Yes it can be negative. The binary interaction parameters (BIP) for a cubic should always be between -1 and 1 however. This is because when using mixing rules in cubic equations of state, such as the SRK, the assumption of one fluid mixing rules is used. cubics struggle with cross interactions. This is a well known fault. And also, in general they struggle with polar components. They are generally inadequate at calculating the attractive energy contribution. i.e., P = RT/(v-b) - (attractive energy contribution). The attractive energy contribution for the SRK is alpha*a/(v(v+b). Anyways, in using mixing rules to calculate mixture properties, the mixing rules often do an insufficient job and need to be modified by a BIP i.e.

• alpha*a = sqrt( [alpha*a]i[alpha*a]j )(1-BIP)

where subscript i represents component i, and subscript j represents component j. Thus there is a BIP (a.k.a. kij) for each pair of components. The larger the BIP, the more the mixing rules needed to be modified... i.e., the poorer they did at predicting the attractive energy contribution.

Anyways, they can be negative or positive, but they are restricted to being between -1 and 1. The sign of the BIP just indicates if it was overestimating or underestimating the attractive energy. Edit: yes, the BIP is just a mathematical coefficient for cubic equations of state. The BIP's in activity coefficient models do have real meaning though.

I recommend the textbook "Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories" by Kontogeorgis and Folas. It covers all of this and much more.

Interesting question. Please specify with more clarification of the used symbols (SRK) and (pvTx) for more large RG members readers.

You can edit your question and modify at any time.

If it's the sum of Redlich-Kister equation for thermodynamic state functions, the Ai coefficients are related to the limiting partial molar quantity at infinite dilution. which can gives an indication of solute-solvent interaction intensity, ...

Partial molar quantities can be negative, .... it's an individual contribution by one of pure component in the mixture, ...

Thank you for your suggestion, Dr Ouerfelli. 

By SRK equation, I was refering to Soave Redlich Kwong equation.

By pvTx, I was refering to pressure, volume, temperature, and mole fraction, which means density calculation.

I was using SRK equation and van der Waals mixing rules to calculate densities. Is the binary interaction coefficient just a "mathematical coefficient"? Or does it have physical meaning?

Yes it can be negative. The binary interaction parameters (BIP) for a cubic should always be between -1 and 1 however. This is because when using mixing rules in cubic equations of state, such as the SRK, the assumption of one fluid mixing rules is used. cubics struggle with cross interactions. This is a well known fault. And also, in general they struggle with polar components. They are generally inadequate at calculating the attractive energy contribution. i.e., P = RT/(v-b) - (attractive energy contribution). The attractive energy contribution for the SRK is alpha*a/(v(v+b). Anyways, in using mixing rules to calculate mixture properties, the mixing rules often do an insufficient job and need to be modified by a BIP i.e.

• alpha*a = sqrt( [alpha*a]i[alpha*a]j )(1-BIP)

where subscript i represents component i, and subscript j represents component j. Thus there is a BIP (a.k.a. kij) for each pair of components. The larger the BIP, the more the mixing rules needed to be modified... i.e., the poorer they did at predicting the attractive energy contribution.

Anyways, they can be negative or positive, but they are restricted to being between -1 and 1. The sign of the BIP just indicates if it was overestimating or underestimating the attractive energy. Edit: yes, the BIP is just a mathematical coefficient for cubic equations of state. The BIP's in activity coefficient models do have real meaning though.

I recommend the textbook "Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories" by Kontogeorgis and Folas. It covers all of this and much more.

Thank you for your suggestions, Mr Kelly. I will look into that book you mentioned. 

Dear Fufang Yang,

it is a very interesting question.

The binary interaction coefficients are adjusting parameters that improve the VLE convergence. So they could be between -1 and 1 and they can be negative.

For example, if you take some original papers of some important EOS you can find negative values for k12 for some important compounds. In the original paper of the PRSVII equation of state, you can find a lot of negative k12.

http://onlinelibrary.wiley.com/doi/10.1002/cjce.5450640225/abstract

I suggest you read the following papers that can help you to understand the effect of k12 on EOS modelling.

Examining the effect of binary interaction parameters on VLE modelling using cubic equations of state  

Relationship between the binary interaction parameters (kij) of the Peng–Robinson and those of the Soave–Redlich–Kwong equations of state: Application to the definition of the PR2SRK model

Hope this answer can help you.

