Stirling s approximation

On applying Stirling s approximation, replacing V/N by kT/P, and using Eq. XVII-20 and setting P = 1 atm, the final result, known as the Sackur-Tetrode equation, is... 

This gives the number of possible permutations, irrespective of whether the solution is a solid or a liquid. Using Stirling s approximation, we may obtain from (46)... 

Stirling s approximation is a mathematical relationship that becomes more accurate as N becomes larger, and becomes very precise for the large number of units we will consider as we work with moles of ideal gas. Substituting equations (10.17) and (10.18) into equation (10.16) gives... 

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/... 

Debye heat capacity equation 572-80 Einstein heat capacity equation 569-72 heat capacity from low-lying electronic levels 580-5 Schottky effect 580-5 statistical weight factors in energy levels of ideal gas molecule 513 Stirling s approximation 514, 615-16 Streett, W. B. 412... 

The quantity Slot is typically referred to as the configurational entropy. It can easily be expressed in terms of the probability P by using Stirling s approximation for large N ... 

In order to solve combinatorial equations like App3. 1., a method called Stirling s approximation for large numbers is used. This gives ... 

Introducing Stirling s approximations, n /je for the factorials, which is legitimate provided both n and n m are large, and consolidating the resulting expression, we obtain... 

On taking logarithms and introducing Stirling s approximation for the factorials,... 

If each solvent molecule may occupy one of the remaining lattice sites, and in only one way, 0 represents also the total number of configurations for the solution, from which it follows that the configurational entropy of mixing the perfectly ordered pure polymer and the pure solvent is given hy Sc —k In Introduction of Stirling s approximations for the factorials occurring in Eq. (7) for fi, replacement of no with Ui+xn[Pg.501]

By Stirling s approximation the logarithm of the factorial of a large number... 

In fact this last approximation is not very accurate and 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) ... 

A more extended expression, using the correct formulation of Stirling s approximation, can be derived as in Section S4.2. 

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