Equations Stirling approximation

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

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 "... 

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 ... 

The factorials in Eq. (5.70) can be simplified by using the Stirling approximation, which is then applied to Eq. (5.71), leading to the following equation ... 

Evidently, this equation of state does not depend on the properties of the molecules. This behavior is not shared by other thermodynamic relations of the ideal gas. For instance, the chemical potential obtained from (1.25) and (1.62) and using the Stirling approximation is... 

Calculations assisted by the Stirling approximation (cf. Equations 2.48 and 2.49) illustrate the important points. The lower panel of Figure 3.11 shows the probability distribution for left-side pressure measurements. N has been arbitrarily set at KT and used as the unit measure = Vj /probileft) by Equation (3.83A). One observes that the distribution is narrower when the left and right compartments are of equal volume contrarily the width is increased when that is, prob(lefl) < probiright). 

Equation 1 is valid for avalanches of any size and it is not a power law by inserting the value k = 1 we can derive the distribution for avalanches of any size in dynamically critical networks. As it will be shown in Sect. 3, this does not lead to a true power law. However, if we consider fairly large avalanches, for which the Stirling approximation holds (while still being V N), it turns out that the distribution indeed approximates a power law with slope -3/2. 

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... 

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... 

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... 

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

Factorial numbers enter often in the equations developed in the previous sections. They are not very conveniently evaluated in the applications considered below. However, there exists an approximation due to Stirling that is... 

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... 

The first approximation of Kasai and Oosawa (1969) has been improved by the efforts of two research groups. Zeeberg et al. (1980) employed an elegant approach with difference equations to the equilibrium exchange problem, and, through the use of a Taylor s series approximation as well as Stirling s approximation, obtained the following solution ... 

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

In doing the algebra, which you should work through yourself, we used Stirling s approximation (Equation 11-11) ... 

By combining Stirling s approximation to this expression with Equation (1) one obtains... 

Using Stirling s approximation,6 Equation (5) can be more conveniently calculated as... 

The solution to the fractional diffusion equation is clearly dependent on fluctuations that have occurred in the remote past note the time lag k in the index on the fluctuations and the fact that it can be arbitrarily large. The extent of the influence of these distant fluctuations on the system response is determined by the relative size of the coefficients in the series. Using Stirling s approximation on the gamma functions determines the size of the coefficients in Eq. (25) as the fluctuations recede into the past, that is, as k — oo we obtain... 

For large numbers, Stirling s approximation (In xl = x In x — x ) can be used to eliminate factorials from the equation ... 

By using Stirling s statistical approximation, Botzmann reached the equation for the isothermal, isobaric, irreversible mixing of two perfect gases having equal volumes ... 

The only difficulty in this equation is finding n . NONMEM in particular does not support the factorial function, so instead Stirling s approximation can be used... 

We now solve these equations using certain approximations employed in Statistical Mechanics, including Stirling s Approximation for large niunbers. If the energy required to form vacancies and Interstitials is greater than kT, and if the defects are only a fraction of the total number of lattice sites, then we get ... 

Substituting equation (6.4) for W into (6.3), and applying Stirling s approximation (6.5) for the factorials of large numbers gives... 

Using Stirling s approximation, In N = N hi N - N, and rearranging in terms of mole fractions. Equation 12.20 becomes... 

The remaining empty cells on the lattice can now be filled by solvent molecules, but as there is only one distinguishable way in which this can be done, Qj = 1, there is no further contribution to Q. and the entropy of the system. The latter can now be calculated from the Boltzmatm equation. The factorials can again be approximated using Stirling s relation and, although this requires considerable manipulation, which will be omitted here, it can eventually be shown that... 

The logarithm can be evaluated by using Stirling s approximation, ln(w ) = m nm — m, for the factorial of a large integer m. Equation (12.12) then leads directly to... 

It is useful to take logarithms to utilize Stirling s approximation for the factorials (equation 2G.11)... 

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