Fitting of the Margules equations to limited data has been discussed in Examples 11.2 and 11.5. Fits to multiple points are preferred, which requires regression of the parameters to optimize the fit. In a few cases, the Gibbs excess function can be rearranged to form the basis for a linear regression. In general, a non-linear regression may be required. Modern computers facilitate either method.
Linear Fitting of the Margules Equation
Eqn. 11.33 can be linearized:.
Margules models can be linearized for fitting of parameters.
Therefore, plotting GE/(x1x2RT) versus x1 gives a slope of (A21 – A12) and an intercept of A12. The value of A21 can also be determined by the value at x1 = 1. Using the data for 2-propanol + water listed in Example 11.8 (ignoring the first mixture point) results in a slope = –0.9289, intercept = 2.001. Thus, A12 = 2.001, A21 = 1.072, slightly different from the single point fit of Example 11.5.
Nonlinear parameter fitting is possible in Excel and MATLAB.
Nonlinear fitting techniques
In general, parameters for excess Gibbs models are nonlinearly related to GE or γ. Even in the cases of the Redlich-Kister and Margules equations, it may be more convenient to simply apply a nonlinear fitting procedure. The parameters can be fitted to the experimental data to optimize the fit to the experimental bubble pressure (P-x-y data) or bubble temperature (T-x-y data). In the case of fitting bubble-pressure data, the parameters can be used to generate a bubble-pressure calculation at each experimental xi as demonstrated with the one-parameter Margules in Example 11.3. The Excel Solver tool or the MATLAB fminsearch or lsqnonlin can provide rapid fits. The spreadsheet Gammafit.xlsx or MATLAB m-file GammaFit.m permit nonlinear fitting of activity coefficient parameters for the Margules equation by fitting total pressure. Either can be easily modified to find parameters for any activity coefficient model. The strategy implemented here is to calculate the activity coefficients using assumed values for the parameters and generate a bubble-pressure calculation using Eqn. 11.2 for each of the experimental points. Then each bubble pressure is compared to the experimental bubble pressure, and 
is calculated. The optimizer is invoked to keep adjusting the parameters until OBJ is minimized.
Example 11.8. Fitting parameters using nonlinear least squares
Measurements for the 2-propanol + water system at 30oC have been published by Udovenko, et al. (1967).a Use the pressure and liquid composition to fit the two-parameter Margules equation to the bubble pressure. Plot the resultant P-x-y diagram.
Solution
In the experimental data, the researchers report experimental vapor pressures. It is best to use experimental values from the same publication to reduce the effect of systematic errors which may be present in the data due to impurities or calibration errors. The solution will be obtained by minimizing the sum of squares of error for the bubble pressures across the composition range.
MATLAB (condensed to show the major steps):
function GammaFit()
% statements omitted to load experiments into matrix ‘Data’
x1 = Data(:,1); %data have been entered into columns of ‘Data’
y1expt = Data(:,2); Pexpt = Data(:,3);
Ps1Calc = 60.7; Ps2Calc = 32.1; %experimental values used for Psat
x2 = 1-x1; % calculate x2
x = [x1 x2]; % create a 2 column matrix of x1 & x2
A = [0 0]; % initial guess for A12 and A21
A = lsqnonlin(@calcError,A); %optimize, calling ‘calcError’ as needed
function obj = calcError(A)
A12 = A(1); %extract coeffs so eqns look like text
A21 = A(2);
Gamma1Calc = exp((x2.^2).*(A12 + 2* (A21 – A12).*x1));
Gamma2Calc = exp((x1.^2).*(A21 + 2* (A12 – A21).*x2));
Pcalc = (x1.*Gamma1Calc)*(Ps1Calc) + …
(x2.*Gamma2Calc)*(Ps2Calc);
obj = Pcalc – Pexpt;
end
The resultant parameters are A12 = 2.173, A21 = 0.9429. The distributed file includes statements to plot the final figure similar to that shown below. Note that fminsearch can be used if lsqnonlin is not available due to the toolboxes on your MATLAB installation. See the fit in Fig. 11.11.
Figure 11.11. Comparison of experimental data with regressed model as explained in Example 11.8.
Excel: The spreadsheet “P-x-y fit P” in the workbook Gammafit.xlsx is used to fit the parameters as shown below. Antoine coefficients are entered in the table for the components shown at the top of the spreadsheet. The flag in the box in the center right determines whether experimental vapor pressures are used in the calculations or values calculated from the Antoine equation.
Experimental data for x1 and Pexpt are entered in columns A and I. Initial guesses for the constants A12 and A21 are entered in the labeled cells in the top table. Solver is then called to minimize the error in the objective function by adjusting the two parameters. Calculated pressures are determined by bubble-pressure calculations.
The results of the fit are shown by the plot on spreadsheet “P-x-y Plot.” See the fit in Fig. 11.11.
