How to Calculate Activity (Physics)

This guide explains how to calculate the activity of a radioactive sample, which is the rate at which its nuclei decay. You enter the decay constant and the number of undecayed nuclei (or derive them from half-life and mass), and you get the activity in becquerels (Bq) or curies (Ci). We’ll cover the formula, a worked example, common pitfalls, and troubleshooting.

Formula

Activity (A) is the product of the decay constant (λ) and the number of radioactive nuclei (N):

A = λ × N

• A – activity, measured in becquerels (Bq) or curies (Ci). 1 Bq = 1 decay per second.
• λ – decay constant, probability of decay per unit time, unit s⁻¹. It relates to half-life (t½) by λ = ln(2) / t½ ≈ 0.693 / t½.
• N – number of undecayed nuclei (dimensionless count).

If you know the mass (m) and molar mass (M) of a pure isotope, N = (m / M) × Avogadro’s number (6.022×10²³ mol⁻¹).

Worked Example

Using the calculator defaults: N = 1×10²² nuclei, half-life = 1000 seconds.

1. Compute λ = 0.693 / 1000 = 6.93×10⁻⁴ s⁻¹.
2. Multiply by N: A = 6.93×10⁻⁴ × 1×10²² = 6.93×10¹⁸ Bq.
3. The calculator shows 6.93×10¹⁸ Bq (rounded to 2 decimals in scientific notation).

That means about 6.93 quintillion decays per second – a very high activity, typical for a large mass of a short-lived isotope.

When to Use This Calculation

Use activity calculations in nuclear physics, radiology, environmental monitoring, and radiometric dating. You need the number of radioactive atoms present and their decay constant (or half-life). If you have a sample that is a mixture of isotopes, you must calculate each isotope’s activity separately and sum them.

Common Mistakes

• Unit mix-ups: Half-life in years but decay constant in seconds – always convert to the same time unit.
• Using ln(2) incorrectly: Some use 0.693, others 0.5 – 0.693 is correct for half-life.
• Forgetting to convert mass to number of nuclei: If you enter grams directly, you get nonsense. Use N = (mass / molar mass) × Avogadro’s number.
• Rounding too early: Rounding λ to 3 significant figures before multiplying can change your answer by a few percent.
• Ignoring decay during measurement: If the half-life is short compared to your measurement time, activity changes; use the initial activity or integrate.

Tips

• Always express half-life in seconds for SI activity (Bq). For curies, you can convert after: 1 Ci = 3.7×10¹⁰ Bq.
• Use scientific notation for very large or small numbers – the calculator handles it.
• If you have a sample mass, look up the molar mass of the exact isotope (e.g., ²³⁸U has molar mass 238 g/mol, not 238.03 for natural uranium).
• For a mixture, calculate each isotope’s activity separately and add – do not average half-lives.
• Remember that activity decreases over time: A(t) = A₀ × e^(-λt). This formula gives the activity at a specific instant.

Troubleshooting

My result seems too high or too low. Check the units: half-life in seconds? If you used years, multiply by 3.156×10⁷ s/year. Also verify N – if you entered a mass, did you convert to nuclei?

The calculator shows “NaN” or “Infinity”. This usually means a zero or negative input. Half-life cannot be zero or negative. Ensure N is positive.

I have a half-life in minutes but want Bq. Convert minutes to seconds (multiply by 60) before entering.

My sample is not pure – it has stable isotopes mixed in. The formula only counts radioactive nuclei. Determine the fraction of radioactive atoms in your sample and multiply N by that fraction.

I get a different answer than a textbook. Check if the textbook uses curies instead of becquerels. Also verify the half-life value – different sources may use slightly different values (e.g., ²³⁸U half-life varies by source).

Frequently Asked Questions

What is the difference between activity and dose? Activity is the number of decays per second. Dose (e.g., sieverts) measures the biological effect of radiation, which depends on energy and tissue type – not directly calculated from activity alone.

How do I convert activity from Bq to Ci? Divide by 3.7×10¹⁰. For example, 1 Ci = 3.7×10¹⁰ Bq.

Can I calculate activity from half-life alone? No, you also need the number of radioactive nuclei. Half-life gives λ, but activity = λN.

What if the half-life is very long, like billions of years? Convert to seconds first. For example, ²³⁸U half-life ≈ 4.468×10⁹ years × 3.156×10⁷ s/year ≈ 1.41×10¹⁷ s. Then λ ≈ 4.9×10⁻¹⁸ s⁻¹.

Does activity depend on temperature or pressure? For most radioactive decay, no – it is a nuclear property independent of chemical environment. Exceptions exist for electron capture, where chemical state can slightly affect half-life, but generally negligible.

Why does the calculator use 0.693 instead of ln(2)? They are the same number (ln(2) ≈ 0.693147). The calculator uses 0.693 for simplicity, which is accurate to three significant figures.

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in N | Number of nuclei | nuclei | 1e22 | 0
in hl | Half-life | s | 1000 | 0
out activity | Activity | Bq | (0.693 / hl) * N | 2
out lambda | Decay constant | s^-1 | 0.693 / hl | 4