Radioactive Decay Calculator
Solve radioactive decay problems using half-life or decay constant. Find remaining amount, elapsed time, half-life, or λ with clear steps.
Calculation Steps
Practical Guide
Radioactive Decay Calculator: Half-Life, Decay Constant, and Remaining Amount
This guide explains how the Radioactive Decay Calculator works, what the outputs mean, and how to use it for real engineering and physics problems—whether you’re estimating remaining isotope mass, elapsed time, half-life, or decay constant.
Quick Start
The calculator is built around the exponential decay law. Use these steps to get correct results quickly and avoid the most common input mistakes.
- 1 Pick what you want to solve for in Solve For: remaining amount \(N(t)\), elapsed time \(t\), half-life \(t_{1/2}\), or decay constant \(\lambda\). The row for the unknown hides automatically.
- 2 Choose your Mode: Use half-life or Use decay constant. Only the relevant input stays visible.
- 3 Enter the initial amount \(N_0\). You can use any consistent unit (mol, g, kg, atoms, or “units” for generic problems).
- 4 Enter the other known value(s): a time \(t\), a remaining amount \(N(t)\), a half-life \(t_{1/2}\), or \(\lambda\), depending on your Solve For choice.
- 5 If you’re using mass units (g/mg/kg), enter the molar mass \(M\). This allows optional activity estimates in Bq.
- 6 Read the main result and scan Quick Stats for derived values like \(\lambda\), \(t_{1/2}\), mean lifetime \(\tau\), and activity \(A\).
- 7 Toggle Show Steps to see the exact equation path and substituted numbers that produced your result.
Tip: Keep \(N_0\) and \(N(t)\) in the same unit. If you switch one unit selector, switch both—otherwise the ratio \(N_0/N(t)\) becomes meaningless.
Watch the sign: \(t\) must be \(\ge 0\), \(\lambda>0\), and \(N(t)\le N_0\). If you enter a larger “remaining” amount than the initial value, the physics doesn’t allow a solution.
Choosing Your Method
Radioactive decay problems usually come in one of three practical “given–unknown” patterns. These map directly to the calculator’s Solve For options and determine which equation form you should rely on.
Method A — Exponential Decay (Direct)
Use this when you know \(N_0\), \(t\), and either \(\lambda\) or \(t_{1/2}\), and you want the remaining amount.
- Most common lab and field prediction.
- Works for any unit of amount or count.
- Simple, stable calculation.
- Requires a decay parameter (\(\lambda\) or \(t_{1/2}\)).
Method B — Log Form (Solve for Time)
Use when you know \(N_0\), a target \(N(t)\), and a decay parameter, and need the elapsed time.
- Best for dating, cooldown windows, and decay scheduling.
- Directly reveals sensitivity to the ratio \(N_0/N(t)\).
- Requires \(N(t)\le N_0\) and positive \(\lambda\).
Method C — Parameter Identification
Use when you have measurements at two times and want \(\lambda\) or \(t_{1/2}\). Engineers see this in decay-curve fitting, shielding validation, or isotope selection.
- Lets you back-calculate unknown decay properties.
- Useful when datasheets are missing or uncertain.
- Measurement noise can dominate when decay is slow.
- Requires reliable \(N_0\) and \(N(t)\).
Engineering note: All three methods assume first-order decay with a constant \(\lambda\). If your isotope is produced or removed by another process (activation, chemical separation, biological uptake), you need a coupled model, not a single-term decay law.
What Moves the Number the Most
The calculator’s outputs are driven by a small set of variables. Understanding these “levers” helps you sanity-check results and interpret real-world scenarios.
Bigger \(\lambda\) means faster decay. A 2× increase in \(\lambda\) halves the characteristic time scale.
Half-life is just another way to express the same rate: \(\lambda=\ln2/t_{1/2}\). Long half-life → slow decay.
Decay is exponential, so early time changes have more impact than equal changes later.
For solving time or parameters, the ratio dominates. Small errors in \(N(t)\) can create large shifts in \(\ln(N_0/N)\).
The math uses normalized SI internally. Mixing units for \(N_0\) and \(N(t)\) is the fastest way to get a wrong answer.