Note that the system is the same used in Example 11.2 on page 417 and Example 11.5 on page 430. The fit in this example using all data is superior. The parameters are also slightly different from the linear fit discussed above because the objective function is different.
a. Udovenko, V.V., Mazanko, T.F. 1967. Zh. Fiz. Khim. 41:1615.
Alternative Objective Functions
An alternative choice of objective function for a given set of data usually results in a slightly different set of parameters. While the total pressure is often measured accurately, it may be desired to include vapor compositions in the objective function. For example, it is not uncommon for the pure component vapor pressures measured by investigators to differ from literature data; this is indicative of an impurity or a systematic error. One method of incorporating additional considerations into the fitting procedure is to use weighted objective functions, where recognition is made of probable errors in measurements. One of the most rigorous methods uses the maximum likelihood principles, which asserts that all measurements are subject to random errors and therefore have some uncertainty associated with them. Such techniques are discussed by Anderson, et al.,13 and Prausnitz, et al.14 The objective function for such an approach takes the form
where σ represents the variance for each type of measurement. The “true” values are calculated as part of the procedure. Typical values for variances are: σP = 2 mmHg, σT = 0.2 K, σx = 0.005, σy = 0.01, and therefore the weight of a measurement depends on the probable experimental error in the value.
11.10. Relations for Partial Molar Properties
Gibbs-Duhem Equation
A useful expression known as the Gibbs-Duhem equation results when we analyze Eqn. 10.40 together with Eqn. 10.42. Consider the differential of Eqn. 10.42 using the product rule,
Substituting for dG in Eqn. 10.40 results in
Simplifying, we obtain the Gibbs-Duhem equation,
Therefore, we conclude at constant T and P:
The relation is typically applied in the context of activity coefficients, as described below.
Gibbs-Duhem Relation for Activity Coefficients
To extend the Gibbs-Duhem equation to excess properties, the excess Gibbs energy can be manipulated in an manner analogous to the derivation above. Therefore,
resulting in
Inserting the relation between excess chemical potential and activity coefficients gives
Technically it is not possible to vary composition for two coexisting phases in a binary without either T or P changing. However, experimental analysis of isothermal P-x-y data or isobaric T-x-y data shows that the SE and VE terms are almost always very small compared to the last term and Eqn. 11.48 is generally an excellent approximation. If we apply the expansion rule, Eqn. 6.17, using one of the mole fractions, and divide by n, in a binary
This equation means that the activity coefficients for a binary system, when plotted versus composition, must have slopes with opposite signs, and the slopes are related in magnitude by Eqn. 11.49. A further deduction is that if one of the activity coefficients in a binary system exhibits a maximum, the other must exhibit a minimum at the same composition. We find this relation useful in: 1) testing data for experimental errors (grossly inconsistent data); 2) generating the activity coefficients in a binary for a second component based on the behavior of the first component in experimental techniques where only one activity coefficient is measured; 3) for development of theories for the Gibbs energy of a mixture, since our model must follow this relation. The Gibbs-Duhem equation is also useful for checking thermodynamic consistency of data; however, the applications are subject to uncertainties themselves because the activity coefficient is itself derived from assumptions, (e.g., modified Raoult’s law).15 Fortunately, developers of activity coefficient models are generally careful to ensure the models satisfy the Gibbs-Duhem equation. An understanding of the restrictions of the Gibbs-Duhem equation is helpful when studying alternative standard states such as Henry’s law or electrolyte models. A particularly useful application of the Gibbs-Duhem equation in a binary mixture is the use of the activity coefficient of one component to calculate the activity coefficient of the other component.16 Often this can be done by fitting a model to the activity of the first component, but the Gibbs-Duhem equation provides a method that does not require the application of a particular mixture model with its associated assumptions.
Relations between Various Excess Properties
For the development of accurate process calculations, a thermodynamic model should accurately represent the temperature and pressure dependence of deviations from ideal solution behavior. The excess functions follow the same relations as the total functions, HE = UE + PVE, GE = HE – TSE, and AE = UE – TSE. The derivative relations are also followed,
resulting in
The Gibbs-Helmholtz relation applies:
Particularly useful is Eqn. 11.53 using the relation with activity coefficients:
Therefore, excess enthalpy data from calorimetry may be used to check the temperature dependence of the activity coefficient models for thermodynamic consistency. Typically, activity coefficient parameters need to be temperature-dependent for representing data accurately, which implies an excess enthalpy. Likewise, any system with a heat of mixing will have temperature-dependent activity coefficients. A simple model modification is to replace the parameters with functions, for example, Aij = aij + bij/T, where aij and bij are constants and T is in K. This sets the stage for computing heats of mixing of any activity model, as shown in Example 11.9.
