If we want to be precise, should rate laws and corresponding ODEs correctly be expressed in terms of concentrations, or in terms of activities? In other words, are reaction rates really functions of concentrations, or of activities (which would make more logical sense) - and if the latter, why can't I find any mention of it online?
You already know the answer to this. It should be in terms of activities. Expressing rates in terms of concentrations is an approximation. As for why you can't find this information on line, I don't know. For ideal gas reactions, using partial pressures or concentrations is valid.
OK, thanks. So the rate law for reaction A -> B should be r=ka_{A}^{n}? Or is it r=kc_{A}^{n} where c_{A}, the concentration of A, is then replaced by concentration as a function of activity. (n is the order wrt to A) Partial pressures should still be valid for non-ideal gases, no? We just need to correct the term using a fugacity coefficient, if converting from concentration... If we had a solid phase or liquid phase reaction, what would we use? Number of moles?
For non-ideal gas solutions, you use the fugacity of the species in the gaseous solution. This replaces the partial pressure. You need to learn how to get the fugacity in gaseous solution. For liquid phase, you still use the fugacity, or, more conveniently, the concentration times the activity coefficient. Again, how to get the activity coefficient needs to be learned.
Huh. This has changed some of my thinking. So partial pressures refer specifically to Concentration*RT, i.e. partial pressure=fugacity for an ideal gas alone. To go from partial pressure to fugacity you multiply by the fugacity coefficient. And in all the real-world scenarios where we are using partial pressures, we should really be using fugacities. Sum of the fugacities = exact total pressure, etc. It's like the accuracy mark-up from concentrations to activities, where instead of an activity coefficient we simply use a fugacity coefficient. That way we can get exact results, whereas working with concentrations or partial pressures is "exact" only for ideal gases or solutions. Is that right? Even if the liquid is pure? We continue to use concentration * activity coefficient? I was under the impression that the activity of all liquids is unity (1), but concentration * activity coefficient definitely looks to be finding the activity of the liquid. If so, is the process to find activity coefficients for liquids (including impure liquids, solvents, pure liquids, etc.) the same as for dissolved substances?
If the pressure of the system is low enough, treating the gaseous solution as an ideal gas is valid, since the fugacity of each species approaches the partial pressure of each species. For a pure component, the fugacity is equal to the pressure times the fugacity coefficient. Getting the fugacity of a species in a gaseous solution takes more doing. If the gaseous solution can be treated as an ideal solution (but not necessarily and ideal gas), the fugacity of each species can be approximated as the fugacity of the pure component at the solution pressure times the mole fraction of the species. Getting the activities or fugacities of each species in liquid solution is a little more complicated. You can find out how to do this in a good Thermo text like Smith and Van Ness or Hougan and Watson. Before you start figuring out what to do thermodynamically for reaction kinetics, you need to study chemical equilibrium thermodynamics, in which the forward and reverse rates of the reaction are equal. This will tell you what concentration parameters you need to use when dealing with non-ideal liquid and gaseous solutions in reaction kinetics.
Ah, thank you. So for gases, we must use fugacities, for liquids and solutions we must use activity. What do we use for solid phase reactions? Thank you for the advice. I appreciate what must be studied here. The transformation from partial pressure to fugacity for the gaseous species and concentration to activity for liquid phase or dissolved species. I then presume that, where we currently use concentration, we would use the corresponding species activity (e.g. r=ka_{A}^{n}), and where partial pressure is used we use the corresponding fugacity; this seems as logical as in my original post, given that the rate is actually a function of activity but approximated in my past experience as a function of molar concentration, to which activity becomes equal in less advanced cases. I would guess then that the main difficulty we face is in finding initial values of the activity or fugacity for each species, which requires the concentration->activity or partial pressure->fugacity transformations of which you speak and which I am planning to study as soon as I can get my hands on a book that covers it. Other than that, if we are simply replacing concentration/partial pressure terms with activity/fugacity terms, it shouldn't be too difficult to solve the ODE system, in fact it should be straightforward as with concentrations.
What you would have to worry about would not be activity coefficients, but their changes during reaction. I never saw anyone worry about this in biochemistry e.g. enzyme kinetics. Although the substances of interest are more often than not charged and I guess must have activity coefficients significantly different from 1. However the substances of interest whose concentration changes are being followed, their change is often a fraction of their total, most often a small fraction. And then much larger concentrations of other substances, buffer ions, would be present. So their activity coefficient would not change significantly during a reaction. In some cases where buffer concentration was low (as when the reaction was measured titrimetrically) one arranged for salt concentration to be enough that ionic strength did not change significantly. I imagine it would often be much the same in many other types of reaction situations. Plus the kinds of discriminations one is making by kinetics might often not call for such great refinement?
For gases, we don't need to use fugacities if we are confident that the gas can be modeled as an ideal gas, and for liquids, we can use concentrations if we are confident that the liquid can be modeled as an ideal solution. The fugacity coefficient of a pure component at the temperature and pressure of the gaseous solution changes with the temperature and pressure, so, if either the temperature, the pressure, or both change, this has to be taken into account. This means that it is not just the initial conditions that determine the fugacity coefficient. It takes a little more work to set up the ODE system than in the case of an ideal gas solution, but that is not much of a constraint in solving the equations numerically.
