Equations solubility coefficient

Fig. 5. Solubility coefficient at 30°C versus boiling point of ester in a low density polyethylene film (18). For unit conversion see equation 6.

The temperature dependence of the permeability arises from the temperature dependencies of the diffusion coefficient and the solubility coefficient. Equations 13 and 14 express these dependencies where and are constants, is the activation energy for diffusion, and is the heat of solution... 

The solubihty coefficients are more difficult to predict. Although advances are being made, the best method is probably to use a few known solubility coefficients in the polymer to predict others with a simple plot of S vs ( poiy perm Y where and are the solubility parameters of the polymer and permeant respectively. When insufficient data are available, S at 25°C can be estimated with equation 19 where k = 1 and the resulting units of cal/cm are converted to kj /mol by dividing by the polymer density and multiplying by the molecular mass of the permeant and by 4.184 (16). 

The Ksp expressions are based on balanced equations for saturated solutions of slightly soluble ionic compounds. The exponents in the K p expressions match the corresponding coefficients in the chemical equation. The coefficient 1 is not written, following chemical convention. 

Empirical equations have been formulated to enable calculation of the Bimsen solubility coefficient for any given temperature and salinity at = 1 atm. These empirical equations are presented in the online appendix on the companion website for the most common gases foimd in seawater but being empirical, they are still subject to refinement. The equilibrium gas concentrations computed from the Bimsen solubility coefficient should be thought of as the gas concentration that a water mass would attain if it were allowed to equilibrate with the atmosphere at its in situ salinity and potential temperature. 

The volume in milliliters of gas dissolved per milliliter of liquid at one atmosphere of partial pressure of the gas at any given temperature. Thus, the Ostwald coefficient (A) differs from the Bunsen solubility coefficient (a) which is based on standard temperature and pressure. The two coefficients are related by the equation A = a(l + 0.00367t) where t is the temperature in degrees Celsius. 

The calculations for problems such as this must be based on the Ksv of the more soluble compound—in this case, the Mg(OH)2. Two simultaneous equilibria are involved the solubility of Mg(OH)2 (horizontal equation), and the dissociation of NH3 (vertical equation). The coefficient of 2 belongs only with the horizontal equation. The concentration of OH" is common to both equilibria. 

Many computational studies of the permeation of small gas molecules through polymers have appeared, which were designed to analyze, on an atomic scale, diffusion mechanisms or to calculate the diffusion coefficient and the solubility parameters. Most of these studies have dealt with flexible polymer chains of relatively simple structure such as polyethylene, polypropylene, and poly-(isobutylene) [49,50,51,52,53], There are, however, a few reports on polymers consisting of stiff chains. For example, Mooney and MacElroy [54] studied the diffusion of small molecules in semicrystalline aromatic polymers and Cuthbert et al. [55] have calculated the Henry s law constant for a number of small molecules in polystyrene and studied the effect of box size on the calculated Henry s law constants. Most of these reports are limited to the calculation of solubility coefficients at a single temperature and in the zero-pressure limit. However, there are few reports on the calculation of solubilities at higher pressures, for example the reports by de Pablo et al. [56] on the calculation of solubilities of alkanes in polyethylene, by Abu-Shargh [53] on the calculation of solubility of propene in polypropylene, and by Lim et al. [47] on the sorption of methane and carbon dioxide in amorphous polyetherimide. In the former two cases, the authors have used Gibbs ensemble Monte Carlo method [41,57] to do the calculations, and in the latter case, the authors have used an equation-of-state method to describe the gas phase. 

Transport through a dense polymer may be considered as an activated process, which can be represented by an Arrhenius type of equation. This implies that temperature may have a large effect on the transport rate. Equations 4.7 and 4.8 express the temperature dependence of the diffusion coefficient and solubility coefficient in Equation 4.5 ... 

The equation shown in Table 27-6 illustrates the complexity of the calculation to correct PO2 to the patient s body temperature. Complexity is unavoidable because at PO2 less than lOOmmHg (SO2 0.95), the hemoglobin-02 dissociation curve is shifted to the left by the decrease in temperature and by the concomitant rise in pH (see Figure 27-3). For temperature corrections of PO2 between 100 and 400mmHg, accurate formulas become even more complicated. The most accurate calculation of the temperature variation of PO2 is made by iterative calculations when the only necessary parameters are the temperature coefficients of the P50 and the solubility coefficient of O2 (a02). Several analyzers perform such calculations. 

Henry s Law states that the amount of gas dissolved by a given liquid, with which, it does not combine chemically, is directly proportional to the partial pressure of the gas if the pressure of a gas is doubled then the amount of gas physically dissolved in the solution is doubled. The constant which converts the proportionality to an equality in the Henry s Law equation is called the Henry s Law constant this constant is the solubility coefficient of the gas in the particular solution. The solubility coefficient varies with the nature of the gas and liquid, the presence of solutes in the liquid, and inversely with the temperature. Thus at a constant pressure, but under hypothermic conditions, more gas can be dissolved in a given amount of fluid (tissue). 

The diffusion coefficient, sometimes called the diffiisivity, is the kinetic term that describes the speed of movement. The solubility coefficient, which should not be called the solubility, is the thermodynamic term that describes the amount of permeant that will dissolve in the polymer. The solubility coefficient is a reciprocal Henry s Law coefficient as shown in equation 3. 

