Activity Decay Calculator: Half-Life & Rate

This calculator works out the remaining activity of a radioactive sample after a given time, using its half-life. You enter the initial activity, the elapsed time, and the half-life, and it returns the remaining activity and the decay constant. The result is given in the same activity unit you entered (becquerel, curie, or any other).

Formula

The activity of a radioactive sample decreases exponentially over time. The formula used by the calculator is:

A(t) = A₀ × (1/2)^(t / T½)

Where:

• A(t) is the remaining activity after time t (in becquerel, curie, or any activity unit).
• A₀ is the initial activity at time zero (same unit as A(t)).
• t is the elapsed time (in seconds, minutes, hours, or any time unit – as long as it matches the half-life unit).
• T½ is the half-life (in the same time unit as t).

The decay constant λ (lambda) is related to the half-life by:

λ = ln(2) / T½

This is the probability per unit time that a nucleus will decay. The activity can also be written as A(t) = A₀ × e^(−λt), which is equivalent to the half-life form.

Worked Example

Using the default values from the calculator above:

• Initial activity (A₀) = 1000 Bq
• Elapsed time (t) = 60 seconds
• Half-life (T½) = 120 seconds

Step 1: Calculate the exponent: t / T½ = 60 / 120 = 0.5

Step 2: Raise 0.5 to that power: 0.5^0.5 = √0.5 ≈ 0.7071

Step 3: Multiply by initial activity: 1000 × 0.7071 = 707.1 Bq

The calculator shows 707.10 Bq, matching the result.

The decay constant λ = ln(2) / 120 ≈ 0.005776 s⁻¹.

When to Use This Calculator

This calculator is useful in several real-world situations:

• Nuclear medicine: estimating how much of a radiopharmaceutical remains in a patient or a vial after a delay.
• Radiation safety: determining when a contaminated sample has decayed to a safe level.
• Research: calculating the activity of a source after shipping or storage.
• Education: checking homework or lab results.

Remember that the calculator assumes a single isotope with a known half-life. If you have a mixture, you need to sum the contributions of each isotope separately.

How to Read the Result

The remaining activity tells you how many disintegrations per second are still occurring. For example, 707 Bq means 707 atoms decay each second. If you started with 1000 Bq, after one half-life (120 s) you would have 500 Bq, after two half-lives 250 Bq, and so on. The calculator interpolates for times that are not exact multiples of the half-life.

The decay constant λ is the probability per second that a given nucleus will decay. A larger λ means a shorter half-life and faster decay. You can use λ to find the mean lifetime (τ = 1/λ) or to compare different isotopes.

Common Mistakes and Unit Issues

The most frequent error is mixing time units. If the half-life is in hours but the elapsed time is in minutes, the result will be wrong. Always convert both to the same unit before entering them. The calculator does not do unit conversion for you.

Another mistake is using the decay constant λ instead of the half-life. The formula with λ is A(t) = A₀ × e^(−λt), which is mathematically equivalent, but if you enter λ where the calculator expects T½, the answer will be off by a factor of ln(2).

Rounding too early can also cause errors, especially when the exponent is large. Keep several significant figures during intermediate steps.

Finally, the activity unit (Bq, Ci, mCi) does not affect the calculation – it just scales the result. You can enter any activity unit, and the output will be in the same unit.

Limitations of the Formula

The exponential decay formula is accurate for a single radioactive isotope with a constant half-life. It assumes:

• The sample is not being replenished (no production of new radioactive atoms).
• The half-life is constant (true for most isotopes, but not for those with decay modes that change under extreme conditions).
• You are measuring the total activity, not the activity of a specific decay branch (if there are multiple decay paths, the total activity still follows the same half-life).

For very short half-lives (fractions of a second) or very long ones (billions of years), the formula still works, but practical measurement may be challenging.

Frequently Asked Questions

What is the difference between activity and dose?

Activity is the number of decays per second (measured in becquerel or curie). Dose is the energy deposited in tissue (measured in sievert or gray). The calculator only deals with activity. To estimate dose, you need additional factors like the type of radiation, energy, and distance.

Can I use this for carbon-14 dating?

Yes, but with caution. Carbon-14 has a half-life of about 5730 years. If you know the initial activity (usually assumed to be the same as living organisms) and measure the current activity, you can solve for the age. However, the calculator gives the remaining activity for a given time, not the age for a given activity. You would need to rearrange the formula or use a dedicated dating calculator.

What if I have a mixture of isotopes?

You need to calculate the activity of each isotope separately and add them together. The total activity is the sum of the individual activities. The calculator handles only one isotope at a time.

Why does the calculator use half-life instead of decay constant?

Half-life is more intuitive and widely published. The decay constant λ is derived from the half-life using λ = ln(2)/T½. Both are equivalent, but half-life avoids the need to remember the natural logarithm.

How do I convert between becquerel and curie?

1 curie (Ci) = 3.7 × 10¹⁰ becquerel (Bq). So to convert Bq to Ci, divide by 3.7 × 10¹⁰. The calculator does not perform this conversion – it assumes you enter the activity in a consistent unit.

What is the mean lifetime and how does it relate to half-life?

The mean lifetime τ is the average time a nucleus survives before decaying. It is τ = 1/λ = T½ / ln(2). For example, if the half-life is 120 s, the mean lifetime is about 173 s. The mean lifetime is longer than the half-life because it averages over all nuclei, including those that survive many half-lives.

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in a0 | Initial activity | Bq | 1000 | 0
in t | Elapsed time | s | 60 | 0
in h | Half-life | s | 120 | 0
out a | Remaining activity | Bq | a0 * 0.5^(t / h) | 2
out lambda | Decay constant | 1/s | ln(2) / h | 6