Exponential Decay

(noun)

When a quantity decreases at a rate proportional to its value.

Examples of Exponential Decay in the following topics:

  • Exponential Decay

    • Exponential decay is the result of a function that decreases in proportion to its current value.
    • The exponential decay of the substance is a time-dependent decline and a prime example of exponential decay.
    • It can also be conveniently inserted into the exponential decay formula as follows:
    • Below is a graph highlighting exponential decay of a radioactive substance.
    • Use the exponential decay formula to calculate how much of something is left after a period of time

  • Exponential Growth and Decay

    • Exponential decay occurs in the same way, providing the growth rate is negative.
    • If $\tau > 0$ and $b > 1$, then $x$ has exponential growth.
    • If $\tau<0$ and $b > 1$, or $\tau > 0$ and $0 < b < 1$, then $x$ has exponential decay.
    • This graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.
    • Apply the exponential growth and decay formulas to real world examples

  • Rate of Radioactive Decay

    • Radioactive decay rate is exponential and is characterized by constants, such as half-life, as well the activity and number of particles.
    • Radioactivity is one very frequent example of exponential decay.
    • Particular radionuclides decay at different rates, so each has its own decay constant, λ.
    • A quantity undergoing exponential decay.
    • Apply the equation Nt=N0e−λt in the calculation of decay rates and decay constants

  • Practice 2: Exponential Distribution

    • Carbon-14 is said to decay exponentially.
    • The decay rate is 0.000121 .
    • We are interested in the time (years) it takes to decay carbon-14.
    • Exercise 5.7.8: Thirty percent (30%) of carbon-14 will decay within how many years?

  • Graphs of Exponential Functions, Base e

    • The function $f(x) = e^x$ is a basic exponential function with some very interesting properties.
    • The basic exponential function, sometimes referred to as the exponential function, is $f(x)=e^{x}$ where $e$ is the number (approximately 2.718281828) described previously.
    • The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable.
    • If the change is positive, this is called exponential growth and if it is negative, it is called exponential decay.
    • For example, because a radioactive substance decays at a rate proportional to the amount of the substance present, the amount of the substance present at a given time can be modeled with an exponential function.

  • Radioactive Decay Series: Introduction

    • Radioactive decay series describe the decay of different discrete radioactive decay products as a chained series of transformations.
    • Radioactive decay series, or decay chains, describe the radioactive decay of different discrete radioactive decay products as a chained series of transformations.
    • Most radioactive elements do not decay directly to a stable state; rather, they undergo a series of decays until eventually a stable isotope is reached.
    • While the decay of a single atom occurs spontaneously, the decay of an initial population of identical atoms over time, $t$, follows a decaying exponential distribution, $e^{-t}$, where $\lambda$ is called the decay constant.
    • Because of this exponential nature, one of the properties of an isotope is its half-life, the time by which half of an initial number of identical parent radioisotopes have decayed to their daughters.

  • Basics of Graphing Exponential Functions

    • The exponential function $y=b^x$ where $b>0$ is a function that will remain proportional to its original value when it grows or decays.
    • This is called exponential growth.
    • When $0>b>1$ the function decays in a manner that is proportional to its original value.
    • This is called exponential decay.
    • This is true of the graph of all exponential functions of the form $y=b^x$ for $x>1$.

  • Derivatives of Exponential Functions

    • The derivative of the exponential function is equal to the value of the function.
    • The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative.
    • If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.
    • Graph of the exponential function illustrating that its derivative is equal to the value of the function.

  • Exponential and Logarithmic Functions

    • Both exponential and logarithmic functions are widely used in scientific and engineering applications.
    • The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.
    • The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.
    • The exponential function $e^x$ can be characterized in a variety of equivalent ways.
    • The derivative (or slope of a tangential line) of the exponential function is equal to the value of the function.

  • The Exponential Distribution

    • The exponential distribution is a family of continuous probability distributions.
    • Another important property of the exponential distribution is that it is memoryless.
    • The exponential distribution describes the time for a continuous process to change state.
    • the time until a radioactive particle decays, or the time between clicks of a geiger counter
    • Reliability engineering also makes extensive use of the exponential distribution.