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Rate of Radioactive Decay
Radioactive decay rate is exponential and is characterized by constants, such as halflife, as well the activity and number of particles.
Learning Objective

Apply the equation N_{t}=N_{0}e^{−λt} in the calculation of decay rates and decay constants
Key Points

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

The decay rate equation is:
$N={N}_{0}{e}^{\lambda t}$ . 
Although the parent decay distribution follows an exponential, observations of decay times will be limited by a finite integer number of N atoms.
Terms

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

nuclide
An atomic nucleus specified by its atomic number and atomic mass.
Full Text
Decay Rate
The decay rate of a radioactive substance is characterized by the following constant quantities:
 The halflife (t_{1/2}) is the time taken for the activity of a given amount of a radioactive substance to decay to half of its initial value.
 The mean lifetime (τ, "tau") is the average lifetime of a radioactive particle before decay.
 The decay constant (λ, "lambda") is 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.
There are also timevariable quantities to consider:
 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. The law of radioactive decay describes the statistical behavior of a large number of nuclides, rather than individual ones. In the following relation, the number of nuclides or nuclide population, N, is of course a natural number. 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:
Particular radionuclides decay at different rates, so each has its own decay constant, λ.
The expected decay
The negative sign indicates that N decreases as time increases, as each decay event follows one after another. The solution to this firstorder differential equation is the function:
Here, N_{0} is the value of N at time t = 0.
The SI unit of radioactive activity is the becquerel (Bq), in honor of the scientist Henri Becquerel. One Bq is defined as one transformation, decay, or disintegration per second. Since sensible sizes of radioactive material contain many atoms, a Bq is a tiny measure of activity; amounts giving activities on the order of GBq (gigabecquerel, 1 x 10^{9} decays per second) or TBq (terabecquerel, 1 x 10^{12} decays per second) are commonly used.
Another unit of radioactivity is the curie, Ci, which was originally defined as the amount of radium emanation (radon222) in equilibrium with one gram of pure radium, isotope Ra226. At present, it is equal, by definition, to the activity of any radionuclide decaying with a disintegration rate of 3.7 × 10^{10} Bq, so that 1 curie (Ci) = 3.7 × 10^{10} Bq. The use of Ci is currently discouraged by the SI. Low activities are also measured in disintegrations per minute (dpm).
Example
Find the decay rate (
To solve, we need to use our equation:
Since we are dealing with the halflife we will use values for N and N_{o} that are equivalent to 0.5.
Now plug in the halflife for the time (t).
Solve for
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Key Term Reference
 Exponential Decay
 Appears in this related concepts: The Integrated Rate Law and The Arrhenius Equation
 SI unit
 Appears in this related concepts: SI Unit Prefixes, SI Units of Pressure, and Molarity
 activity
 Appears in this related concepts: Transition Metals, The Equilibrium Constant, and Medical Solutions: Colligative Properties
 atom
 Appears in this related concepts: Early Ideas about Atoms, Stable Isotopes, and John Dalton and Atomic Theory
 decay
 Appears in this related concepts: Radioactive Decay Series: Introduction, Models Using Differential Equations, and Sensory Registers
 differential equation
 Appears in this related concepts: Direction Fields and Euler's Method, PredatorPrey Systems, and Resistors and Capacitors in Series
 element
 Appears in this related concepts: The Law of Definite Composition, The Periodic Table, and Elements and Compounds
 equilibrium
 Appears in this related concepts: Humidity, Evaporation, and Boiling, Homogeneous versus Heterogeneous Solution Equilibria, and Open Economy Equilibrium
 integer
 Appears in this related concepts: Scientific Notation, Total Number of Subsets, and Finding a Specific Term
 isotope
 Appears in this related concepts: Exponential Decay, Structural Determination, and Basics of DNA Replication
 mole
 Appears in this related concepts: MoletoMole Conversions, Mole Fraction and Mole Percent, and Converting between Mass and Number of Moles
 radioactive decay
 Appears in this related concepts: Alpha Decay, Nuclear Stability, and Tracers
 radioactivity
 Appears in this related concepts: The Atomic Bomb, Nuclear Binding Energy and Mass Defect, and Discovery of Radioactivity
 solution
 Appears in this related concepts: Electrolyte and Nonelectrolyte Solutions, Using Molarity in Calculations of Solutions, and Forming Your Thesis
 substance
 Appears in this related concepts: Complex Ion Equilibria and Solubility, Substances and Mixtures, and Types of Synthetic Organic Polymers
 volume
 Appears in this related concepts: Volumes, Cylindrical Shells, and Shape and Volume
Sources
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Cite This Source
Source: Boundless. “Rate of Radioactive Decay.” Boundless Chemistry. Boundless, 19 Nov. 2014. Retrieved 26 Apr. 2015 from https://www.boundless.com/chemistry/textbooks/boundlesschemistrytextbook/nuclearchemistry19/radioactivity134/rateofradioactivedecay5367522/