Molecular, incompressible volume

The molecules of liquids are separated by relatively small distances so the attractive forces between molecules tend to hold firm within a definite volume at fixed temperature. Molecular forces also result in tlie phenomenon of interfacial tension. The repulsive forces between molecules exert a sufficiently powerful influence that volume changes caused by pressure changes can be neglected i.e. liquids are incompressible. 

Properties of Different Solvents. In discussing molecular dipoles in Sec. 25, we estimated the force of attraction between an atomic ion and a dipole having the most favorable orientation and found this attraction to be very strong. In any ionic co-sphere those molecular dipoles which have a favorable orientation will bo attracted, while those that have the opposite orientation will be repelled. Since the former are more numerous the solvent in the co-sphere is, on the whole, attracted toward the ion. Since the liquid is not incompressible, we must expect that this will lead to a contraction in each co-sphere. In any ionic solution the sum of the contractions that have taken place in the co-spheres of the positive and negative ions will be apparent if we measure accurately the volume of the solution. 

An attempt has been made by Spiering et al. [39,40] to relate the magnitude of the interaction parameter F(x) as derived from experiment to the elastic interaction between HS and LS ions via an image pressure [47]. To this end, the metal atoms, inclusive of their immediate environments, in the HS and LS state are considered as incompressible spheres of radius /"h and Tl, respectively. The spheres are embedded in an homogeneous isotropic elastic medium, representing the crystal, which is characterized by specific values of the bulk modulus K and Poisson ratio a where 0 < a < 0.5. The change of molecular volume A Fas determined by X-ray diffraction may be related to the volume difference Ar = Ph — of the hard spheres by ... 

The IAD I(a,t) in a chemical reactor is specially interesting and it does not seem that sufficient attention has been paid to the possibilities offered by this function. For instance, let us consider a semi-batch reactor, and let Q(t) be the feed flowrate of an incompressible fluid. The instantaneous fluid volume is V = /q Q(tf)dtf, from which the IAD is written I(a,t) = Q(t-a)/V. I(a,t) can be used to calculate the chemical conversion in different segregation states. Consider a species of concentration C produced with the rate t. If the mixture is assumed to be well mixed at the molecular scale, one obtains the familiar mass balance equation... 

When solutions are incompressible, the partial molecular volumes... 

The following example shows how important this distinction is. In fact, let us consider an ideal incompressible solution of molecules with the same molecular volume. We have [see (5.1.60)]... 

No corresponding simple equation of state exists for the liquid phase other than one based on the use of a known pure species liquid density and the assumptions of incompressibility and additive volumes. When a vapor is not an ideal gas, formulation of an accurate equation of state becomes difficult because of the necessity to account for molecular interactions. 

The free energy of n micelles in a volume (which is assumed to be incompressible) that has a solvent, copolymers, etc. j = 1,2,...) as its molecular species, is given by F = Q.n + ZjfijUj where nj and ilj are the number and chemical potential of molecules of type j. The optimization of the free energy with respect to the number of micelles leads to the condition of equilibrium for micelles, i.e., Q. = 0 [36, 92]. The SCF model gives access to the grand potential Q.m of the micelle that... 

Due to such densely packed molecularly interpenetrated structures, rubbers are incompressible under deformation. Each chain takes a Gaussian conformation following the Flory theorem for screened excluded-volume interaction. On the basis of these characteristics, we can derive the elastic properties of rubbers from a microscopic point of view. 

The reduction of thread PRISM with the R-MMSA closure for the idealized fully symmetric block copolymer problem to the well-known incompressible RPA approach " is reassuring. However, in contrast with the blend case, for copolymers that tend to microphase separate on a finite length scale, the existence of critical or spinodal instabilities is expected to be an artifact of the crude statistical mechanical approximations. That is, finite N fluctuation effects are expected to destroy all such spinodal divergences and result in only first-order phase transitions in block copolymers [i.e., Eq. (7.3) is never satisfied]. Indeed, when PRISM theory is numerically implemented for finite thickness chain models using the R-MMSA or R-MPY/HTA closures spinodal divergences do not occur. Thus, one learns that even within the simpler molecular closures, the finite hard-core excluded volume constraint results in a fluctuation effect that destroys the mean-field divergences. 

In general the net macroscopic pressure tensor is determined by two different molecular effects One pressure tensor component associated with the pressure and a second one associated with the viscous stresses. For a fluid at rest, the system is in an equilibrium static state containing no velocity or pressure gradients so the average pressure equals the static pressure everywhere in the system. The static pressure is thus always acting normal to any control volume surface area in the fluid independent of its orientation. For a compressible fluid at rest, the static pressure may be identified with the pressure of classical thermodynamics as may be derived from the diagonal elements of the pressure tensor expression (2.189) when the equilibrium distribution function is known. On the assumption that there is local thermodynamic equilibrium even when the fluid is in motion this concept of stress is retained at the macroscopic level. For an incompressible fluid the thermodynamic, or more correctly thermostatic, pressure cannot be deflned except as the limit of pressure in a sequence of compressible fluids. In this case the pressure has to be taken as an independent dynamical variable [2] (Sects. 5.13-5.24). 

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