Volume fraction mixture

V, = molar volume of component i Xj, = volume fraction of component i Xj = mole fraction of component i l = conductivity of the component i in the liquid state A, = conductivity of the mixture in the liquid state... 

Figure C2.1.10. (a) Gibbs energy of mixing as a function of the volume fraction of polymer A for a symmetric binary polymer mixture = Ag = N. The curves are obtained from equation (C2.1.9 ). (b) Phase diagram of a symmetric polymer mixture = Ag = A. The full curve is the binodal and delimits the homogeneous region from that of the two-phase stmcture. The broken curve is the spinodal.

Figure C2.6.10. Phase diagram of colloid-polymer mixtures polymer coil volume fraction vs particle...

This result should be compared with Eq. (8.28) for the case of the ideal mixture. It is reassuring to note that for n = 1, Eq. (8.36) reduces to Eq. (8.28). Next let us consider whether a change of notation will clarify Eq. (8.36) still more. Recognizing that the solvent, the repeat unit, and the lattice site all have the same volume, we see that Ni/N is the volume fraction occupied by the solvent in the mixture and nN2/N is the volume fraction of the polymer. Letting be the volume fraction of component i, we see that Eq. (8.36) becomes... 

The polyamides are soluble in high strength sulfuric acid or in mixtures of hexamethylphosphoramide, /V, /V- dim ethyl acetam i de and LiCl. In the latter, compHcated relationships exist between solvent composition and the temperature at which the Hquid crystal phase forms. The polyamide solutions show an abmpt decrease in viscosity which is characteristic of mesophase formation when a critical volume fraction of polymer ( ) is exceeded. The viscosity may decrease, however, in the Hquid crystal phase if the molecular ordering allows the rod-shaped entities to gHde past one another more easily despite the higher concentration. The Hquid crystal phase is optically anisotropic and the texture is nematic. The nematic texture can be transformed to a chiral nematic texture by adding chiral species as a dopant or incorporating a chiral unit in the main chain as a copolymer (30). 

Eig. 10. The variation of the density of carbon-fiber reinforced epoxy resin with the fiber volume fraction, based on the rule of mixtures. 

The difference between the bounds defined by the simple models can be large, so that more advanced theories are needed to predict the transverse modulus of unidirectional composites from the constituent properties and fiber volume fractions (1). The Halpia-Tsai equations (50) provide one example of these advanced theories ia which the rule of mixtures expressions for the extensional modulus and Poisson s ratio are complemented by the equation... 

Siddiqi-Lucas suggested that component volume fractions might be used to correlate the effects of concentration dependence. They found an average absolute deviation of 4.5 percent for nonpolar-nonpolar mixtures, 16.5 percent for polar-nonpolar mixtures, and 10.8 percent for polar-polar mixtures. 

A composite material for a car-repair kit consists of a random mixture of short glass fibres in a polyester matrix. Estimate the maximum toughness of the composite. You may assume that the volume fraction of glass is 30% the fibre diameter is 15 pm the fracture strength of the fibres is 1400 MPa and the shear strength of the matrix is 30 MPa. 

The premise that discontinuous short fibers such as floating catalyst VGCF can provide structural reinforcements can be supported by theoretical models developed for the structural properties of paper Cox [36]. This work was recently extended by Baxter to include general fiber architecture [37]. This work predicts that modulus of a composite, E can be determined from the fiber and matrix moduli, Ef and E, respectively, and the fiber volume fraction, Vf, by a variation of the rule of mixtures,... 

Concentrations of moderator at or above that which causes the surface of a stationary phase to be completely covered can only govern the interactions that take place in the mobile phase. It follows that retention can be modified by using different mixtures of solvents as the mobile phase, or in GC by using mixed stationary phases. The theory behind solute retention by mixed stationary phases was first examined by Purnell and, at the time, his discoveries were met with considerable criticism and disbelief. Purnell et al. [5], Laub and Purnell [6] and Laub [7], examined the effect of mixed phases on solute retention and concluded that, for a wide range of binary mixtures, the corrected retention volume of a solute was linearly related to the volume fraction of either one of the two phases. This was quite an unexpected relationship, as at that time it was tentatively (although not rationally) assumed that the retention volume would be some form of the exponent of the stationary phase composition. It was also found that certain mixtures did not obey this rule and these will be discussed later. In terms of an expression for solute retention, the results of Purnell and his co-workers can be given as follows,... 

Vs) is the total volume of stationary phase in the column and (a) is the volume fraction of phase (A) in the stationary phase mixture... 