Example 11.9. Heats of mixing with the Margules two-parameter model
Fitting the VLE of methanol + benzenea in the range of 308–328 K with the Margules two-parameter model and then fitting the parameters to Aij = aij + bij/T gives A12 = 0.1671 + 714/T and A21 = 2.3360 – 247/T. Estimate the heat of mixing at 318 K and 50 mol% benzene.
Solution
The Margules two-parameter model is,
The relation between GE and HE is given by Eqn. 11.53. Noting the right side Margules parameters for the problem statement are simple functions of (1/T), we can manipulate the derivative for this function. Since d(1/T) = –T–2dT,
Thus,
At 318 K and x1 = x2 = 0.5, HE = 8.314(0.5)(0.5(0.5·714 – 0.5·247) = 485 J/mol. Note that direct measurement of excess enthalpy is recommended when possible. Phase equilibria data must be very precise to provide an accurate enthalpy of mixing.
a. Gmehling J., et al., 1977-. VLE Data Collection. Frankfurt/Main: DECHEMA; Flushing, N.Y.: Distributed by Scholium International.
This example illustrates what it means for the “activity coefficient parameters to be temperature-dependent” and the manner of taking the derivative of GE. Though we have calculated the excess Gibbs energy in the previous example, the parameters may also be used to calculate temperature dependence of activity coefficients. Activity coefficients are strong functions of composition but weak functions with respect to temperature. This becomes apparent as you study more systems.
Recalling that heat of mixing for an ideal solution is zero, we note that the heat of mixing and the excess enthalpy are one and the same. The heat capacity for liquid methanol is about 80 J/mol-K and about 130 J/mol-K for benzene. In an adiabatic mixing process, we would thus expect this equimolar mixture to be colder after mixing by roughly 5°C. (Note: The excess heat capacity, CpE = Cp – Cpis, is the temperature derivative of the excess enthalpy; can you determine its value for this mixture?) In general, we could write that the enthalpy of any given composition is given by,
In this way, we can represent the enthalpy of any stream to perform energy balances.
11.11. Distillation and Relative Volatility of Nonideal Solutions
To illustrate the impact of activity coefficients on practical applications, it is helpful to revisit our discussion of distillation. The relative volatility of the light to heavy key, αLH, is important to distillation, as discussed in Section 10.6. Since αLH may not be constant over an entire distillation column, it is common to estimate the average value by the geometric mean of the bottom and top.
Recalling the definition of αLH from Eqn. 10.32, substituting Eqn. 11.18, and canceling pressures,
Suppose in a binary mixture that we specify splits so that the top is xLKtop = 0.99, and xLKbot = 0.01. Then recognizing that the activity coefficients go to unity near purity,
Note: It is required that αLH > 1 at both ends of the column in order to avoid an azeotrope. In other words, Eqns. 10.35 and 11.57 CANNOT be applied unless αLH > 1 at both ends of the column.
Example 11.10. Suspecting an azeotrope
Make a preliminary estimate of whether we should suspect an azeotrope in the system benzene (B) + 2-propanol (I) at 80°C. Assume the MAB model. A convenient feature of Margules one-parameter models (including the MAB model) is that the infinite dilution activity coefficients are equal. (Note that “convenient” may not equate to “accurate.”)
Solution
Note that this problem is isothermal rather than a distillation column design, but we can evaluate the relative volatility at either end of the composition range. Antoine.xlsx gives vapor pressures of PBsat = 757 mmHg and PIsat = 683 mmHg at 80°C, so benzene is the LK. For the MAB model,
A12 = (9.23 – 0.63)(11.86 – 2.24)(89.8 + 76.8)/[4(8.314)353] = 1.174;
γi∞ = exp(1.174) = 3.235
Using the component key assignments, PLKsat/PHKsat = 757/683 = 1.108. Therefore, at the end rich in LK, αLH = (PLKsat)/(γHK∞PHKsat) = 1.108/3.235 = 0.343, and the end rich in HK, αLH = (γLK∞PLKsat)/(PHKsat) = 3.235·1.108 = 3.58. MAB predicts an azeotrope since (αLH–1) changes sign. The prediction should be validated with experimental data and/or more accurate models because of the approximations in the MAB model.
11.12. Lewis-Randall Rule and Henry’s Law
In this chapter, we have thus far introduced the standard state using the pure component properties at the state of the system (e.g., same T, P, and liquid state). What if the pure liquid substance does not exist at these conditions? For example, in liquid-phase hydrogenation reactions, H2 is far above its critical temperature, yet exists in liquid solution at small concentrations. A pure standard state of liquid H2 is impractical. Similarly, salts dissolve as ions in aqueous solution, but the ions cannot exist as pure liquids. A model for dilute liquid solutions would be convenient, particularly if it is possible to model the liquid as some type of ideal solution. To develop models for this behavior, we first consider the general compositional behavior of the component fugacities. Based on these observations, we introduce Henry’s law to model the solution behavior relative to an ideal solution at dilute concentrations.
