Stirlings Approximation

The approximation described as Stirling s approximation above and in Section S3.2 is not very accurate. It is several percent in error even for values of N as large as 1010. The correct expression for Stirling s approximation is 

It is of interest to derive the relationship between defect numbers and configurational entropy using the correct form of Stirling s approximation. The principle can best be illustrated with respect to the population of vacancies in a monatomic crystal (Section 2.1). Substituting from the more accurate Eq. (S4.1)  

Assuming that N is much greater than n v, this can be solved to give 

When n becomes large, it is difficult to evaluate n and Stirling s approximation is often employed. To find the value for , we take the logarithms of both sides of equation (A 1.36) to obtain 

This is the form in which we will use Stirling s approximation/ 

For the large values of n that we will be using, equation (A1.41) also reduces to equation (A 1.40). 

The ability to measure temperature and temperature differences accurately and reproducibly is essential to the experimental study of thermodynamics. A thermometer constructed with an ideal gas as its working fluid yields temperatures that correspond to the fundamental thermodynamic temperature scale. However, such thermometers are extremely difficult to use, are not amenable to miniaturization, and are very expensive. Therefore, other means to measure temperatures that reproduce the ideal gas or thermodynamic temperature scale (Kelvin) have had to be developed. The international temperature scale represents a method to determine temperatures over a wide range with measuring devices that are easier to use than the ideal gas thermometer. The goal is to make temperature measurements that correspond to the thermodynamic temperature as accurately as possible. 

The international temperature scale is based upon the assignment of temperatures to a relatively small number of fixed points , conditions where three phases, or two phases at a specified pressure, are in equilibrium, and thus are required by the Gibbs phase rule to be at constant temperature. Different types of thermometers (for example, He vapor pressure thermometers, platinum resistance thermometers, platinum/rhodium thermocouples, blackbody radiators) and interpolation equations have been developed to reproduce temperatures between the fixed points and to generate temperature scales that are continuous through the intersections at the fixed points. 

The first hypothesis of microscopic thermodynamics is that of the atomic structure of matter and the application of laws of quantum mechanics. 

If we consider, for example, a phase contained in volume V limited by boundaries called the system, usually a collection with a large number of objects is needed to describe this system, since these objects are so small that it is often necessary to apply the laws of quantum mechanics. 

In fact, these objects are of very different natures molecules, atoms, electrons, nuclei, protons, and neutrons with interaction bonds existing between different objects, i.e. very local or global interaction energies. 

We therefore see that, from the description above, the system can be described as a number of objects but the definition of elementary objects is not required a priori. 

Phase Modeling Tools. Applications to Gases, First Edition. Michel Soustelle. ISTE Ltd 2015. Published by ISTE Ltd and John Wiley Sons, Inc. 

The value of N1 is practically impossible to calculate for large N. For example, if N is the number of molecules of a macroscopic system containing 1 mole, there is no direct way to calculate the term 6.023 x 10 31. 

James Stirling (1692-1770) devised the following approximation to compute In iVl  

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It is also instructive to start from the expression for entropy S = log(g(A( m)) for a specific energy partition between the two-state system and the reservoir. Using the result for g N, m) in section A2.2.2. and noting that E = one gets (using the Stirling approximation A (2kN)2N e ). 

The calculations were performed by numerical integration, after replacing the factorial by a lower end Stirling approximation... 

This is recognized as the standard entropy of mixing that is lost when a total of N particles is partitioned into = xN and = (1 — x)N particles in two phases. The simple form of the result shows that the calculation leading to Eq. (23) is essentially a generalization of the Stirling approximation. 

Substituting Eq. (A.4) into the Boltzmann relation, and using the Stirling approximation for the factorials, one obtains... 

When later we replace a< by the Stirling approximation, we will be following the generally accepted procedure of Boltzmann. Requirement (b) of note 119 will be especially important in that case. 

If the system consists of a very large number of molecules, as is the case under normal conditions, the various numbers N, N29 ., Ni,. .., are also very large. It is then possible to make use of the Stirling approximation for the factorials of large numbers thus,... 

For certain later purposes, it is convenient to write this equation in a somewhat different form by utilizing the Stirling approximation... 

In applications of these concepts to many particle systems, for example in statistical mechanics, we encounter the need to approximate discrete distributions such as in Eq. (1.10), in the limit of large values of their arguments by continuous functions. The Stirling Approximation... 

Here it must be noted that we have made an imprecise description of what under more formal terms is known as the Stirling approximation. Within the context of our characterization of the configurational entropy, the Stirling approximation of In N allows us to rewrite the entropy as... 

To obtain an explicit analytic expression for the entropy of mixing as shown in eqn (3.89), we exploited the Stirling approximation. Plot both the exact and the approximate entropy (as gotten from the Stirling approximation) for systems of size N = 10, 100 and 1000. 

Further, we substitute Equation 5.20 into Equation 5.7 and, using the Stirling approximation, we determine the surface free energy corresponding to the van der Waals model "... 

Within the Stirling approximation, we have partial cancellation leading to... 

By taking logarithms and using Stirlings approximation for large numbers, we have... 

The result is not encouraging now the calculation already stops at N= 143 (rather than at JV= 171) because of numerical overflow in the term Nn+ of the Stirling formula. While we might want to improve on the numerical accuracy of Nl by using additional terms in the exponent of (8.12-1), we will still be limited to JV< 142. Obviously, the Stirling approximation runs into overflow problems before the exact formula does. 

For an ideal gas of Nindistinguishable point particles one has = Q IN = V V IN.. For large None can again use the Stirling approximation for N. and obtain the Helmholtz free energy... 

To evaluate this expression we take advantage of the Stirling approximation When N is very large, then... 

FIGURE 12.1. Percentage difference between the Stirling approximation and the actual factorial value as a function of n. 

It is, nevertheless under certain conditions, possible to measure them, as was done for instance for H on tungsten [194], The nuclear partition function is unity and the problem left is to estimate the electronic partition function of the adsorbed atom since that now includes in this scheme the adsorption energy of the atom. Inserting the expression for the partition functions from Equations (4.41) and (4.10) in Equation (4.43) we find straightforwardly, utilizing the ideal gas law and Stirlings approximation ... 

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