## Radioactive decay rate is exponential and is characterized by constants (such as half-life) as well the activity and number of particles.

#### Key Points

• The law describes the statistical behavior of a large number of nuclides, rather than individual ones.

• The decay rate equation is: $N={N}_{0}{e}^{-\lambda t}={N}_{0}{e}^{-\frac {t }{\tau }}$.

• Although the parent decay distribution follows an exponential, observations of decay times will be limited by a finite integer number of N atoms.

#### Terms

• The time required for half of the nuclei in a sample of a specific isotope to undergo radioactive decay.

• An atomic nucleus specified by its atomic number and atomic mass.

#### Figures

1. ##### Exponential Decay

A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constants of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5.

The decay rate of a radioactive substance are characterized by the following:

Constant quantities:

• The half-life—t1/2, is the time taken for the activity of a given amount of a radioactive substance to decay to half of its initial value;
• The decay constant— λ, "lambda" the inverse of the mean lifetime.

Although these are constants, they are associated with statistically random behavior of populations of atoms. Predictions using these constants are less accurate for small number of atoms. In principle the reciprocal of any number greater than one— a half-life, a third-life, or even a (1/√2)-life can be used in exactly the same way as half-life; but the half-life, t1/2, is adopted as the standard time associated with exponential decay.

Time-variable quantities:

• Total activity— A, is number of decays per unit time of a radioactive sample.
• Number of particles—N, is the total number of particles in the sample.
• Specific activity—SA, number of decays per unit time per amount of substance of the sample at time set to zero (t = 0). "Amount of substance" can be the mass, volume or moles of the initial sample.

Radioactivity is one very frequent example of exponential decay, shown in this figure Figure 1. The law describes the statistical behavior of a large number of nuclides, rather than individual ones. In the following formalism, the number of nuclides or nuclide population N, is of course a discrete variable (a natural number) - but for any physical sample N is so large (amounts of L = 1023, avogadro's constant) that it can be treated as a continuous variable. Differential calculus is needed to set up differential equations for modelling the behavior of the nuclear decay.Consider the case of a nuclide A decaying into another B by some process A → B.  Given a sample of a particular radioisotope, the number of decay events, −dN, expected to occur in a small interval of time dt is proportional to the number of atoms present N, that is:

$-\frac { dN }{ dt } \alpha N$

Particular radionuclides decay at different rates, so each has its own decay constant λ.  The expected decay −dN/N is proportional to an increment of time, dt:

$-\frac { dN }{ N } =\quad \lambda dt$

The negative sign indicates that N decreases as time increases, as each decay event follows one after another. The solution to this first-order differential equation is the function:

$N={N}_{0}{e}^{-\lambda t}={N}_{0}{e}^{-\frac {t }{\tau }}$

where N0 is the value of N at time t = 0.  Although the parent decay distribution follows an exponential, observations of decay times will be limited by a finite integer number of N atoms and follow Poisson statistics as a consequence of the random nature of the process.  We have for all time t:

${ N }_{ total }= { N }_{ A } + { N }_{ B } = {N}_{{A}_{0}}$

where Ntotal is the constant number of particles throughout the decay process, clearly equal to the initial number of A nuclides since this is the initial substance.  If the number of non-decayed A nuclei is:

${ N }_{ A } = {N}_{{A}_{0}}{e}^{-\lambda t}$

then the number of nuclei of B, i.e. number of decayed A nuclei, is:

${ N }_{ B}={ N }_{ {A}_{0} }-{N}_{A}={N{{A}_{0}}}-{N}_{{A}_{0}}{ e }^{ -\lambda t } ={ N }_{ { A }_{ O } }(1-{ e }^{ -\lambda t })$

We should make a quick mention of the units of decay.  The SI unit of radioactive activity is the becquerel (Bq), in honor of the scientist Henri Becquerel.  One Bq is defined as one transformation (or decay or disintegration) per second.  Since sensible sizes of radioactive material contains many atoms, a Bq is a tiny measure of activity; amounts giving activities on the order of GBq (gigabecquerel, 1 x 109 decays per second) or TBq (terabecquerel, 1 x 1012 decays per second) are commonly used.  Another unit of radioactivity is the curie, Ci, which was originally defined as the amount of radium emanation (radon-222) in equilibrium with one gram of pure radium, isotope Ra-226. At present it is equal, by definition, to the activity of any radionuclide decaying with a disintegration rate of 3.7 × 1010 Bq, so that 1 curie (Ci) = 3.7 × 1010 Bq. The use of Ci is currently discouraged by the SI.  Low activities are also measured in disintegrations per minute (dpm).

#### Key Term Glossary

activity
In chemical thermodynamics, activity (symbol a) is a measure of the “effective concentration” of a species in a mixture, meaning that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution.
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atom
the smallest possible amount of matter that still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons
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constant
Consistently recurring over time; persistent
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decay
To change by undergoing fission, by emitting radiation, or by capturing or losing one or more electrons.
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differential equation
an equation involving the derivatives of a function
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equilibrium
the state of a reaction in which the rates of the forward and reverse reactions are the same
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Exponential Decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its value.
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finite
limited, constrained by bounds, impermanent
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integer
An element of the infinite and numerable set {...,-3,-2,-1,0,1,2,3,...}.
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isotope
Any of two or more forms of an element where the atoms have the same number of protons but a different number of neutrons within their nuclei. As a consequence, atoms for the same isotope will have the same atomic number but a different mass number (atomic weight).
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Isotope
Isotopes are variants of a particular chemical element. While all isotopes of a given element share the same number of protons, each isotope differs from the others in its number of neutrons.
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mass
The quantity of matter that a body contains, irrespective of its bulk or volume. It is one of four fundamental properties of matter. It is measured in kilograms in the SI system of measurement.
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mole
In the International System of Units, the base unit of the amount of substance; the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. Symbol: mol. The number of atoms in a mole is known as Avogadro’s number.
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nuclide
An atomic nucleus specified by its atomic number and atomic mass.
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Any of several processes by which unstable nuclei emit subatomic particles and/or ionizing radiation and disintegrate into one or more smaller nuclei.
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Spontaneous emission of ionizing radiation as a consequence of a nuclear reaction, or directly from the breakdown of an unstable nucleus.
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SI unit
The International System of Units (abbreviated SI from French: Système international d'unités) is the modern form of the metric system, used extensively in the sciences.
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solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
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Solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
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standard
something used as a measure for comparative evaluations
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substance
Physical matter; material.
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volume
A unit of three-dimensional measure of space that comprises a length, a width, and a height. It is measured in units of cubic centimeters in metric, or cubic inches or cubic feet in English measurement.