Mariano

http://www.sciencedirect.com/science/article/pii/S0378381211001695

http://www.sciencedirect.com/science/article/pii/S0378381210001822

Dear Fufang Yang 

The kij should be between 0 and 1, but in most of calculations it is between -1 and 1 and negative values exists in most of them. I think the limitations of models lead to this. For example, if you do not consider the polarity or hydrogen bond for some cases, the results of phase equilibrium calculations are not good at all and the kij will be negative or unusual amount of kij can improve the results. I think that two thing affect this values:

1) mixing rule

2) the parameters of pure components and contributions of EOS, especially the energy parameter in cubic EOS. For example, some authors work on correction of  the energy parameter in cubic EOS.

For more help, I suggest you to refer to "Pvt and Phase Behaviour of Petroleum Reservoir Fluids" by Ali Danesh

see non-random mixing rule for VPT EOS. You see that by using appropriate mixing rule and correction of energy parameter of water, all kij are between 0 and 1.  Pages 158-160 and 362.

See this article. It is helpful.

Yes, k12 could be negative. k12 is an adjustable parameter, value depending on EOS (i.e. SRK ) and mixing rules ( One fluid van der Waals ) and to a certain extent on regression's data and method. It could be interpreted in terms of London-Mie theory for dispersion forces, because of the geometric mean rule for the molecular cross energy parameter ( as shown in the article of Coutinho et al. ) . Refer also to the following RG question:https://www.researchgate.net/post/What_is_the_physical_meaning_of_the_EOS_energy_interaction_parameter_ie_combining_rule_parameter_for_the_binary_in_a_multi_component_mixture

In the linked article it is indicated by deriving combining rule for the attractive intermolecular potential that a negative binary interaction parameter is in fact possible, considering the contributions of polar forces :http://www.sciencedirect.com/science/article/pii/S0378381208000502

The kij can be negative.

Please have a look at the many papers linked with the PPR78 model.

Usually a negative kij means that you have negative deviations to ideally i.e. that you have strong interactions between the 2 molecules.

The kij cal also be HIGHER THAN 1 (the opposite is often wrongly written).

Please feel free to contact me.

You can have negatives kij for mixtures of polar/non polar.

This is not a problem.

Professor Jaubert,

You have said that the kij can be higher than 1. While this is true since kij values are very often empirical, and can for this reason take on any value, could you elaborate on instances where it is necessary to have kij's that are greater than the absolute value of 1 when they are treated as purely empirical parameters, rather than using a different model and/or mixing rules?

If an empirical kij is greater than the abs(1) then either the mixing rule is a poor one, or the model is fundamentally ill equipped to handle deviations due to interactions, such as those caused by hydrogen bonds for instance. Wouldn't kij values that are greater than one strongly indicate that a different model and/or mixing rules needs to be used for the problem at hand?

Where would you draw the line for kij values at which the attempt to model a systems thermodynamic properties becomes more a practice of statistical parameter fitting and less an attempt at predictive modelling. It is my understanding that in general a kij less than the absolute value of 1 is the common line that is drawn.

I am very interested in your much more educated opinion!

Thanks,

Braden

Dear Braden,

thanks for your kind comment. I have to develop several points.

1) As you know the classical van der waals mixing rules write for the attractive parameter (a_mixture):

a_mixture = double sum[xi*xj*square_root(ai*aj)*(1-kij)]    (Equation 1)

Moreover a_mixture has to be POSITIVE (in the Van der Walls theory, the pressure is the sum of a repulsive and of an attractive pressure. And the attractive pressure has to be negative).

It is wrongly often concluded that if kij is greater than 1 in equation (1), then a_mixture is negative. This is wrong. To convince you, develop the double sum. For a binary system, you get:

a_mixture = x1*x1*a1 + x2*x2*a2 + 2x1*x2*square_root(a1*a2)*(1_k12)

It is thus easy to find example for which k12>1 and for which a_mixture remains positive. Indeed the sum contains 3 terms, two of which are positive and the last negative.

2) I agree with you, if the fitted kij is higher than 1 this OFTEN means that the mixing rules are not the best ones. However, this is very usual when with model systems containing quantic compounds like H2.

Kind regards.

Jean-Noël.

"Can the binary interaction coefficient be negative? Does a negative coefficient have a physical meaning? "

It has; it is associated to attractive interactions between molecules and to negative deviations of the activity coefficients in a liquid mixture with respect to the Raoult law. Zero kij values are typical of regular solutions, with slightly positive activity coefficients.