Stirling's Approximation

[Allcock-409]


is needed to solve


which is needed for Boltzmann's Relation.

Your calculator can't handle N! if N is greater than 69 because overflow error occurs. 
 1! = 1          =  1

 2! = 1*2        =  2

 3! = 1*2*3      =  6
   
 4! = 1*2*3*4    = 24

... and so on to ...

68! ...        = 2.48 * 10^+96

69! ...        = 1.71 * 10^+98

70! ...        = overflow error on many calculators
For most polymer problems (especially those on tests...), N! values are greater than N=69.

  • An example application is provided below that uses numbers that your calculator can determine so you can compare the results to the correct answer:
      Let N(0)! = 49!; N(1)! = 23!; N(2)! = 14!
    First, solve the problem to get the true answer: 
    
   49!
    
--------- = 2.698995 * 10^+29   the accuracy limit here calculator roundoff
    
 23! 14!
    The calculator answer is 2.698995 * 10^+29

    Now, use Stirling's approximation to calculate each N!, multiply them together, and see how close the answer this generates is to the true answer.

    You need to recall the rules for     convolution.

  • Continuing with the problem, use Stirling's approximation to calculate each ln (number)! for the problem:



    Last Update, June 30, 1995- wld