The Henry’s Law Standard State
Consider the shape of the component-2 fugacity versus composition that results when Eqn. 11.13 is used along with an activity coefficient model developed in this chapter. If the liquid-phase model parameters provide positive deviations from Raoult’s law, the shape of the component fugacity curve is represented by the curve in Fig. 11.12. In the figure, we follow the widely used convention for dilute binary solutions, where the solvent is designated as component-1, and the dilute solute is component-2. The use of a pure component property as a standard state creates an ideal solution line (Eqn. 10.68) known as the Lewis-Randall rule ideal solution line. Raoult’s law is a special case of the Lewis-Randall ideal solution where we use the vapor pressure to approximate the standard state fugacity.17 Thus, the activity coefficient models that we have developed previously are relative to the Lewis-Randall ideal solution. For a Lewis-Randall ideal solution, the activity coefficients approach one as the concentration approaches purity for that component, and the activity coefficients are usually farthest from unity at infinite dilution.
Figure 11.12. Schematic representation of the fugacity of component 2 in a binary mixture.
Consider that the fugacity curve in Fig. 11.12 is nearly linear at low concentrations. Thus, we could express the component fugacity as proportional to concentration using a tangent line near infinite dilution,
which is the behavior of an ideal solution given by Henry’s law.
The Henry’s law constant, hi, is usually determined experimentally, and depends on temperature, pressure, and solvent. The fact that it depends on solvent makes it very different from a pseudo-vapor pressure because a vapor pressure would be independent of solvent.
Looking at Fig. 11.12, note that Henry’s law fails at high concentrations of a component unless an activity coefficient method is developed. Introducing a Henry’s law activity coefficient to represent non-idealities,
Henry’s law for non-ideal solution.
where γi* is the Henry’s law activity coefficient. For the fugacity curve shown in Fig. 11.12 the Henry’s law activity coefficient needs to be less than one at high concentrations, and the Henry’s law activity coefficient goes to one at infinite dilution (compare relative to the Henry’s law ideal solution line). In contrast, Fig. 11.12 shows that the activity coefficient relative to the Lewis-Randall ideal solution would be greater than one. This can be confusing if you have grown accustomed to activity coefficients less than one meaning that the components “like” each other. The component will have negative deviations from Henry’s law and positive deviations from the Lewis-Randall rule.
Although the Henry’s law activity coefficient goes to one at infinite dilution, it is inaccurate to designate the Henry’s law standard state at that composition. In fact, the correct standard state designation is a hypothetical pure component fugacity (often not experimentally accessible) selected in a manner such that the infinite dilution activity coefficient goes to one. Applying the definition of activity, Eqn. 11.23, we see that 
. Comparing with Eqn. 11.61, we see that the standard state is 
. So the important activity coefficient value is at infinite dilution, but the standard state composition is a hypothetical pure state. This perspective is especially useful for electrolyte solutions.
Relating γi for Henry’s Law and Lewis-Randall Rule
Note in Fig. 11.12 that both Henry’s approach and the Lewis-Randall approach must represent the same fugacity. Equating the two approaches,
Taking the limit at infinite dilution where γ2* approaches one, and we see
resulting in the relation between the Henry’s and the Lewis-Randall fugacity and activity coefficient,
Relation between Henry’s Law constant and Lewis-Randall fugacity.
In Fig. 11.13, look at the right side where lnγ2 approaches lnγ2∞ and lnγ2* approaches zero. The difference in the intercept at x2 = 0 is ln(h2/f2). To model the Henry’s law activity coefficient, the restriction that the activity coefficients must follow the Gibbs-Duhem Eqn. 11.49 remains; thus, the slope of the logarithm of the Henry’s law activity coefficient must be the same as the slope of the logarithm of the Lewis-Randall activity coefficient—the shift is independent of composition. The shift is illustrated in Fig. 11.13. We may adapt any activity model developed for the Lewis-Randall rule to Henry’s law by shifting the intercept values for the components modeled by Henry’s law. Thus,
Figure 11.13. Schematic illustration of the relation of the Henry’s law activity coefficient compared to the Lewis-Randall rule activity coefficient.
Formula to shift a Lewis-Randall activity model to a Henry’s law activity model.
where any Lewis/Randall model can be used for γi and the same model is used for γi∞. Usually the activity coefficient model is manipulated to obtain the infinite dilution activity coefficient expressed in terms of the activity model parameters, and the difference is expressed analytically. For the one-parameter Margules equation, 
. If applied to component 2 (dilute solute) of a binary, with component 1 (rich solvent) represented by the Lewis-Randall rule,
Unsymmetric activity coefficients for Henry’s law based on the one-parameter Margules equation.
