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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