0.0001209681
1/unit
3.8332475961e-12
s^-1
8,266.642584
unit
260,875,400,018.10876
s
180,825,048,000
s
0.0001209681
1/unit
3.8332475961e-12
s^-1
8,266.642584
unit
260,875,400,018.10876
s
180,825,048,000
s
The Decay Constant Calculator converts a radioactive isotope's half-life into its decay constant $$\lambda$$, the fundamental rate parameter governing radioactive decay. The decay constant represents the probability per unit time that any individual atom will undergo decay, and it is the key parameter in the exponential decay law $$N(t) = N_0 e^{-\lambda t}$$.
This calculator accepts half-life values in any common time unit and provides the decay constant in both the native unit and per second (SI standard). It also computes the mean life and converts the half-life to seconds for cross-referencing with nuclear data tables. Understanding the decay constant is essential for activity calculations, radiation safety assessments, and nuclear engineering applications.
The decay constant is derived from the half-life using the fundamental relationship:
$$\lambda = \frac{\ln(2)}{t_{1/2}} = \frac{0.693147}{t_{1/2}}$$
This arises from setting $$N(t_{1/2}) = N_0/2$$ in the decay equation:
$$\frac{N_0}{2} = N_0 \cdot e^{-\lambda t_{1/2}}$$
$$\frac{1}{2} = e^{-\lambda t_{1/2}}$$
$$\ln(2) = \lambda \cdot t_{1/2}$$
To convert between time units, the calculator uses standard conversion factors. For the SI decay constant (s⁻¹), the half-life is first converted to seconds:
$$\lambda_{\text{SI}} = \frac{\ln(2)}{t_{1/2} \text{ (in seconds)}}$$
The mean life follows as $$\tau = 1/\lambda = t_{1/2}/\ln(2)$$.
The Decay Constant in Input Unit gives $$\lambda$$ in the same time unit as your half-life input, useful for calculations in that time scale. The Decay Constant in s⁻¹ is the SI-standard value used in scientific literature and nuclear databases. The Mean Life is the average atom survival time ($$1/\lambda$$). The Half-Life in Seconds facilitates comparison with reference databases that typically report half-lives in seconds.
Inputs
Results
C-14 has λ ≈ 1.21×10⁻⁴ yr⁻¹ or 3.84×10⁻¹² s⁻¹, reflecting its very slow decay rate over thousands of years.
Inputs
Results
Tc-99m, the most widely used medical isotope, has λ ≈ 0.115 hr⁻¹, meaning about 11.5% of remaining atoms decay each hour.
The decay constant $$\lambda$$ is the probability per unit time that a single radioactive atom will decay. It is the fundamental rate parameter in the exponential decay law $$N(t) = N_0 e^{-\lambda t}$$. A larger $$\lambda$$ means faster decay and a shorter half-life.
The decay constant has units of inverse time (time⁻¹). In SI units, it is expressed in s⁻¹ (per second). It can also be expressed in min⁻¹, hr⁻¹, day⁻¹, or yr⁻¹ depending on the context and the magnitude of the half-life.
The factor $$\ln(2) \approx 0.6931$$ appears because the half-life is defined as the time for the quantity to reduce by half. Setting $$e^{-\lambda t_{1/2}} = 1/2$$ and solving gives $$\lambda t_{1/2} = \ln(2)$$. This natural logarithm of 2 is the bridge between the base-e exponential and the factor-of-2 definition.
Yes. If the half-life is less than $$\ln(2) \approx 0.693$$ in the chosen time unit, then $$\lambda > 1$$. For example, a half-life of 0.1 seconds gives $$\lambda = 6.93 \text{ s}^{-1}$$. This simply means the decay probability per unit time is high, not that probabilities exceed 100% (which is the probability over an infinitesimal interval).
Activity equals $$A = \lambda N$$, where $$N$$ is the number of radioactive atoms. The decay constant directly scales the conversion from atom count to disintegration rate. Higher $$\lambda$$ means more decays per second for the same number of atoms.
Uranium-238 has a half-life of 4.468 billion years, giving $$\lambda = 1.551 \times 10^{-10} \text{ yr}^{-1}$$ or $$4.916 \times 10^{-18} \text{ s}^{-1}$$. This extremely small value reflects U-238's exceptional stability among radioactive isotopes.
Yes, for radioactive decay specifically. The decay constant $$\lambda$$ is mathematically identical to the first-order rate constant $$k$$ in chemical kinetics. Both describe the rate of an exponential process where the rate is proportional to the current quantity.
For extremely short-lived states ($$\tau < 10^{-22}$$ s), the energy-time uncertainty principle becomes significant: $$\Delta E \cdot \tau \geq \hbar/2$$. In this regime, the decay constant relates to the resonance width $$\Gamma$$ through $$\lambda = \Gamma/\hbar$$, and the concept of a precise decay time becomes quantum-mechanically fuzzy.
Multiply or divide by the appropriate conversion factor. For example, to convert from yr⁻¹ to s⁻¹, divide by the number of seconds in a year (3.156 × 10⁷). This calculator handles these conversions automatically.
While theoretically possible, it would be an extraordinary coincidence. Each isotope's decay constant is determined by its unique nuclear structure, quantum states, and decay channels. In practice, no two different isotopes share identical decay constants.
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