Prediction of Binary Diffusion Coefficients In Non-Ideal Mixtures From NMR Data: Hexane-Nitrobenzene near Its Consolute Point

Prediction of binary diffusion coefficients in non-ideal mixtures from NMR data: hexane-nitrobenzene near its consolute point

C. D'Agostino, M.D. Mantle, L.F. Gladden and G.D. Moggridge*

Department of Chemical Engineering and Biotechnology, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA, UK.

* corresponding author; +44 1223 334763, gdm14@cam.ac.uk.

Short Abstract

Pulsed field gradient nuclear magnetic resonance was used to measure the tracer diffusivity of the species in mixtures of nitrobenzene and n-hexane close to the consolute point.  Measurements were taken over a wide range of composition (including the consolute composition, x₁ = 0.422) at temperatures between the consolute temperature (19.4°C) and 35°C.  NMR-derived tracer diffusivities are compared with literature values for the binary diffusion coefficient under the same conditions.  It is shown that it is possible to calculate the binary diffusion coefficient, even very close to the consolute point, from the NMR-derived tracer diffusivities using a fairly simple thermodynamic correction factor, of a form similar to those reported in the literature.  The necessary thermodynamic parameters are calculated by fitting vapour-liquid equilibrium data for the system under the same conditions, which is available in the literature.  The ability to predict binary diffusion coefficients from NMR measurements has significant potential, for example in studying mass transport in porous solids or packed beds, situations where conventional diffusion measurements are impossible to make.

Introduction

In ideal solutions diffusion is well described by Fick's Law, which gives the driving force for diffusion as a concentration gradient.  However, from a thermodynamic perspective, the driving force is more correctly considered to be a gradient of chemical potential.  The simplest analysis (Schreiner, 1922) for a binary mixture suggests:

                                                                      (1)

 is the Fickian diffusion coefficient and  is a different diffusion coefficient defined for a chemical potential gradient.  For non-ideal mixtures  is a function of composition; it is hoped that  is independent of composition, or at least a much less strong function of composition than .

                                                                                                                                                                            (2)

so                                                                                           (3)

 may be identified as some sort of molecular mobility, whilst the term in square brackets is a thermodynamic correction factor, taking account of the “force” on the diffusing molecules due to the gradient of excess chemical potential. 

There is a consensus (Cussler, 2009) that equation 3 underpredicts measured binary diffusion coefficients near the consolute point, but there is not general agreement about the appropriate form of an improved equation. 

Scaling laws, based on dynamic concentration fluctuations near critical points, suggest that the temperature dependence of the diffusion coefficient at the consolute composition should be:

                                                                                                    (4)

where   is the consolute temperature,  is a temperature-independent constant and  is a parameter, expected to be around two thirds. 

If the excess Gibbs energy is independent of temperature, then the temperature dependence of the Schreiner thermodynamic correction factor at the consolute composition is given by:

                                                                                             (5)

Thus we might speculate that for temperatures close to the consolute point, the Schreiner equation can be corrected by the factor a to give both temperature and composition dependence as follows:

                                                                                              (6)

where  is again some sort of molecular mobility at the relevant temperature and composition. 

Pulsed field gradient nuclear magnetic resonance (PFG-NMR) allows measurement of the mean square distance moved per unit time for individual species in a mixture (effectively their tracer diffusivities).  These can be averaged to give .

In order to obtain the thermodynamic correction factor (the term in square brackets in Equation 6), it is necessary to construct an activity coefficient model valid for temperatures close to the consolute temperature.  This was done on the basis of vapour pressure data from Neckel and Volk (1964), who report the total vapour pressure of liquid mixtures of nitrobenzene and hexane over a range of compositions at 21°C, 25°C and 35°C. 

In this paper we examine the validity of Equation 6 for predicting binary diffusion coefficients from NMR data near the consolute point in the system hexane-nitrobenzene.  Predicted values are compared to measured binary diffusion coefficients reported by several authors over a range of temperatures and compositions near the consolute point (Claesson and Sundelof, 1957; Haase and Siry, 1968; Wu et al., 1988)

Results

Figure 1.  Binary diffusion coefficients predicted by equation 6 (with a = 0.64) at 20°C, compared to measured values from the literature. 

