decay process, $e¯λt$ .

How is effective half-life different from physical half-life?

Physical half-life is the decay time due to nuclear instability alone. Effective half-life is shorter because it includes both physical decay and the biological elimination of the substance from an organism.

What is the elimination rate constant k in pharmacokinetics?

In pharmacokinetics, the elimination rate constant $k$ describes how quickly a drug is removed from the body. It is mathematically equivalent to the decay constant λ in physics, and they are related by the formula $k = ln(2) / T1/2(elimination)$ .

What if the decay is not first-order?

If the decay is not first-order, the half-life is not constant and the formulas on this page do not apply. For example, zero-order reactions decay at a constant rate instead of a constant fraction per unit of time.

How many half-lives until a substance is "gone"?

A substance undergoing exponential decay never technically reaches exactly zero. However, it is considered virtually eliminated after $5$ to $7$ half-lives, when the remaining fraction is well below $1%$ .

How do I convert the decay constant $\lambda$ to half-life T1/2?

To convert the decay constant λ (units of $1/time$ ) into the half-life $T1/2$ (units of $time}$ ), use the formula $T1/2 = ln(2)/λ$ .

Important Disclaimers and Privacy Note

Educational Use Only: The calculations provided by this tool are for educational, general planning, and estimation purposes only. They are not intended for clinical dosing, nuclear waste management, or radiation safety assessments.

Safety: For any application involving radioactive materials or patient health, always follow official institutional protocols, regulatory guidance (e.g., NRC, FDA), and professional medical or engineering advice.

Privacy: All calculations are performed in-browser (client-side). We do not collect, store, or transmit your input values or results to our servers. Your privacy is guaranteed.

Authoritative References

Reference Category	Source Title & Author	Direct/Publisher Link	Purpose in Content
Nuclear Physics	Knoll, Glenn F. Radiation Detection and Measurement. 4th ed., Wiley, 2010.	Publisher: Wiley Online Library – Radiation Detection and Measurement, 4th Edition	Core decay physics, activity concepts, and $\text{Bq}$ / $\text{Ci}$ unit handling.
Pharmacokinetics	Gibaldi, Milo, and Donald V. Perrier. Pharmacokinetics. 2nd ed., Marcel Dekker, 1982.	Publisher: Routledge – Pharmacokinetics, Second Edition	Biological and Effective Half-Life definitions and the kinetics (λ $\lambda \leftrightarrow k$ ) context.
Fundamental Constants	National Institute of Standards and Technology (NIST). The NIST Reference on Constants, Units, and Uncertainty.	Direct Link: NIST Fundamental Physical Constants	Standard physical constants, SI unit definitions, and conversion factors.
Nuclear Safety/Data	International Atomic Energy Agency (IAEA). Safety Standards Series.	Direct Link: IAEA Safety Standards	Context for nuclear medicine applications, radiation safety, and general decay data reliability.