Readers should recognize that we have been careful to distinguish between the two standard states in the presentation here, including distinct symbols for the different activity coefficients. When one component is represented with Henry’s law and the other represented by the Lewis-Randall rule, the overall model is described as using the unsymmetrical normalization convention for the activity coefficients.
Henry’s Law on Molal Activity Scale
Eqn. 11.61 suggests that the units for the Henry’s law constant should be pressure, but other conventions also exist. For example, a common way of presenting Henry’s constants for gases is to express the liquid phase concentration in molality and provide a constant inverted relative to hi. The result is
where the change in units for concentration requires a change in the activity coefficient and a change in units of the Henry’s law constant.18 Details on the molal activity coefficient 
are deferred until Chapter 18, but like 
it goes to one at infinite dilution. For a gas phase component, we have seen that 
, and we may use the vapor phase fugacity in the Henry’s law calculation. Many Henry’s law constants in the NIST Chemistry WebBook follow the KH convention for molality concentration units. The relation between molality and mole fraction in water is xi = mixw0.001Mw, w as we derive later (cf. Eqn. 18.147) where Mw,w is the molecular weight of water, and xw is the mole fraction of water.
Molal activity coefficients and Henry’s law.
Example 11.11. Solubility of CO2 by Henry’s Law
Carbon dioxide solubility in water plays a critical role in biological physiology and environmental ocean chemistry, affects the accuracy of acid-base titrations in analytical chemistry, and makes many beverages fizzy. The Henry’s law constant for CO2 in water is listed on the NIST Chemistry WebBooka as KH = 0.035 mol/kg-bar at 298.15 K. Estimate the mole fraction of CO2 in water at 0.7 MPa total pressure and 298.15 K. Treat the vapor phase as an ideal gas and the liquid as an ideal solution with the Henry’s law standard state. Neglect formation of ionic carbonate species.
Solution
KHyCO2 P = mCO2 where mCO2 is mol/(kgH2O). First find yCO2 by using Raoult’s law for water. Taking Pwsat from the steam tables, Raoult’s law for water gives
yw = Pwsat/P = 0.00317/0.7 = 0.0045 => yCO2 = 0.9955.
The solubility of CO2 is thus,
The ionic species ignored here in this binary system are sufficient to lower the pH, and though essential for comprehensive understanding, the concentrations are small relative to the molecular CO2 modeled here. In physiology or ocean chemistry, many other salts are involved which make the equilibrium more complicated. Chapter 18 addresses several issues of ionization.
Dilute Solution Calculations Using Hypothetical Lewis-Randall Fugacities
Dissolved gas solubilities can be modeled by treating the liquid phase and vapor phase both with direct use of an equation of state (to be discussed in Chapter 15). However, Eqn. 11.63 suggests that we can model dilute solutions relative to the Lewis-Randall rule. Looking at Eqn. 11.63, you can appreciate why the Henry’s law constant depends on solvent—the Lewis-Randall γi∞ will be different for every solvent. The activity coefficient models we have developed can take γi∞ into account. What we need is a manner to correlate the fugacity of hypothetical liquids above the critical point. A prevalent model for light gases in petrochemicals is the Grayson-Streed model (and the closely related Chao-Seader and Prausnitz-Shair models).19
Note that the shortcut vapor pressure equation yields finite numerical results for the Lewis-Randall fugacity even when T > Tc. Close to the critical temperature, it is sensible to simply extrapolate the shortcut equation. In that case, the shortcut result is more properly referred to as an estimate of fL since the liquid vapor pressure is not rigorously defined at T > Tc. Common experience is that increasing temperature decreases gas solubility. At temperatures where T > 2Tc, however, a surprising thing happens. The solubility of the gas increases with increasing temperature. In other words, fugacity of the hypothetical gaseous component decreases with increasing temperature. This behavior is predicted by equations of state (Chapter 15), but for simple correlations the previously cited researchers developed correlations of the hypothetical Lewis-Randall fugacities that matched experiments together with simple mixture models (predominantly the regular solution models discussed in the next chapter). We introduce the concepts here using the MAB model.
Fig. 11.14 shows several generalized estimates for fL as a function of reduced temperature. The Grayson-Streed estimates vary substantially depending on whether the general correlation is applied (GS-0) or specific correlations as for methane (CH4) or hydrogen (H2). The Grayson-Streed estimates have limits in their range of temperature that are reflected by their ranges in Fig. 11.14. The Prausnitz-Shair correlation gives a single curve for all compounds regardless of their acentric factor. For our purposes, we would like to have a generalized correlation that improves qualitatively on the shortcut equation at high temperatures. This goal can be achieved with the following correlation for the Lewis-Randall fugacity:
Figure 11.14. Comparison of correlations for liquid fugacity at high temperature as described in the text. GS = Grayson/Streed, PS = Prausnitz/Shair, SCVP = shortcut vapor pressure equation., SCVP+ = extended shortcut vapor pressure equation. (ω = 0.21 for SCVP and SCVP+)
SCVP+ model for extrapolating Lewis-Randall fugacity above Tc.
