2012B13 Using an example of exponential decay, define and outline the
important features of an exponential function.

 

List:

     Exponential function

     Natural exponential function

     Exponential decay

 

Exponential function:

Equation

   y = ax

   Output equals a constant to the power of the input

Graph

Runaway curve

 

Natural exponential function:

Equation

   y=Aex = δy/δx (where e = 2.718281828)

   i.e. rate of change of Y is proportional to amount of Y

Graphs

Use

Used to model time-dependent exponential processes

   Drug metabolism

   Alveolar filling

   Pre-oxygenation

   Radioactive decay

 

Exponential decay:

Equation

N = N0e-kt

Where

   N = amount of isotope

   N0 = amount of isotope at t=0

Graph

Rate constant (k)

     Proportion of isotopes decaying per unit time, had the initial rate of decay continued

   Units: min-1

Time constant (τ)

   Time taken for completion had the initial rate of decay continued

   τ = 1/k

   Units: minutes

   ~95% completion at 3 x τ

Half life (t1/2)

   Time taken for 50% decay

   t1/2 = τ x loge2

   ~97% completion = 5 x t1/2

 

 

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