Expansion factor

No simple relationship exists between the two expansion parameters and as-However, for small expansions and ctj are close and one can use an average expansion factor a instead of or,- and ctj for simplicity. 

Expansion Factor The second virial coefficient, A2, is a measure of solvent-polymer compatibility. Thus, a large positive value 0/A2 indicates a good solvent for the polymer favoring expansion of its size, while a law value (sometimes even negative) shows that the solvent is relatively poor. The value of A2 will thus fell 

Apart from this excluded volume effect, coil size in solutions is affected by interaction between a coil and a solvent. Solvent molecules penetrate inside the coil, which thus swells and increases in size. The change in Gibbs free energy with coil size increase, caused by swelling, is expressed by the equation (Flory, 1953) 

The total change in Gibbs free energy accompanying the change in coil size is, from Eqs. (3.88) and (3.89)  

Equation (3.93) indicates that the size of a polymer coil increases with increasing molecular weight of the polymer. According to this equation, at a temperature equal to the Flory temperature (i.e., T = 0), - a = 0, i.e., a = 1. Thus, 

The value of Aj will tell us whether or not the size of the polymer coil, which is dissolved in a particular solvent, will be permrbed or expanded over that of the unpermrbed state, but the extent of this expansion is best estimated by calculating the expansion factor a. 

The extent of this coil permrbation by long-range effects is measured by an expansion factor a, introduced by Hory. This relates the permrbed and imperturbed dimensions by 

In good solvents (large, positive Aj), the coil is more extended than in poor solvents (low Aj), and a is correspondingly larger. Because a is solvent- and temperature-dependent, a more characteristic dimension to measure for the polymer is S, which can be calculated from hght scattering in a theta solvent, or indirectly 

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The relationship required is the gas expansion factor (E), and is defined for a given quantity (mass or number of moles) of gas as... 

In gas reservoir engineering, the gas expansion factor, E, is commonly used. However, in oil reservoir engineering it is often more convenient to refer to the gas formation volume factor which is the reciprocal E, and is expressed in units of scf/stb (using field units). The reason for this will become apparent in Section 8. 

In a good solvent, the end-to-end distance is greater than the 1q value owing to the coil expansion resulting from solvent imbibed into the domain of the polymer. The effect is quantitatively expressed in terms of an expansion factor a defined by the relationship... 

We shall defer a quantitative discussion of this expansion factor until the discussion of solutions in Chaps. 8 and 9. 

Although the emphasis in these last chapters is certainly on the polymeric solute, the experimental methods described herein also measure the interactions of these solutes with various solvents. Such interactions include the hydration of proteins at one extreme and the exclusion of poor solvents from random coils at the other. In between, good solvents are imbibed into the polymer domain to various degrees to expand coil dimensions. Such quantities as the Flory-Huggins interaction parameter, the 0 temperature, and the coil expansion factor are among the ways such interactions are quantified in the following chapters. 

We saw in Sec. 1.11 that coil dimensions are affected by interactions between chain segments and solvent. Both the coil expansion factor a defined by Eq. (1.63) and the interaction parameter x are pertinent to describing this situation. 

Conditions in which the effects of item (2) exactly compensate are called 0 conditions. The expansion factor a gives the ratio of coil dimensions under non-0 conditions to those under 0 conditions. 

Next we consider the situation of a coil which is unperturbed in the hydro-dynamic sense of being effectively nondraining, yet having dimensions which are perturbed away from those under 0 conditions. As far as the hydrodynamics are concerned, a polymer coil can be expanded above its random flight dimensions and still be nondraining. In this case, what is needed is to correct the coil dimension parameters by multiplying with the coil expansion factor a, defined by Eq. (1.63). Under non-0 conditions (no subscript), = a(rg)Q therefore under these conditions we write... 

Next we shall examine the molecular weight dependence of the coil expansion factor a to see if the latter can explain the observations of a s greater than 0.5. 

Our primary objective in undertaking this examination of the coil expansion factor was to see whether the molecular weight dependence of a could account for the fact that the Mark-Houwink a coefficient is generally greater than 0.5 for T 0. More precisely, it is generally observed that 0.5 < a < 0.8. This objective is met by combining Eqs. (9.55) and (9.68) ... 

What is especially significant about Eq. (9.68) is the observation that the coil expansion factor a definitely increases with M for good solvents, meaning that-all other things being equal longer polymer chains expand above their 0 dimensions more than shorter chains. Even though the dependence of a on... 

SAN resins show considerable resistance to solvents and are insoluble in carbon tetrachloride, ethyl alcohol, gasoline, and hydrocarbon solvents. They are swelled by solvents such as ben2ene, ether, and toluene. Polar solvents such as acetone, chloroform, dioxane, methyl ethyl ketone, and pyridine will dissolve SAN (14). The interactions of various solvents and SAN copolymers containing up to 52% acrylonitrile have been studied along with their thermodynamic parameters, ie, the second virial coefficient, free-energy parameter, expansion factor, and intrinsic viscosity (15). 

Table 4. Enamel Oxide Coefficient of Expansion Factors ...

X Y Value of expression Expansion factor Dimensionless Dimensionless... 

For the flow of gases, expansion factor Y, which allows for the change in gas density as it expands adiabaticaUy from pi to po, is given by... 

For the flow of liquids, expansion factor Y is unity. The change in potential energy in tne case of an inclined or vertical venturi meter must be allowed for. Equation (10-20) is accordingly modified to give... 

FIG. 10-16 Values of expansion factor Y for orifices, nozzles, and venturis. 

Rate of discbarge through an orifice meter is given by Eq. (10-8) for either hquids or gases. For the case of subsonic flow of a gas (/ < / < 1.0), the expansion factor Y for orifices is approximated by... 

The expansion factor e takes into account the compressibility effects of the fluid. It is close to unity in most industrial ventilation applications. 

For flow of compressible fluids use the net expansion factor Y (see later discussion) [3] ... 

For the discharge of compressible fluids from the end of a short aiping length into a larger cross section, such as a larger pipe, vessel, or atmosphere, the flow is considered adiabatic. Corrections are applied to the Darcy equation to compensate for fluid property changes due to the expansion of the fluid, and these are known as Y net expansion factors [3]. The corrected Darcy equation is ... 

Figure 2-38A. Net expansion factor, Y, for compressible flow through pipe to a larger flow area. By permission, Crane Co., Technical Paper U410, Engineering Div., 1957. Also see 1976 edition.

Y = net expansion factor from Figures 2-38A or 2-38B AP = differential pressure (equal to inlet gauge pressure when discharging to atmosphere)... 

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