Osmotic equation

In the assumed dilute-solution limit, we can also assume that VA V, the total volume of solution. Under these conditions, (7.72) reduces to the Van t Hoff osmotic equation... 

The pressure required to drive water out of mixtures of various concentrations of alcohol against pure water at 30°C is calculated from the osmotic equation... 

This relation can be inverted to yield fip(n) as function of c.p(nr) (eh Appendix A 4.1). Substituting the result into 77[/xp], we find the osmotic pressure as function of the concentration, which is the standard form of the osmotic equation of state. Also all the other thermodynamic quantities can be calculated from n[fip]. The excess free energy due to the solute, for instance, takes the form... 

Equation 5.21a is known in literature [10] as the Sips or Bradleys isotherm equation. This isotherm equation describes fairly well the experimental data of adsorption in zeolites and other microporous materials [26], The linear form of the osmotic equation can be expressed as follows... 

These approximations can then be used in the osmotic equation of state to obtain the compressibility factor. Monte Carlo simulations using the above-discussed Monte Carlo techniques have been performed to assess the approximations inherent in the generalized Flory theory of hard-core chain systems. This theory does quite well in predicting the equations of state of hard-core chains at fluid densities. The question then arises, why does it do so well since the theory typically only incorporates information from a dimer fluid as a reference state ... 

The differential Eq. (14) relates the insertion probability with the pressure. When integrated for P, it gives a continuous analogue of Eq. (12) ( osmotic equation of... 

Equation (12) is an exact expression for the pr ure in terms of the insertion probabilities. It is a form of the osmotic equation of state. The underlying physical picture is one of building up the system of N chains by adding chains one by one in the volume V. To make Eq. (12) practically useful, we need a way to calculate the insertion probabilities p (i,V,T). 

To relate macroscopic observables to forces acting between particles is the objective. The relevant measurable properties are (a) the phase diagram, (b) scattering of photons, neutrons, and X-rays, (c) the osmotic equation of state, and (d) rheological behaviour. This Report covers (a), (b), and (c) with emphasis on the transition between ordered and disordered states. Discussion of (b) is limited to light-scattering. Current problems in relation to (d) are set out (up to mid-1980). 

Distribution and Correlation Functions.—We consider a single spherical particle with position r and velocity v at time t in a concentrated dispersion of mean number density p. The distribution function measures the probability of finding a particle (the same or another particle) with position r" and velocity v" at time t". The osmotic equation of state is related to a time-averaged distribution function that depends on r alone, whereas the dynamic behaviour depends on time-dependent functions. A basic premise of statistical mechanics is that a time-average is equivalent to an ensemble average at fixed time the ensemble average is denoted by angular brackets (...). 

A completely different nonstandard technique to obtain a first overview of the equation of state was recently proposed by Addison et al. [269], whereby a gravitation-like potential is applied to the system, and the equilibrium density profile and the concentration profile of the center of mass of the polymers is computed to obtain the osmotic equation of state, fii this sedimentation equilibrium method one hence considers a system in the canonical MVT ensemble using a box of linear dimensions L x L x H, with periodic boundary conditions in x and y directions only, while hard walls are used at z = 0 and at z = H. An external potential is applied everywhere in the system ... 

From the partition function the free energy Fjy follows and hence all thermodynamic quantities of interest can be estimated (entropy, chemical potential, osmotic pressure...). Ottinger applied this technique to test the osmotic equation of state for dilute and semidilute polymer solutions for N <60. Extension of this technique to off-lattice systems has also been made. ... 

In polymer science, weight-based concentrations are almost always more useful than molar units, and accordingly the osmotic equation of state is customarily written in the form... 

The validity of the virial series expansion of the osmotic equation of state becomes questionable for moderately concentrated (or semidilute ) solutions, in which the overall number density N c/wo... 

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