Figure 1 shows an example of the results obtained: the concentration dependence at 20°C (the consolute temperature is 19.4°C) of experimental binary diffusion coefficients, and the predictions of Equation 6 from NMR data.  Clearly the fit is excellent using a = 0.64.  Predictions of other models, based on cluster diffusion are also included.

Results at other temperatures and temperature dependant data at constant composition are similarly consistent with Equation 6 with a = 0.64.  Alternative models fit the data less well and require composition dependent parameters, and so are clearly less satisfactory.

Discussion

The results presented here suggest that it should be possible to predict from NMR data, with good accuracy, the binary diffusion coefficients in non-ideal liquid mixtures over a wide range of temperatures and compositions.  The case addressed here, of a separating mixture close to its consolute point, is the most difficult case to deal with, since the thermodynamic correction factor and therefore the binary diffusion coefficient, becomes very small. 

Such an NMR method will allow the measurement of diffusion coefficients in practically relevant situations, such as within pores or in packed beds, where other techniques for measuring diffusion coefficients may be very difficult or impossible. 

The results also demonstrate that a simple thermodynamic correction factor, as suggested by Schreiner, is inadequate to model the difference between molecular mobility (as measured by NMR) and the binary diffusion coefficient near the consolute point, at least for the system studied here.  The results are consistent with a model based on dynamic concentration fluctuations. 

It is significant that most models, including that of Wu et al., for diffusion in non-ideal liquids predict only the trend with temperature or composition and do not allow the calculation of absolute values.  Thus the models normally allow, for example, the prediction of a binary diffusion coefficient at one temperature if the coefficient is already known at another temperature.  By contrast, we have shown here that it is possible to predict the difference between measured (by NMR) tracer diffusion coefficients and the measured (values taken from the literature) mutual diffusion coefficient at the same temperature and composition, using only a thermodynamic correction factor.  For each condition it is necessary to model the absolute value of the difference between two measured quantities, not merely their trend with temperature or composition, a significantly more challenging proposition.  This makes it all the more impressive, and encouraging for future applications, that it has been possible to do this using a relatively simple thermodynamic model and a previously reported, and theoretically sound, a parameter.

Conclusion

We have demonstrated the practicality of calculating binary diffusion coefficients from NMR measurements of tracer diffusion, even in the case of the most highly non-ideal mixtures, those close to a consolute point.  This has been done on the basis of a relatively simple thermodynamic correction factor, given in equation 10.  The required non-thermodynamic parameter (a = 0.64) is consistent with that reported by Wu et al. (1988) for the temperature dependence of the diffusion coefficent in the same system, and with the theoretical value expected from semi-empirical scaling laws, describing the influence of dynamic concentration fluctuations.  This could have significant applications, notably in measuring diffusion coefficients in practical situations such as porous catalysts or packed beds; further work is required to ascertain if different diffusion mechanisms require more complex analysis in such solid-liquid systems.

References

Claesson, S. and Sundelof, L.-O., 1957.  Diffusion libre au voisinage de la temperature critique de miscibilité.  J. Chim. Physique 54, 914-919.

Cussler, E.L, 2009.  Diffusion: mass transfer in fluid systems, 3^rd edition.  Cambridge University Press, Cambridge.

Haase, R. and Siry, M., 1968.  Diffusion im kritischen entmischungsgebiet binärer flüssiger systeme.  Zeit. Phys. Chem. 57, 56-73.

Neckel, A. and Volk, H., 1964.  Zur thermodynamik des systems n-hexan–nitrobenzol. Mh. Chem. 95, 822-841.

Schreiner, E., 1922.  Om anvendelsen av bjerrums elektrolytiske teori paa elektrolytdiffusjonen og diffusjonpotensialet.  Tidsskrift for Kemi og Bergvaesen 2(10), 151.

Wu, G., Fiebig, M. and Leipertz, A., 1988.  Messung des binären diffusionskeoffizienten in einem entmischungssystem mit hilfe der photonen-korrelationsspektroskopie.  Wärm. Stoffüb. 22, 365-371.