If you enter mass, the calculator estimates atoms via \(n=m/M\) to compute activity \(A=\lambda N\). If \(M\) is wrong, activity is wrong, but remaining fractions are still correct.
Worked Examples
These examples mirror typical homework and field calculations. Follow the steps and compare with the calculator’s result/steps view.
Example 1 — Remaining Amount from Half-Life
- Initial amount: \(N_0 = 100\) g of I-131
- Half-life: \(t_{1/2} = 8.0\) days
- Elapsed time: \(t = 24\) days
- Solve for: Remaining amount \(N(t)\)
In the calculator: set Solve For to “Remaining amount,” Mode to “Use half-life,” enter \(N_0=100\) g, \(t=24\) day, \(t_{1/2}=8\) day. You should see about 12.5 g remaining.
Example 2 — Time to Reach a Target Activity
- Initial amount: \(N_0 = 2.0\times10^{20}\) atoms
- Decay constant: \(\lambda = 1.2\times10^{-6}\ \text{s}^{-1}\)
- Target remaining: \(N(t) = 5.0\times10^{19}\) atoms
- Solve for: Elapsed time \(t\)
In the calculator: choose Solve For “Elapsed time,” Mode “Use decay constant,” enter \(N_0\), \(N(t)\), and \(\lambda\). The result should be ~13.4 days (or its equivalent in your chosen unit).
Common Layouts & Variations
Real work rarely matches a single textbook pattern. This table summarizes typical configurations and how to interpret calculator results in each.
| Scenario / Configuration | Typical Inputs | Best Solve For | Notes & Pros/Cons |
|---|---|---|---|
| Medical isotope dose planning (e.g., I-131, Tc-99m) | \(N_0\) (activity or mass), \(t\), \(t_{1/2}\) | \(N(t)\) | Assumes no biological clearance. If body eliminates isotope, decay is faster in practice. |
| Radiological cooldown / storage window | \(N_0\), target \(N(t)\), \(t_{1/2}\) or \(\lambda\) | \(t\) | Good for planning when shielding or access becomes safe enough. |
| Isotope identification from two measurements | \(N_0\), \(N(t)\), \(t\) | \(\lambda\) or \(t_{1/2}\) | Noise sensitive. Use averaged measurements if possible. |
| Long-term waste classification | \(N_0\), \(t\), \(t_{1/2}\) | \(N(t)\) | Exponential decay dominates; small \(t_{1/2}\) errors compound over decades. |
| Activation products in materials | \(N_0\), \(t\), \(\lambda\) | \(N(t)\) | Only valid after activation stops. During activation you need a source term. |
- Check that \(t\) and \(t_{1/2}\) are in the same time base internally (calculator converts for you).
- Verify \(N(t)\le N_0\) unless you are modeling growth from a source term (not covered here).
- If you use mass units, confirm molar mass and isotopic purity.
- Round results only after the final step; don’t round \(\lambda\) early.
Specs, Logistics & Sanity Checks
The decay law is simple, but using it responsibly requires checking assumptions and measurement context. These notes help ensure your calculator result matches physical reality.
Assumptions Built Into the Calculator
- Single-isotope, first-order decay with constant \(\lambda\).
- No production term (no ongoing activation or breeding).
- No removal term (chemical separation, biological clearance, ventilation).
- Amounts and counts refer to the same isotope over time.
Measurement Reality Checks
- Detector dead time and pile-up can bias \(N(t)\) high or low.
- Background subtraction matters most at low activity.
- Mass-based \(N_0\) assumes known isotopic fraction.
- Temperature/pressure don’t affect nuclear decay, but can affect sampling.
Engineering Safety Notes
- Use conservative inputs when planning access or shielding.
- Consider regulatory limits on dose and transport.
- Document the half-life source (datasheet, IAEA/NNDC, etc.).
- For mixed isotopes, model each separately and sum activities.
Sanity shortcut: after \(n\) half-lives, the remaining fraction is \(2^{-n}\). If your calculator says 12.5% remaining, that’s exactly 3 half-lives.
When this tool is not enough: If your system has a source term \(S\), your governing equation becomes \(dN/dt = S – \lambda N\). That requires a different calculator or a spreadsheet model.