This correlation is designed to match the shortcut vapor pressure (SCVP) equation at T < Tc. It provides a reasonable match of the Grayson-Streed estimates for CH4 at T > Tc and a fairly accurate match to the Prausnitz-Shair correlation when a value of ω = 0.21 is applied to Eqn. 11.68. It also provides reasonable results for all temperatures. We refer to this as the SCVP+ equation.
Example 11.12. Henry’s constant for CO2 with the MAB/SCVP+ model
The solubility for CO2 in water at 298 K and 7 bar can be estimated as xCO2 = 0.0044. Treating the gas phase as an ideal gas and neglecting any aqueous ionic species, (a) fit γi∞ using the Lewis-Randall rule and the SCVP+ equation for pure CO2 and determine the one-parameter Margules parameter; (b) estimate A12 of the MAB model for CO2 in water and γi∞ and compare to part (a); (c) predict Henry’s constant at 311 K using the MAB and the SCVP+ equation.
Solution
a. First use the SCVP+ equation to predict the hypothetical liquid fugacity, log10(fL / 73.82) = 7(1 + 0.228)(1 – 304.2/298)/3 – 3exp(–3·304.2/298) ⇒ fCO2L = 73.82·10–0.200 = 47 bar.
(FYI: The Lewis-Randall standard state by the SCVP model would be 64 bar instead of 47 bar.) Referring to Example 11.11, the ideal gas vapor fugacity has been calculated there, and we can equate it with Henry’s law and use the fugacity just calculated with the experimental xCO2,
yCO2P = 7·0.9955 = xCO2γCO2∞fCO2L ⇒ γi∞ = 7·0.9955/(0.0044·47) = 34.
lnγCO2∞ = A12 = ln(34) = 3.52
b. For MAB the default estimate is A12 = (α2 – α1)(β2 – β1)(V1 + V2)/(4RT)
A12 = (1.87 – 50.13)(0 – 15.06)(44/1.18 + 18/1)/(4(8.314)298) = 4.05
γi∞ = exp(4.05) = 57.4
The MAB prediction for A12 is approximately (100%)(4.05 – 3.52)/3.52 = 15% too high.
c. At 311 K, the fitted MAB model suggests that A12 = 3.52(298/311) = 3.37 = lnγCO2∞. So, γCO2∞ = 29.
By Eqn. 11.68, log10(fL/73.82) = 7(1 + 0.228)(1 – 304.2/311)/3 – 3exp(–3·304.2/311) = –0.0968
By Eqn. 11.64, hCO2 = γCO2∞ fCO2L = 29(10–0.0968) = 23 bar
Note that γi∞ > 10 for CO2 in H2O with the SCVP+ model of Henry’s law, suggesting that CO2 and water are not very compatible. In fact, the CO2+H2O system does exhibit VLLE, affirming that this approach to Henry’s law maintains consistency with the Lewis-Randall perspective. In a similar manner, other activity models, compounds, and conditions can be characterized.
11.13. Osmotic Pressure
Semi-permeable membranes exhibit the remarkable ability to sort molecules at the nanoscale. Semi-permeable membranes are used in reverse-osmosis water purification where water can permeate but salts cannot and in dialysis membranes where blood is purified. Cell walls and cell membranes in biological systems also have selective permeability to many species. Consider the membrane shown in Fig. 11.15(a) where pure W is on the left and a mixture of W + C is on the right. (Often the solvent is water but in polymer chemistry organic solvents can be used.) The membrane is permeable to W but not to C. If the solutions are at the same pressure, P, then component W spontaneously flows from the left chamber (higher chemical potential because higher mole fraction) to the right chamber (lower chemical potential because lower mole fraction) in the condition of osmosis. If the pressure on the right side is increased, the degree of flow can be decreased. When the pressure has been increased by the osmotic pressure, Π, the sides achieve phase equilibrium and flow stops. If the pressure on the right side is increased by more than the osmotic pressure, a condition of reverse osmosis exists and component W flows from the right to the left. Reverse osmosis is on the verge of becoming the largest scale chemical engineering unit operation in the world as populations grow and water becomes scarce.
Figure 11.15. (a) Illustration of a semipermeable membrane. The membrane is permeable to component W, but not component C. The label of W is convenient because water is a typical solvent. (b) Illustration of the path used in calculation of the chemical potential in the mixture at the osmotic pressure.
