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 / loge2

·   ~97% completion =  5 x t1/2

 

 

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