Radioactive Decay Equation: Formula and Examples

The radioactive decay equation is N(t) = N₀e^−λt, where N₀ is the starting number of radioactive nuclei and λ is the decay constant. This calculator estimates the number remaining after a chosen time: enter the initial number of nuclei, the decay constant, and elapsed time to get the remaining amount and the fraction that has decayed.

Formula

The number of undecayed nuclei after time t is N(t) = N₀e^−λt. The calculator also reports the fraction remaining, e^−λt, and the fraction decayed, 1 − e^−λt. Here N(t) and N₀ are counts of nuclei, t is elapsed time in seconds, and λ is the decay constant in inverse seconds (s⁻¹). The product λt must be dimensionless, so the time and decay constant must use matching units.

The decay constant is related to half-life by λ = ln(2) / T_1/2, where T_1/2 is the time for half the nuclei in a large sample to decay. If half-life is given in years, for example, either convert it to seconds before using this calculator or express both time and decay constant using years consistently. The calculator accepts seconds, so convert other time units to seconds first.

Worked Example

Using the defaults in the calculator:

• Initial count: N₀ = 1,000,000 nuclei.
• Decay constant: λ = 0.000001 s⁻¹.
• Elapsed time: t = 693,147 s.

First calculate the exponent: −λt = −(0.000001 s⁻¹)(693,147 s) = −0.693147. The fraction remaining is e^−0.693147 ≈ 0.500000. Therefore:

• Remaining nuclei: N(t) = 1,000,000 × 0.500000 ≈ 500,000.00.
• Fraction remaining: ≈ 0.500000.
• Fraction decayed: 1 − 0.500000 ≈ 0.500000.

This elapsed time is approximately one half-life because ln(2) is about 0.693147 and λ is 10⁻⁶ s⁻¹. The displayed count is a mathematical expectation; an actual sample’s number of decays fluctuates statistically.

Assumptions

• The sample contains one radioactive nuclide with a constant decay probability per unit time.
• Radioactive decay is random for individual nuclei, while the exponential law describes the expected behavior of a population.
• The entered starting count, decay constant, and time are valid and use compatible units; time is nonnegative and the decay constant cannot be negative.
• The model does not account for radioactive material entering or leaving the sample, chemical separation, or contributions from other nuclides.

How Radioactive Decay Works

For a single nucleus, the exact time of decay cannot be predicted by this equation. Instead, λ expresses the probability of decay per unit time for each nucleus, under the usual constant-rate model. A large population therefore declines predictably in proportion to how many unstable nuclei remain. In differential form, this idea is written dN/dt = −λN; solving it gives the exponential equation.

Radioactive decay may produce an alpha particle, a beta particle, or other radiation, depending on the nuclide and decay process. The simple equation tracks the parent nuclide disappearing. If a daughter nuclide is radioactive too, its amount may grow and then decline according to its own decay constant. That situation requires coupled equations rather than treating the parent equation as a complete description of every isotope in the sample.

Half-Life, Activity, and Units

Half-life is often easier to find in reference tables than λ. Convert it using λ = ln(2)/T_1/2, keeping units consistent. For example, if half-life is stated in days, λ calculated from that value has units of day⁻¹. To use the calculator’s seconds input, first convert the half-life to seconds; one day is 86,400 seconds.

Activity is not the same as the number of nuclei. It is the rate of decays, with SI unit the becquerel (Bq), equal to one decay per second. For a single nuclide, its expected activity is A(t) = λN(t). Thus the calculator’s remaining count can be used to find activity if λ is known in s⁻¹; the resulting unit is s⁻¹, or Bq. Activity may also depend on how much material is present and its isotopic composition.

Using the Result and Avoiding Errors

The remaining fraction is useful when the absolute number of nuclei is unknown or when comparing how much of a sample survives. Multiply that fraction by the initial count to obtain the expected remaining count. The decayed fraction describes the proportion of original parent nuclei that have decayed; it does not necessarily equal the amount of a particular daughter product currently present, especially if that daughter is itself unstable.

• Match time units. A decay constant in s⁻¹ requires time in seconds. Mixing years and seconds can change the result by millions.
• Use the right nuclide’s half-life. Isotopes of the same element can have very different decay constants.
• Distinguish decay from activity. N is a number of nuclei; activity is a rate. They are related by A = λN, not interchangeable.
• Interpret counts sensibly. The equation gives an expected population. A fractional result does not mean a fraction of one physical nucleus exists.
• Do not confuse radioactive decay with ordinary loss. Leakage, dilution, chemical removal, and biological clearance are different processes and need additional models.

Limits and Related Models

The exponential equation applies when the decay constant is effectively constant and the system is described by a single parent nuclide. For a chain of parent and daughter isotopes, use the Bateman equations or another appropriate chain model. If the material is a mixture of isotopes, calculate each component separately and combine the results only when the relevant initial amounts and decay constants are known.

Radioactive decay is not meaningfully accelerated or slowed by ordinary changes in temperature, pressure, or chemical form for most commonly encountered nuclear transitions. A small number of specialized electron-capture cases can show measurable environmental effects, so precise work should use data appropriate to the nuclide and conditions. The calculator is a mathematical tool, not a radiation measurement or a substitute for radiation-safety procedures.

Frequently Asked Questions

What is the equation for radioactive decay?
The standard single-nuclide equation is N(t) = N₀e^−λt. It gives the expected number of parent nuclei remaining after elapsed time t.

How do I calculate the decay constant from half-life?
Use λ = ln(2)/T_1/2. The half-life unit determines the inverse unit of λ; convert units if you will use a different time unit in the decay equation.

How much remains after two half-lives?
One quarter remains: the surviving fraction is (1/2)² = 0.25. After k half-lives, the fraction remaining is (1/2)^k.

Is radioactive decay linear or exponential?
For a constant decay rate per nucleus, it is exponential. The number decaying in a given interval depends on how many radioactive nuclei remain at the start of that interval.

Does the equation calculate radiation dose?
No. It estimates the number of parent nuclei remaining. Dose requires additional information, including radiation type, emitted energy, geometry, exposure, and how radiation interacts with tissue or another material.

Related

• Percent Yield Calculator: Actual Yield ÷ Theoretical Yield × 100
• Mole Calculator From a Chemical Equation: Moles of Product From the Mole Ratio

in n0 | Initial radioactive nuclei | nuclei | 1000000 | 0
in lam | Decay constant | 1/s | 0.000001 | 0
in t | Elapsed time | s | 693147 | 0
out n | Nuclei remaining | nuclei | n0 * exp(-lam * t) | 2
out f | Fraction remaining | - | exp(-lam * t) | 6
out d | Fraction decayed | - | 1 - exp(-lam * t) | 6