At the pressure (P + Π) on the right side, inward flow of W stops and the chemical potential is balanced. Let us create a convenient pathway to relate the chemical potential for the pure fluid at P to the mixture at P + Π. We can consider pressurizing the pure fluid and then mixing, or we can consider mixing the fluid and then pressurizing at fixed composition. Following historical derivation, it is common to use P as the standard state pressure for the mixing. The mixing process can be represented by the activity, aW = xWγW (cf. Eqn. 11.23)
For W in a mixture, the pressure effect on chemical potential at constant T is 
. Because the liquid is nearly incompressible, for the pressure step 
and overall,
The calculation path is illustrated in Fig. 11.16(b). The initial state represents the left side of the membrane and the final state represents the right side. Equating the chemical potential expressions for the two sides of the membrane results in
Figure 11.16. (a) Osmotic pressure in mmHg for solutions of bovine serum albumin (BSA) in water at 25°C at different pH values. Data from Vilker, V.L.; Colton, C.K.; Smith, K.A. J. Colloid Int. Sci. 1981. 79:548. (b) Fits of osmotic pressure at pH 5.4 as explained in Example 11.13.
Leading to the relation between osmotic pressure and activity of the permeable species,
Relation between osmotic pressure and activity of the permeable species.
The activity can be calculated from any activity coefficient model. Note that because the solution is very nearly pure W on a molar basis, we calculate activity relative to the Lewis-Randall rule for W, and it is common to replace the partial molar volume with the volume of pure W. A method known as the McMillan-Mayer framework20 is used frequently in biology to express lnaW, writing the logarithm of the activity as an expansion in terms of the molar concentration of the solute, [C]. Using the molar volume of W to normalize the expression per mole of W, the McMillan-Mayer framework results in
where B2(T) and B3(T) are functions of temperature known as the osmotic virial coefficients. Combining the two expressions, eliminating the molar volume, and rewriting the expression using solute generic subscript i, Ci the solute mass density (in units of grams/(volume of solution)), and Mw,i molecular weight, results in the form which is common in presenting data:
The osmotic virial coefficients are explicitly given temperature dependence, though they also depend on pH for biological molecules that change charge as a function of pH. Note that a plot of Π/(RTCi) will have an intercept related to the reciprocal of molecular weight and the plot can be used to determine molecular weight of solutes. Experimental data for osmotic pressure for the pig blood protein bovine serum albumin (BSA) in water at various pH values are shown in Fig. 11.16. The pH effects on charge are explained when we discuss electrolytes in Chapter 18.
Example 11.13. Osmotic pressure of BSA
Bovine serum albumin (BSA) has a molecular weight of 66399 g/mol. The osmotic pressure of an aqueous solution at 25°C and pH 5.4 is 74 mmHg when the concentration is 130 g/L and 260 mmHg at 234 g/L.a Using only these data, determine the second and third osmotic virial coefficients and estimate the pressure needed to concentrate a solution to 450 g/L across a membrane with pure water on the other side.
Solution
Since two points are given, let us linearize the equation for osmotic pressure to relate the coefficients to the slope and intercept. Defining a variable s to hold the rearranged variables,
Converting the osmotic pressure to MPa, 
is
Then at 234 g/L, si = 1.269×10-2L/g. The third coefficient is given by the slope of si versus Ci,
The second coefficient is found using the third coefficient with either of the original data points. From the point at 130 g/L:
Now at 450 g/L,
Therefore, we must apply a minimum estimated pressure of 1400 mmHg to concentrate the BSA to 450 g/L. The original paper cited gives a value of approximately 1500 mmHg. The estimate is within 10%. The prediction is sensitive to noise in the data points selected. A better method is to collect a few more data and regress a best fit.
Fig. 11.16(b) shows three fits of the data. For the “Linear Fit”, the data are linearized following the procedure in this example, and then linear regression is used over all points. For the ‘“Non-Linear Fit”, the error in the osmotic pressure prediction of Eqn. 11.73 is minimized using nonlinear regression. The “Example 11.13” curve uses the coefficients fitted in this example. The second osmotic coefficient for this data set is sensitive to the regression method. For the linear fit [B2 B3] = [1.93E-4 5.352E-5], for the nonlinear fit, [–3.57E-3 6.360E-5]. Careful analysis of the regression statistics shows that the uncertainty in the value of B2 is larger than the value—the uncertainties for the 95% confidence limit of the nonlinear fit are ±[5.25E-3 1.25E-5].
a. Vilker, V.L., Colton, C.K., Smith, K.A. 1981. J. Colloid Int. Sci. 79:548. Note the original paper uses a molecular weight of 69000 g/mol.
