Radioactive Decay Activity Formula and Calculator

This calculator estimates the activity of a radioactive sample from its initial activity, elapsed time, and half-life. Enter those three values to get the remaining activity and the fraction of the original activity left, using the exponential decay law.

Formula

The activity after time t is the initial activity multiplied by an exponential decay factor:

A(t) = A₀e^−λt = A₀ × 2^−t/T_1/2

Here, A(t) is activity at elapsed time t, in becquerels (Bq); A₀ is activity at the start, in Bq; λ is the decay constant, in s⁻¹ when time is entered in seconds; and T_1/2 is half-life, in seconds. The relationships are λ = ln(2)/T_1/2 and e is the base of natural logarithms. The calculator uses 0.6931471805599453 as ln(2). Its second output, f, is the dimensionless fraction remaining.

Worked Example

Use the calculator’s defaults: initial activity 1000 Bq, elapsed time 86,400 s, and half-life 604,800 s. Since the half-life is seven days and the elapsed time is one day, the sample has aged by one-seventh of a half-life.

First calculate the exponent: −0.6931471805599453 × 86,400 / 604,800 = −0.0990210257943.

Then calculate the remaining fraction: e^{−0.0990210257943} ≈ 0.9057237. Thus, activity is 1000 Bq × 0.9057237 ≈ 905.72 Bq, which displays as 905.72 Bq to two decimal places. The fraction output displays as 0.9057 to four decimal places.

Assumptions

• The sample contains a single radionuclide with a constant, known half-life.
• Decay follows the standard exponential law; no production or addition of the radionuclide occurs during the interval.
• The initial activity is specified at the start of the elapsed time, and all time values use seconds.
• The activity is the nuclear decay rate, not a detector count rate or a dose rate.

What Activity Means

Radioactive activity is the expected number of nuclear transformations per unit time. One becquerel is one transformation per second. The older unit, the curie, is defined as exactly 3.7 × 10¹⁰ Bq. Activity describes how frequently decays occur; it does not directly state how much radiation reaches a person or detector.

For a sample with N radioactive nuclei, activity is also written A = λN. As nuclei decay, N declines, so activity declines at the same exponential rate. A measured sample may contain multiple radionuclides, in which case each component has its own decay law and the total activity is the sum of their activities.

How to Use the Calculator

• Initial activity: enter the activity at the reference time. Use Bq; convert other units before entering them.
• Elapsed time: enter the nonnegative time from that reference point, in seconds.
• Half-life: enter the radionuclide’s half-life, in seconds. Half-life values depend on the isotope, so consult a reliable nuclear data reference.
• Read the result: the first output is activity after the elapsed time. The second is the proportion remaining; multiply by 100 to express it as a percentage.

The calculator accepts any nonnegative activity and elapsed time, and a positive half-life. For example, if a half-life is given in days, multiply days by 86,400 to convert to seconds. If activity is in kilobecquerels, multiply by 1,000 to obtain Bq. Keeping units consistent is essential: the ratio of elapsed time to half-life must be dimensionless.

Interpreting Results and Related Quantities

After one half-life, the activity is half its starting value; after two, one quarter; after three, one eighth. For any interval, the exponential formula gives a continuous result rather than rounding time to a whole number of half-lives. A short interval relative to the half-life produces a small decrease, while a long interval can make the remaining activity extremely small.

Activity is not the same as the number of radioactive atoms, although the two are connected by A = λN. If the decay constant is known, the estimated atom count is N = A/λ. Nor is activity the same as absorbed dose, which concerns energy deposited per unit mass, or effective dose, which also accounts for biological considerations. Shielding, radiation type, distance, geometry, and exposure duration matter when assessing radiation fields; activity alone cannot determine a person’s dose.

A detector’s counts per second are also not automatically equal to Bq. A detector usually registers only some decays, and its efficiency, geometry, background, and calibration affect the measured count rate. Converting counts to activity requires appropriate measurement information. For bench work, follow your lab method, the safety data sheet (SDS) and your supervisor.

Common Errors and Limits

• Mixing time units: entering an elapsed time in days while the half-life is in seconds gives a wrong exponent. Convert both to the same unit.
• Using a half-life for the wrong isotope: elements can have many isotopes with very different half-lives. Verify the isotope, not just the element name.
• Confusing activity with mass or dose: a given mass does not have a universal activity; it depends on the radionuclide and its amount. Dose requires additional physical and biological information.
• Overinterpreting tiny results: the equation predicts a mean activity. At very low numbers of remaining atoms, actual decays are statistical and may vary substantially from that mean.
• Ignoring uncertainty: initial activity, elapsed time, and published half-life may each carry uncertainty. The displayed digits are calculation precision, not necessarily measurement accuracy.

The model describes radioactive decay itself. It does not account for chemical removal, leakage, shielding, detector response, environmental transport, or a changing supply of radionuclides from another decay. In a decay chain, a daughter isotope may be produced as its parent decays; predicting total activity then requires the relevant chain equations, not this single-isotope calculation alone.

Frequently Asked Questions

What is the formula for radioactive decay activity?
A(t) = A₀e^−λt, or equivalently A(t) = A₀ × 2^−t/T_1/2. Activity is in Bq when it represents decays per second.

How do you calculate the decay constant?
Divide the natural logarithm of 2 by the half-life: λ = ln(2)/T_1/2. The time unit used for λ is the inverse of the unit used for half-life.

How much activity remains after one half-life?
Half of the initial activity remains. After two half-lives one quarter remains, and after three one eighth remains, assuming the single-isotope decay model.

Is radioactive activity the same as radiation dose?
No. Activity counts nuclear transformations per second. Dose concerns energy absorbed and, for effective dose, biological weighting; it cannot be calculated from activity alone.

Can I use this formula for a mixture of isotopes?
Calculate each isotope’s activity separately using its own initial activity and half-life, then add the activities at the time of interest. A decay chain may need a more complete model because daughter nuclides are produced as parents decay.

Related

• What Does the Rydberg Equation Calculate? Hydrogen Line Wavelength Calculator
• Weighted Average Atomic Mass Calculator With Abundance
• Isotope Atomic Mass Calculator: Mass of One Atom From amu

in a0 | Initial activity | Bq | 1000 | 0
in t | Elapsed time | s | 86400 | 0
in th | Half-life | s | 604800 | 0
out a | Activity after elapsed time | Bq | a0 * exp(-0.6931471805599453 * t / th) | 2
out f | Fraction remaining | - | exp(-0.6931471805599453 * t / th) | 4