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# 物理代写|热力学代写Thermodynamics代考|How Do We Obtain Activity Coefficients?

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## 物理代写|热力学代写Thermodynamics代考|How Do We Obtain Activity Coefficients?

The experimental methods used to acquire values of the activity coefficient have been alluded to in Question 4.6.5. Other methods rely on the use of Equation 4.178. For the majority of cases, it is necessary to have experimental values of the activity coefficients for substance $\mathrm{B}$ in a binary mixture. A typical experimental determination of the activity coefficient therefore requires measurements of the total pressure $p$, as well as the mole fractions, $y_{\mathrm{B}}$ and $x_{\mathrm{B}^{\prime}}$ in the vapor and liquid phase, respectively, for a binary mixture at vapor-liquid equilibrium at a particular temperature.

Measurements as a function of mole fraction for the liquid phase are used to determine the parameters in a suitable activity-coefficient model. As an example, Figure 4.2 shows the measured $(p, x, y)_T$ at $T=318.15 \mathrm{~K}$ for (nitromethane + tetrachloromethane) while, in Figure 4.3, the corresponding activity coefficients of both components are shown also as a function of liquid composition. The mixture (nitromethane + tetrachloromethane) is not ideal and, as expected, the activity coefficients for both substances are greater than unity.

The Poynting factor given by Equation 4.101 is set equal to unity, which is a reasonable assumption provided the pressure does not differ significantly from the vapor pressure of the pure components.

## 物理代写|热力学代写Thermodynamics代考|Activity Coefficient Models

The first model of this type was reported by Margules (1895) and represented the logarithm of the activity coefficient by a power series in composition for each component. van Laar (1910 and 1913) proposed a model based on van der Waals’s equation of state with two adjustable parameters; predictive capabilities of that scheme have been found to be limited.
Typically, the model requires the measurement of (vapor + liquid) equilibria at a given temperature for all possible binary mixtures formed from the components of the fluid. The parameters of the activity coefficient model are then fitted to experimental data for binary mixtures. The resulting model can be applied to predict the activity coefficients of a multicomponent mixture over a range of temperature and pressure. For binary mixtures, the model is used to extrapolate the measured values with respect to temperature and pressure. For multicomponent mixtures, the model also exploits extrapolation of the composition. Examples of this approach are Wilson (1964), T-K-Wilson (Tsuboka and Katayama 1975), the Non-Random Two-Liquid model (NRTL) of Renon (1968 and 1969) and UNIQUAC (Abrams and Prausnitz 1975). Certainly, the most reliable procedure for the determination of parameters in any activity-coefficient model involves a fit to experimental data over a range of liquid compositions. The solution of the model for the parameters which best represent the data is a matter for nonlinear regression analysis. However, whatever the solution eventually found, it must still conform to the Gibbs-Duhem Equation 4.174. A description of activity coefficient models has been given by Assael et al. (1996), or Kontogeorgis and Folas (2010).

The requirement to measure the (vapor + liquid) equilibria for all binary mixtures can be rather onerous and it will be no surprise to learn that engineers have created other approximate routes that either reduce or eliminate recourse to specific measurements. In the absence of sufficient measurements, the model parameters are often estimated from Equations 4.157 to 4.159 . In this case, the activity coefficients of each of the components A and B in a binary mixture in the limit as their mole fractions approach unity (often called infinite-dilution) are used, because access to the parameters of an empirical model is simplified. For example, the Wilson method may be implemented from the two infinite-dilution activity coefficients for a binary pair. Other models of this type have been proposed by Pierotti et al. (1959) and Helpinstill and van Winkle (1968) for polar mixtures. Thomas and Eckert (1984) proposed the Modified Separation of Cohesive Energy Density (given the acronym MOSCED) model for predicting infinite-dilution activity coefficients from pure component parameters only.

In the absence of specific measurements, the parameters of the activity-coefficient model can be estimated using a group-contribution method which assumes that groups of atoms within a molecule contribute in an additive manner to the overall thermodynamic property for the entire molecule. Thus, a methyl group may make one kind of contribution while a hydroxyl group makes another contribution. Once the contributions to the property from each group of the molecule have been determined the activity coefficient of the molecule can be obtained from the contributions of the groups it contains. Schemes of this type ultimately rely on (vapor + liquid) equilibria measurements that are used with definitions of the groups within molecules to determine the parameters of a model for the molecular group by regression. Examples of this approach are the Analytical Solution of Groups (ASOG) (Wilson and Deal 1962; Wilson 1964; Kojima and Toshigi 1979) and the Universal Functional Group Activity Coefficients (UNIFAC) (Fredenslund et al. 1975; 1977) models; the UNIFAC method is widely used (Kontogeorgis and Folas 2010).

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