Starling equation

Other pressure—volume—temperature (PVT) relationships may be found in the Hterature ie, Benedict, Webb, Rubin equations of state (4—7) the Benedict, Webb, Rubin, Starling equation of state (8) the Redlich equation of state (9) and the Redlich-Kwong equation of state (10). 

Now, let us consider a model in which the association site is located at a distance slightly larger than the hard-core diameter a. The excess free energy for a hard sphere fluid is given by the Carnahan-Starling equation [113]... 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

Fig. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan-Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC configuration. Spring stiffness is K = 70,000, granular temperature is 9 = 1.0, and coefficient of normal restitution is e = 1.0. The system is driven by rescaling.

Li et al. [189] assumed that a pair of deformable droplets has the shape of truncated sphere separated by a planar film and used the improved Carnahan-Starling equation to describe the repulsion term as ... 

In eq 3.1, the activity coefficients appear as a result of the hard-sphere repulsions among the droplets. Since the calculations focus on the most populous aggregates, the hard-sphere repulsions will be expressed in terms of a single droplet size corresponding to the most populous aggregates. One can derive expressions for the activity coefficients y ko of a component k in the continuous phase O starting from an equation for the osmotic pressure of a hard-sphere fluid,3-4 such as that based on the Carnahan—Starling equation of state (see Appendix B for the derivation) ... 

In obtaining the expression for the activity coefficient part of the chemical potential, we have considered droplets of a single size represented by the most populous size (corresponding to the maximum in the size distribution). A more formal equation allowing for droplets of various sizes can be written according to the Mansoori—Carnahan—Starling equation of state for mixtures of hard spheres.26 The results based on such an expression are not expected to be essentially different from those obtained on the basis of a single droplet size. 

Figure 7.12 Excess chemical potential of the hard-sphere fluid as a function of density. The open and filled circles correspond to the predictions of the primitive quasi-chemical theory and the self-consistent molecular field theory, respectively. The solid and dashed lines are the scaled-particle (Percus-Yevick compressibility) theory and the Carnahan-Starling equation of state, respectively (Pratt and Ashbaugh, 2003).

The equilibrium density profile can then be obtained by finding minimum of the grand potential as defined in eq.(la). Tarazona used the Camahan-Starling equation to derive... 

Table 1. Parameters for the Carnahan - Starling equation of state at 298. IS K...

Here p(r) is the smoothed density and A is the thermal de Broglie wavelength. The repulsive part of the Helmholtz free energy is usually calculated by the Carnahan-Starling equation derived for the hard sphere fluid [80] ... 

The Carnahan-Starling equation of state agrees well with the result of computer simulations over the range shown in fig. 2.10 and is used in all further calculations presented here. 

Estimate the value of PV/ Nk- T) for a hard-sphere fluid using the Carnahan and Starling equation of state assuming a concentration of 10 M and a hard-sphere diameter of 300 pm. 

The data designated ( ) were obtained using the pressure equation (2.9.9), those designated ( ) using the compressibility equation (2.9.10), and the smooth curve, using the Carnahan-Starling equation (2.9.11). 

Estimate the packing fraction for a hard-sphere liquid with a density of 21.25 atoms nm and a hard-sphere diameter of 350 pm. Use this result to calculate the Percus-Yevick product for the system at 85 K using the Carnahan-Starling equation of state (equation (2.9.11)). 

While the Starling equation of fluid transport is useful for understanding the factors involved in fluid shifting between compartments, it is not a practical tool for use in the clinical setting. 

The factors involved in fluid shifting between the intravascular and interstitial spaces are described by the modified Starling equation ... 

The balance of forces across the capillary wall is expressed succinctly by the Starling equation ... 

Fig. 21. Entropy versus log-temperature diagram for the hard-sphere model. The solid curves give the computer simulation values for the supercooled fluid, glass, and crystal. The dashed curves have the following bases (a) a calculation from the virial equation using the known first seven coefficients and higher coefficients obtained from the conjectured closure (the plot corresponds quite closely with that calculated from the Camahan-Starling equation ) and (i>) an extrapolation of higher temperature behavior such as that used by Gordon et al., which implies a maximum in the series of virial coefficients. The entropy is defined in excess of that for the ideal gas at the same temperature and pressure. Some characteristic temperatures are identified 7, fusion point 7 , upper glass transition temperature T/, Kauzmann isoentropic point according to closure virial equation.

The ensemble average in the Widom formula, exp -p v g )), is sometimes loosely referred to as the insertion probability . It becomes very low for dense fluids. For example, for hard spheres, we can use the scaled-particle theory [104] or the Camahan-Starling equation of state [105] to estimate it (see figure B3.3.8. ... 

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