Katz et al. tested the theory further and measured the distribution coefficient of n-pentanol between mixtures of carbon tetrachloride and toluene and pure water and mixtures of n-heptane and n-chloroheptane and pure water. The results they obtained are shown in Figure 17. The linear relationship between the distribution coefficient and the volume fraction of the respective solvent was again confirmed. It is seen that the distribution coefficient of -pentanol between water and pure carbon tetrachloride is about 2.2 and that an equivalent value for the distribution coefficient of n-pentanol was obtained between water and a mixture containing 82%v/v chloroheptane and 18%v/v of n-heptane. The experiment with toluene was repeated using a mixture of 82 %v/v chloroheptane and 18% n-heptane mixture in place of carbon tetrachloride which was, in fact, a ternary mixture comprising of toluene, chloroheptane and n-heptane. The chloroheptane and n-heptane was always in the ratio of 82/18 by volume to simulate the interactive character of carbon tetrachloride. 

The column was operated in the normal phase mode using mixtures of n-hexane and ethanol as the mobile phase. Equation (13) is validated by the curves relating the corrected retention volume to the reciprocal of the volume fraction of ethanol in Figure 19. It is seen that an excellent linear relationship is obtained between the corrected retention volume and the reciprocal of the volume fraction of ethanol. 

Figure 23. A 3-D Set of Curves Relating the Retention Volume of 4-Phenyl-2-oxazolidinone to the Column Temperature and the Volume Fraction of Ethanol in the Solvent Mixture...

The numerical constants were obtained over the temperature range of 5°C to 45°C and a concentration range of 0 to 0.5 volume fraction of ethanol inn-hexane.The effect of temperature and solvent composition on solute retention can, again, be best displayed by the use of 3-D graphs, and curves relating both temperature and solvent composition to the retention volume of the (S) enantiomer of 4-benzyl-2-oxazolidinone are shown in Figure 23. Figure 23 shows that the volume fraction of ethanol in the solvent mixture has the major impact on solute retention. 

If the volume fraction of methanol in the original mixture was (a), then the volume fraction of water would be (1-a). The molar volume of a substance is the ratio of the molecular weight to the density and thus the molar concentration of methanol and... 

Thus the weight (mi) of the mixture originally containing a volume fraction (a]) of methanol and a volume fraction (1-cti) of water will be... 

Employing an iterative computer program, in conjunction with the above equations, Katz et al. examined a wide range of values for (k) and (Vmw) nd calculated the volume change on mixing of a series of methanol/water mixtures having assumed volume fractions of methanol. The results for each selected values of (k) and (Vmw) and each volume fraction of methanol were compared with experimentally determined values of (vj) and the specific values of (k) and (Vmw) that gave the minimum error... 

The results obtained by Katz et al. [15] are shown as experimental points on the curves relating the distribution coefficient of the solute against volume fraction of methanol added to the original mixture in Figure 31. Due to the difficulty of measuring the distribution coefficient of each solute between pure water and hexadecane (because of their extremely high retention), the values were obtained from a polynomial curve fit to the data which gave a value for (K) at a = 0. 

Figure 31. Graph of the Distribution Coefficient of the Solutes between Methanol/Water Mixtures and Hexadecane against Volume Fraction of Methanol in the Original Mixture...

Scott and Beesley [2] measured the corrected retention volumes of the enantiomers of 4-benzyl-2-oxazolidinone employing hexane/ethanol mixtures as the mobile phase and correlated the corrected retention volume of each isomer to the reciprocal of the volume fraction of ethanol. The results they obtained at 25°C are shown in Figure 8. It is seen that the correlation is excellent and was equally so for four other temperatures that were examined. From the same experiments carried out at different absolute temperatures (T) and at different volume fractions of ethanol (c), the effect of temperature and mobile composition was identified using the equation for the free energy of distribution and the reciprocal relationship between the solvent composition and retention. 

Consider the effect of the abstraction of individual solvents on consequent solute concentration. Let the volume fraction of ethanol, ethyl acetate and n-hexane in the original mobile phase mixture be (ae), (cxea) and (an), respectively. If a given solute is placed on the TLC plate at a concentration (cs) then, assuming for the... 

The mixture cohesive energy density, coh-m> was not to be obtained from some mixture equation of state but rather from the pure-component cohesive energy densities via appropriate mixing rules. Scatchard and Hildebrand chose a quadratic expression in volume fractions (rather than the usual mole fractions) for coh-m arid used the traditional geometric mean mixing rule for the cross constant ... 

This is an important relationship. It states that the modulus of a unidirectional fibre composite is proportional to the volume fractions of the materials in the composite. This is known as the Rule of Mixtures. It may also be used to determine the density of a composite as well as other properties such as the Poisson s Ratio, strength, thermal conductivity and electrical conductivity in the fibre direction. 

The volume fraction of the gas in the mixture, or void fraction or voidage, as it is also called, is defined as... 

---