Activity Equation: Definition, Formula & Units

This calculator determines the activity of a radioactive sample using the fundamental decay equation. You enter the number of radioactive atoms (or mass) and the half-life, and it returns the activity in becquerels (Bq), curies (Ci), or disintegrations per second, along with the decay constant. The activity equation is central to nuclear physics, radiochemistry, and medical imaging, and this tool simplifies the calculation while explaining the underlying physics.

Formula

The activity A of a radioactive sample is the rate at which nuclei decay, given by:

A = λN

where:

• A = activity (becquerels, Bq, or disintegrations per second, dps)
• λ = decay constant (s^-1)
• N = number of radioactive atoms present

The decay constant is related to the half-life (T_1/2) by:

λ = ln(2) / T_1/2

Substituting gives the combined equation:

A = N * ln(2) / T_1/2

Units: if N is dimensionless (count) and T_1/2 is in seconds, then A is in becquerels (1 Bq = 1 decay/s). For half-life in minutes, convert to seconds first (multiply by 60).

Worked Example

Using the default values from the calculator: N = 1,000,000 atoms, T_1/2 = 100 seconds.

1. Calculate the decay constant: λ = ln(2) / 100 = 0.6931 / 100 = 0.006931 s^-1
2. Multiply by the number of atoms: A = 0.006931 * 1,000,000 = 6,931 Bq

The calculator shows 6,931.47 Bq (rounding to two decimals). This means about 6,931 disintegrations occur every second.

When to Use the Activity Equation

The activity equation is used whenever you need to know how many decays per second a sample emits, which is essential for:

• Radiation safety: estimating exposure from a known quantity of a radioisotope.
• Medical imaging: calculating the dose of a radiopharmaceutical (e.g., technetium-99m) needed for a scan.
• Carbon dating: determining the remaining activity of carbon-14 in organic material.
• Nuclear power: monitoring fuel rod activity and fission product inventory.
• Research: measuring the purity of a radioactive tracer.

The equation applies to any sample where the decay follows first-order kinetics, which is true for all radioactive isotopes. It assumes a pure isotope with a known half-life; mixtures require summing activities of each component.

How to Read the Result

The activity value tells you the number of nuclear disintegrations per second. In SI units, 1 becquerel (Bq) equals 1 decay per second. A common alternative unit is the curie (Ci), where 1 Ci = 3.7 × 10¹⁰ Bq (originally defined as the activity of 1 gram of radium-226). For practical purposes, activities are often expressed in millicuries (mCi) or microcuries (µCi).

Remember that activity decreases over time. The equation gives the activity at the moment you measure N. If you need activity at a future time, use the decay equation: A(t) = A₀ * (1/2)^(t / T_1/2).

Common Mistakes and Misconceptions

Here are frequent errors when using the activity equation:

• Unit mismatch: Forgetting to convert half-life to seconds. If your half-life is in hours, multiply by 3600 before using the formula.
• Confusing N with mass: The equation requires the number of atoms, not the mass. To get N from mass, use N = (mass / molar mass) * Avogadro’s number (6.022 × 10²³ mol^-1).
• Using ln(2) incorrectly: The decay constant is ln(2) divided by half-life, not half-life divided by ln(2).
• Ignoring daughter products: If the daughter is also radioactive, the total activity may be higher than the parent alone.

A common misconception is that activity is the same as the number of atoms. In reality, two samples with the same number of atoms can have very different activities if their half-lives differ—a short half-life means more decays per second.

Limits of the Equation

The activity equation is valid for a single radioactive isotope under these conditions:

• The sample is large enough that statistical fluctuations are negligible (typically > 10⁴ atoms).
• The half-life is much shorter than the age of the sample, so the activity is measurable.
• The decay is not affected by external factors (temperature, pressure, chemical state) – true for most isotopes, but electron capture can be slightly affected by chemical environment.
• For extremely short half-lives (femtoseconds), measurement becomes challenging, but the equation still holds.

For mixtures, you must sum the activities of each isotope. For samples with a constant production rate (e.g., in a reactor), you need a more complex equation that includes production and decay.

Frequently Asked Questions

What is the difference between activity and count rate?

Activity is the total number of decays per second in the sample, while count rate is the number of decays detected by an instrument per second. The count rate is always lower than activity because detectors have limited efficiency and geometry (not all emitted particles hit the detector). The relationship is: count rate = activity × detection efficiency.

How do I convert activity from Bq to Ci?

To convert becquerels to curies, divide by 3.7 × 10¹⁰. For example, 1 mCi = 3.7 × 10⁷ Bq. Conversely, multiply curies by 3.7 × 10¹⁰ to get Bq. Always check which unit your source uses—medical doses are often in mCi, while SI standards use Bq.

Can I use this equation for a sample with multiple isotopes?

No, the equation as written applies to a single isotope. For a mixture, calculate the activity of each isotope separately using its own N and half-life, then sum the results. If the isotopes decay into each other (a decay chain), the calculation becomes more complex and requires solving differential equations.

Why is the decay constant λ equal to ln(2)/half-life?

Because activity decreases exponentially: N(t) = N₀e^-λt. At the half-life, N(T_1/2) = N₀/2. Substituting gives e^-λT_1/2 = 1/2, so -λT_1/2 = ln(1/2) = -ln(2), hence λ = ln(2)/T_1/2.

What is the activity of 1 gram of radium-226?

Radium-226 has a half-life of 1600 years (about 5.05 × 10¹⁰ seconds). One gram contains approximately 2.67 × 10²¹ atoms. Using the equation, A = (2.67 × 10²¹ × 0.693) / (5.05 × 10¹⁰) ≈ 3.66 × 10¹⁰ Bq, which is very close to the historical definition of 1 curie (3.7 × 10¹⁰ Bq).

How does activity change over time?

Activity decreases exponentially: A(t) = A₀ × (1/2)^(t/T_1/2). After one half-life, activity is half; after two, a quarter, and so on. To find the time for activity to drop to a certain level, use t = (T_1/2 / ln(2)) × ln(A₀/A).

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in n | Number of atoms | atoms | 1000000 | 0
in t_half | Half-life | seconds | 100 | 0
out A | Activity | Bq | n * (ln(2) / t_half) | 2
out lambda | Decay constant | 1/s | ln(2) / t_half | 4