Stirling’s formula

Setting the chemical potentials equal, /[Pg.116]

According to Stirling s formula In AH m NlnN — N so we obtain for the entropy ... 

One now recognizes that in situations of chemical interest Ni is very large so that Nj in Equation 4.53 can be well approximated by Stirling s formula... 

Application of Stirling s formula to equation 5.191 and comparison with the configurational state of pure components lead to the definition of a configurational entropy of mixing term in the form... 

Tii is the number of components of i and N is the total number of components in the system. For one mole of components, N is equal to Avogadro s number. From Stirling s formula, 5 is now equal to... 

When we use Stirling s formula for the binomial number, we have... 

It will be easier to find the distribution that maximizes In W instead of finding the maximum in W itself. Because N, and Nj, etc., are such large numbers, we can take advantage of Stirling s formula In nl = n In n — n. Taking the natural log of Eq. 8.28 gives... 

To evaluate Eq. 10.122, additional simplifying assumptions are necessary. Consider the vibrational degeneracy in the limit that v is very large. Applying Stirling s formula, we can rewrite Eq. 10.115 as... 

By using Stirling s formula (see, for example, [S. Chandrasekhar (1943)]), the probability of this special sequence is... 

The Gaussian integral over is easily carried out. Using Stirling s formula to evaluate Mnl after a little calculation we find an expression for the density of the free energy in the large volume limit ... 

Stirling s formula is fairly accurate for values of N greater than 10 for still larger N% where JV and (N/c)N are very large numbers, the factor /27rN is so near unity in proportion that it can be omitted for most purposes, so that we can write Nl simply as (N/e)N. Adopting this approximation, we can rewrite Eq. (2.3) as... 

Since N0 is very large, we take advantage of Stirling s formula for the logarithms of factorials of large numbers ... 

PROBLEM 5.2.2. Using Stirling s formula, Eq. (2.20.7), show that, for large Ni and fpo becomes... 

Equation (25) follows after differentiation and application of Stirling s formula (n = N/Nm, nd = Nd/Nm, Nm = Avogadro s number /f = NmAg°d). It is worthy to note that the strict result (see l.h.s. of Eq. 25) which is formally valid also for higher concentrations, is of the Fermi-Dirac type. This is due to the fact that double occupancy is forbidden and hence the sites are exhaustible similar as it is the case for the quantum states in the electronic problems. 

Using Stirling s formula, the definition of the Helmholtz free energy F = -kT In and standard thermodynamic relations, several useful formulae may be deduced from the expression for Zn in Equation 12. The chemical potential (/x) is given by... 

Of particular interest is the molar mass distribution of high molar mass hyperbranched polymers that are produced when the reaction of AB/ i monomers is driven close to completion. The number fraction of molecules [Eq. (6.9)] can be approximated for large N, using Stirling s formula ... 

NONMEM code for Stirling s formula would thus appear as... 

For q 3> 1 and for a fixed value of k, one can simplify the right-hand side of (101) using Stirling s formula for the gamma functions and the easily verified asymptotic formula... 

The central difference formula of Stirling thus furnishes the same result as the ordinary difference formula of Newton. We get different results when the higher orders of differences are neglected. For instance, if we neglect differences of the second order in formulae (7) and (20), Stirling s formula would furnish more accurate results, because, in virtue of the substitution A1 = A1 - JA2 v we have really retained a portion of the second order of differences. If, therefore, we take the difference formula as far as the first, third, or some odd order of differences, we get the same results with the central and the ordinary difference formulae. One more term is required to get an odd order of differences when central differences are employed. Thus, five terms are required to get... 

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