Cell membranes are excellent examples of semipermeable membranes, especially when considering water permeability. One implication of this property is that altering the osmotic pressure in the cellular environment can make the cells “uncomfortable.” Specifically, a higher salt concentration outside the cell might cause dehydration. On the other hand, zero salt concentration outside the cell might cause the cell to swell or rupture. This property extends to cell aggregates like skin, or the epithelium of the eye. For example, one requirement to minimize discomfort caused by eye drops is to make the solution isotonic, meaning that the osmotic pressure of water in the solution is the same as that of the reference cellular material, the eyes in this case.21
A common situation in pharmaceutical preparation is that the drug concentration is determined by the treatment protocol. However, the delivery solution should be isotonic with bodily fluids. Therefore, the solution must be supplemented with sodium chloride to make it isotonic. Fortunately, solute concentrations in cells are so low that B2 and B3 of Eqn. 11.73 can be neglected. This results in the interesting observation that osmotic pressure is independent of the nature of the compound as long as the molar concentration is the same. In other words, the concentrations of all constituents can simply be added up until the isotonic concentration is achieved. A property that follows this rule of adding up the constituents regardless of chemical nature is called a colligative property, of which osmotic pressure is an example (as long as the concentration is sufficiently low that B2 and B3 may be neglected).22 As a point of reference, human blood is in the concentration range where colligative properties can be assumed and isotonic with any solution of 0.308 mol/L solute.
Example 11.14. Osmotic pressure and electroporation of E. coli
E. coli are bacteria commonly used to express desired proteins through genetic modification because they replicate and express whatever intracellular DNA they find. Introducing foreign DNA requires weakening the cell membrane by washing twice briefly (~10 min.) with pure water at 4°C, followed by a wash with 10wt% glycerol solution, centrifuging to isolate the cell pellet from the medium before washes. After the cells are rendered “electro-competent” through washing, all but 1 ml of the glycerol solution is removed and the aliquots are frozen for storage until the “electroporation” step (electrically shocking the cells) is conducted. What concentration of glycerol (wt%) is necessary to make a solution that is isotonic with human blood? Describe what happens to the water in the cells and the glycerol outside the cells when the medium is replaced with 10 wt% glycerol.
Solution
The molecular weight of glycerol can be found from the NIST Chemistry WebBook as 92.1. This means that a 0.308 mol/L solution has 0.308·92.1 g/L of glycerol. Assuming 1000g/L as the density (the same as water since the concentration is low), this gives a weight fraction of 0.308·92.1/1000 = 0.0284 = 2.84 wt%. Therefore, the 10 wt% is hypertonic. The activity of water is too low to be isotonic. The driving force is for water to come out of the cells, diluting the glycerol outside the cells. The cells will shrink and shrivel.
11.14. Summary
The strategies for problem solving remain much the same as the strategy set forth at the end of Chapter 10 and a review of that strategy is suggested. Use Table 10.1 on page 373 and the information in Sections 10.1–10.8 to identify known variables and the correct routine to use. Then apply the valid approximations.
The introduction of activity coefficients is new in this chapter. First, we showed that there are three different stages in working with activity coefficients: obtaining them from experiments; fitting a model to the experiments; and using the models to extrapolate to new compositions or different temperatures and pressures.
We provided several methods of fitting activity coefficient models to experiments, and we demonstrated bubble-pressure and bubble-temperature calculations. We presented the strategy for relating the non-idealities to the excess Gibbs energy. We hypothesized models to fit the correct shape of the excess Gibbs energy and we differentiated the models to obtain expressions for the activity coefficients. We related the nature of the non-idealities to the chemical structures in the mixture through the concepts of acidity and basicity.
We introduced the concept of activity. The foundation is laid here for relating the fugacity of a component in a mixture to its pure component fugacity. Subtle details pertain to the characterization of standard state, as discussed in the introductions to Henry’s law and osmotic pressure and we superficially introduced the molality scale for Henry’s law. We will refer back to this discussion in the context of electrolytes.
Recognize that the primary difference between this chapter and Chapter 10 is the γi used to calculate Ki. We also pointed out the need for a greater understanding of solution behavior to assist in developing theory-based activity models, which will lead us into the next two chapters.
Important Equations
The starting point for many phase equilibrium problems is Eqn. 11.13 on page 425:
The various activity models alter the method of computing γi, but do not alter this basic equation. Eqn. 11.13 will appear in the simplified form for modified Raoult’s law:
Another significant equation can be summarized as the Redlich-Kister expansion (Eqn. 11.32 on page 429), in that this implicitly represents all the Margules models. When a GE model is combined with Eqn. 11.28, the activity coefficients can be derived at any composition and substituted into modified Raoult’s law to solve a wide variety of problems. The relations between GE and activity coefficient are given by
The simplest binary phase equilibrium equation to keep in mind is the bubble pressure,
Through this equation, it is very easy to compute the implications of non-ideality and assess qualitatively whether process complications like azeotropes or LLE are likely. A simple equation to guide your assessment is the MAB estimate of A12 in the Margules one-parameter model.
When considering distillation applications you must first check that αLH > 1 at top and bottom:
We also developed Henry’s law,
We showed how to relate Henry’s law to the Lewis-Randall rule used for modified Raoult’s law and how to predict the solubilities of supercritical gases in liquid solvents with the SCVP+ model.
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