Exponential Decay
Exponential decay is the result of a function that decreases in proportion to its current value.
Learning Objective

Use the exponential decay formula to calculate how much of something is left after a period of time
Key Points
 Exponential decrease can be modeled as:
$N(t)=N_0e^{\lambda t}$ where$N$ is the quantity,$N_0$ is the initial quantity,$\lambda$ is the decay constant, and$t$ is time.  Oftentimes, halflife is used to describe the amount of time required for half of a sample to decay. It can be defined mathematically as:
$t_{1/2}=\frac{ln(2)}{\lambda }$ where$t_{1/2}$ is halflife.  Halflife can be inserted into the exponential decay model as such:
$N(t)=N_0(\frac{1}{2})^{t/t_{1/2}}$ . Notice how the exponential changes, but the form of the function remains.
Terms

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. As a consequence, atoms for the same isotope will have the same atomic number but a different mass number (atomic weight).

halflife
The time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacological, physiological, biological, or radiological activity.
Full Text
Introduction
Just as it is possible for a variable to grow exponentially as a function of another, so can the a variable decrease exponentially. Consider the decrease of a population that occurs at a rate proportional to its value. This rate at which the population is decreasing remains constant but as the population is continually decreasing the overall decline becomes less and less steep.
Exponential rate of change can be modeled algebraically by the following formula:
where
Half Life
The time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacological, physiological, biological, or radiological activity is called its halflife. The exponential decay of the substance is a timedependent decline and a prime example of exponential decay.
As an example let us assume we have a
Use of Half Life in Carbon Dating
Halflife is very useful in determining the age of historical artifacts through a process known as carbon dating. Given a sample of carbon in an ancient, preserved piece of flesh, the age of the sample can be determined based on the percentage of radioactive carbon13 remaining. 1.1% of carbon is C13 and it decays to carbon12. C13 has a halflife of 5700 years—that is, in 5700 years, half of a sample of C13 will have converted to C12, which represents approximately all the remaining carbon. Using this information it is possible to determine the age of the artifact given the amount of C13 it presently contains, and comparing it to the amount of C13 it should contain.
Halflife can be mathematically defined as:
It can also be conveniently inserted into the exponential decay formula as follows:
Thus, if a sample is found to contain 0.55% of its carbon as C13 (exactly half of the usual 1.1%), it can be calculated that the sample has undergone exactly one halflife, and is thus 5,700 years old.
Below is a graph highlighting exponential decay of a radioactive substance. Using the graph, find that halflife.
Graph depicting radioactive decay
The amount of a substance undergoing radioactive decay decreases exponentially, eventually reaching zero. Since there is 50% of the substance left after 1 year, the halflife is 1 year.
Key Term Reference
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 exponent
 Appears in these related concepts: Logarithms of Products, Logarithms of Powers, and Logarithms of Quotients
 exponential
 Appears in these related concepts: Exponential Growth and Decay, Solving General Problems with Logarithms and Exponents, and Population Growth
 function
 Appears in these related concepts: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 graph
 Appears in these related concepts: Graphical Representations of Functions, Graphing Equations, and Reading Points on a Graph
 proportional
 Appears in these related concepts: Graphs of Exponential Functions, Base e, Combined Variation, and Direct and Inverse Variation
 rate of change
 Appears in these related concepts: Increasing, Decreasing, and Constant Functions, Rates of Change, and Linear Mathematical Models
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
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