(189c) 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,

.

* 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.