Ah, I see. So what we need to be able to do is express the fugacity coefficient of each gas as a function of the total pressure, temperature, and constants which vary from gas to gas. Then fugacity coefficient*partial pressure is the fugacity. Not that complicated in principle. Activity coefficients are more complicated, the method for determining them is based on ionic strength for solutes/solvents (and also varies with temperature if I recall correctly - being in solution, it will not vary with pressure which is a gaseous property), and entirely separate for pure liquids and solid phase species; nonetheless concentration*activity coefficient is the activity of that substance in its phase. Am I on track so far?
Use of ionic strength applies only to aqueous electrolyte solutions. It doesn't come into play for solutions of organics, or for aqueous systems where ionization does not occur.
OK, so even further research is needed to ascertain different methods of finding activity, depending on the exact phase. This will in the end come down to activity coefficients, which can be a function of temperature, pressure and concentration of species, multiplied by concentration, for the activity of that species. Fugacity coefficients are a function of temperature and pressure "only", and multiplying by the partial pressure we find fugacity of the gaseous species in question. Any corrections to make?
For the gas phase, the fugacity of a species is equal to the fugacity coefficient of the pure species at the same temperature and total pressure as the solution multiplied by the partial pressure, only for an ideal gaseous solution. For a non-ideal solution of gases, the treatment must be more like that of a liquid. Also, the partial pressure of a species is not a directly measurable quantity, and must be calculated as the mole fraction times the total pressure. Fugacity also comes into play for liquids. For a liquid species in ideal solution below its critical temperature and pressure, the starting point for getting the fugacity of a liquid species is the equilibrium vapor pressure of the pure species at the same temperature. This can be used to determine the fugacity of the same species in the gas phase at the equilibrium vapor pressure. Since, for a pure species, its free energy in the gas phase is equal to the free energy in the liquid phase, this necessarily means that the fugacity of the pure species in the liquid phase at saturation is equal to its fugacity in the gas phase. Therefore, we know the fugacity of the pure species in the liquid at the solution temperature and equilibrium vapor pressure. The fugacity of the pure liquid species can then be calculated at the same temperature and total pressure of the solution by integrating RTdlnf=dp/ρ between the saturation vapor pressure and the total solution pressure, where ρ is the molar density of the liquid (nearly a constant). Then, for an ideal liquid solution, the fugacity can be obtained by multiplied the fugacity of the pure liquid component at the same temperature and pressure of the solution by the mole fraction in the liquid. For non-ideal liquid solutions, however, further correction is necessary involving activity coefficients. See Smith and Van Ness for more details.
According to Wikipedia (http://en.wikipedia.org/wiki/Fugacity) a fugacity coefficient is defined by fugacity/partial pressure, so that fugacity = partial pressure * fugacity coefficient. How can this change for a non-ideal gas solution? And if it does change, what is the new expression; or perhaps the better question is, what it is the new fugacity coefficient a function of. (Noting that fugacity coefficient for this gas * partial pressure for this gas must = fugacity for this gas according to the Wikipedia page.) So that is the method for calculating activity/fugacity for liquids. Perhaps here I should ask, what is the activity coefficient a function of? (Since, again, according to Wikipedia, activity coefficient * concentration = activity, it is best to collect an understanding of what factors affect the activity coefficient) So far I'm counting temperature, total pressure, and sometimes concentrations of and charges on the species in the system (or even the activities themselves? an iterative calculation), or even perhaps fugacities of some gaseous species in the system.
A pressing question for me is, how do we phrase rate laws in terms of activity? Do we simply replace concentration terms with activity terms for each species, e.g. where we previously wrote d([A])/dt=-k[A]^{x}[ B]^{y}, now we write d(a_{A})/dt=-k*a_{A}^{x}*a_{B}^{y}? And equivalently for fugacities. If so, what is the main issue with this field? I've heard it's actually quite difficult but this seems to be a standard procedure, not different from solving using concentrations and partial pressures except for different initial values. So if the correct expression we should use instead of d([A])/dt=-k[A]^{x}[ B]^{y} is not d(a_{A})/dt=-k*a_{A}^{x}*a_{B}^{y}, can you explain how to find the correct expression.
I've tried to encourage you to study the non-ideal solution behavior sections of Smith and Van Ness, and Hougan and Watson. You need to understand the equilibrium thermodynamics development first, before getting to the kinetics. All your questions are answered in these books. The transition to the kinetics will not be difficult once you understand the non-ideal solution thermodynamics. The development is too long to present in responses to physics forums. You need to go through several chapters in these books (or other equivalent developments). You may need to look at other books as well to learn about Debye-Huckel or other more accurate forms of the activity coefficients for aqueous electrolyte solutions. Chet
Why? And if that's the case this will not be a solvable set of equations. We will have twice the variables that we do equations.
You always have to know the relation between activity and concentration, so this doesn't double the number of variables. The point is that a rate basically is the change of particle number with time. This change may depend in a complex way on the concentrations of all other substances present and I even doubt that it is generally sufficient to use activities on the RHS, as these refer to local equilibrium situations, only. However the LHS is always the absolute change of particle number or concentration if related to total volume.