The solubility coefficient must have units that are consistent with equation 3. In the literature S has units cc(STP)/(cm3-atm), where cc(STP) is a molar unit for absorbed permeant (nominally cubic centimeters of gas at standard temperature and pressure) and cm 3 is a volume of polymer. When these units are multiplied by an equilibrium pressure of permeant, concentration units result. In preferred SI units, S has units of nmol /(m3GPa). 

In the mass units for flavor, aroma, and solvent molecules, the solubility coefficient has units kg/( m3Pa). Equation 6 shows how to convert from the mass units to the molar units. 

Table 10 contains some selected permeability data including diffusion and solubility coefficients for flavors in polymers used in food packaging. Generally, vinylidene chloride copolymers and glassy polymers such as polyamides and EVOH are good barriers to flavor and aroma permeation whereas the polyolefins are poor barriers. Comparison to Table 5 shows that the laige molecule diffusion coefficients are 1000 or more times lower than the small molecule coefficients. The solubility coefficients are as much as one million times higher. Equation 7 shows how to estimate the time to reach steady-state permeation / if the diffusion coefficient and thickness of a film are known. 

Carbon Dioxide Transport. Measuring the permeation of carbon dioxide occurs far less often than measuring the permeation of oxygen or water. A variety of methods are used however, the simplest method uses the Permatran-C instrument (Modem Controls, Inc.). In this method, air is circulated past a test film in a loop that includes an infrared detector. Carbon dioxide is applied to the other side of the film. All the carbon dioxide that permeates through the film is captured in the loop. As the experiment progresses, the carbon dioxide concentration increases. First, there is a transient period before the steady-state rate is achieved. The steady-state rate is achieved when the concentration of carbon dioxide increases at a constant rate. This rate is used to calculate the permeability. Figure 18 shows how the diffusion coefficient can be determined in this type of experiment. The time lag is substituted into equation 21. The solubility coefficient can be calculated with equation 2. 

Equations 4-8 indicate that P is a product of a diffusion coefficient (a kinetic factor) and of a solubility coefficient (a thermodynamic factor). A common method of inferring the mechanism of permeation is to determine the dependence of P on p (or c) and on the temperature. This requires, in turn, a knowledge of the dependence of D and S on these variables. As is shown below, this dependence is very different above and below the glass-transition of the polymer. 

For a thorough analysis of aroma transport in food packaging, the permeability (P) and its component parts - the solubility coefficient (SJ and the diffusion coefficient or diffusivity (D) are needed. These three parameters are related as shown in Equation 1. 

The solubility coefficients make interesting contributions to the permeabilities. The solution process of these gas phase permeants can be separated for analysis into two components -condensation and mixing. Table III contains the heats of condensation for the experimental esters. Table III also contains the heats of solution in the polymer films from linear fits of the data in Figures 10 and 15. Finally, Table III contains the heats of mixing for the esters in the films. The heats of mixing were calculated using equation 9. 

For slightly soluble gases in membranes, the concentration is proportional to the partial pressure, P. The solubility coefficient, S, can be combined with the diffusivity to give the permeability coefficient, [PM]. Recognizing that at the surface, n = 0 and the membrane is in contact with a high vacuum, we reduce Equation 11 to... 

To evaluate the mass transport equation, MTE (Equation 15), with the proper boundary conditions, it is necessary to obtain a functional relationship between the concentrations and partial pressures of the gas dissolved in the fluid. For water, 02, N2, and C02 can be assumed to be linear—i.e., have a constant solubility coefficient, S. ... 

Partial pressure values are calculated by dividing the concentration c by the solubility coefficient a). To obtain a differential equation for c in the capillaries, we refer to Equation 1. For a stationary case and Q(c) = 0, we get... 

The importance of this equation is that it demonstrates that 7 is a linear function of the test pressure P, as long as the transition pressure between diffusive flow and capillary flow is not reached or exceeded. Other variables that must be controlled in diffusion testing include (a) the filter membrane area, because it defines the effective area of pores or void fraction (b) the temperature, because it defines the solubility of gas in liquid and (c) the composition of the liquid phase, because the presence of solutes affects the solubility coefficient. 

When evaluating the behavior of individual nonpolar components, is usually used partition coefficient or Bunsen and Henry solubility coefficients. For pure water in standard conditions these parameters are tied between themselves by equation... 

The value of the Sechenow coefficient, as well as the solubility and Bunsen coefficients, depends on temperature. This is why attempts were undertaken to express the correlation of Bunsen solubility coefficient vs. temperature and salinity by one common equation. For instance, Ray Weiss (1970) gathered published data on solubility, put them into one dimension and using the least square technique came up with empiric exponential series of the correlation between Bunsen solubility coefficient (mole-k bar"0 vs. temperature and water salinity ... 

Partial pressure of a gas component in water under the same conditions may be determined using Henry or Bunsen solubility coefficients identified for partial pressures of 1 bar. If the value of Henry solubility coefficient is available, partial pressure of a component i is calculated from equation (2.280). Values of Henry solubility coefficients may be found in reference literature (Table 2.27). However, these values are applicable only... 

Much more often used is Bunsen solubility coefficient. Then partial pressure is determined from equation... 

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