Sterling s approximation

For high molecular weight polymers, n is large and the logarithm of large factorials is accurately given by Sterling s approximation. 

Taking logarithms of both sides in Eq. (1.26) and applying Sterling s approximation gives... 

Using Sterling s approximation ( nM = M nM-M for M large), the Gibbs energy of mixing becomes... 

This quantum expression can be simplified by assuming that the spacing is very small so that n and m are very large. With this assumption we can apply Sterling s approximation for the factorial functions, which is h = h /e. If in addition the number of quanta is much larger than the number of oscillators, that is (n — m) >> s, then (n — m + s — 1) can be replaced by (n — m) (but not in the exponent ). The probability then reduces to... 

Sterling s approximation is useful to simplify the factorial of large numbers with less than 1% error when N > 100 ... 

The reader can easily check the validity of the integration for small values of n. The inverse of Sterling s approximation for In n gives, after some elementary algebra, a second rendition of Q ... 

Taking the logarithm, dividing by iV, and using Sterling s approximation InAT. NAnNaf we obtain... 

Substituting this equation in Equation 1, we get Equation 3. Sterling s approximation... 

Equation 15 64 is an exact answer to our problem but can be simplified when n and s are very large numbers. Using Sterling s approximation... 

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