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

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

halflife
The time required for half of the nuclei in a sample of a specific isotope to undergo radioactive decay.
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:
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.
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
Key Term Reference
 Exponential Decay
 Appears in these related concepts: The Integrated Rate Law and The Arrhenius Equation
 SI unit
 Appears in these related concepts: SI Unit Prefixes, Using Molarity in Calculations of Solutions, and Molarity
 activity
 Appears in these related concepts: Medical Solutions: Colligative Properties, Some Polycyclic Heterocycles, and The Equilibrium Constant
 atom
 Appears in these related concepts: Overview of Atomic Structure, Description of the Hydrogen Atom, and Stable Isotopes
 decay
 Appears in these related concepts: Radioactive Decay Series: Introduction, Models Using Differential Equations, and Sensory Registers
 differential equation
 Appears in these related concepts: Direction Fields and Euler's Method, Separable Equations, and Chemical Kinetics and Chemical Equilibrium
 element
 Appears in these related concepts: Development of the Periodic Table, Elements and Compounds, and The Periodic Table
 equilibrium
 Appears in these related concepts: Homogeneous versus Heterogeneous Solution Equilibria, Diffusion, and Second Condition
 integer
 Appears in these related concepts: Scientific Notation, Binomial Expansions and Pascal's Triangle, and Finding a Specific Term
 isotope
 Appears in these related concepts: Exponential Decay, Modes of Radioactive Decay, and Isotopes
 mole
 Appears in these related concepts: Avogadro's Number and the Mole, Molar Mass of Compounds, and Concept of Osmolality and Milliequivalent
 radioactive decay
 Appears in these related concepts: Alpha Decay, Nuclear Stability, and Tracers
 radioactivity
 Appears in these related concepts: The Atomic Bomb, Nuclear Binding Energy and Mass Defect, and Discovery of Radioactivity
 solution
 Appears in these related concepts: Electrolyte and Nonelectrolyte Solutions, Turning Your Claim Into a Thesis Statement, and What is an Equation?
 substance
 Appears in these related concepts: Substances and Mixtures, Types of Synthetic Organic Polymers, and Complex Ion Equilibria and Solubility
 volume
 Appears in these related concepts: Volumes, Volume and Density, and Shape and Volume
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