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#1
Jun113, 04:41 PM

P: 343

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?



#2
Jun113, 05:50 PM

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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.



#3
Jun213, 06:28 AM

P: 343

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 nonideal 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? 


#4
Jun213, 11:53 AM

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Rate laws
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. 


#5
Jun213, 12:32 PM

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#6
Jun213, 10:49 PM

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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 nonideal liquid and gaseous solutions in reaction kinetics. 


#7
Jun313, 04:41 AM

P: 343

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. 


#8
Jun313, 01:27 PM

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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? 


#9
Jun313, 10:31 PM

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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. 


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Jun413, 03:46 AM

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#11
Jun413, 11:24 PM

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#12
Jun513, 12:10 PM

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#13
Jun513, 04:54 PM

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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 nonideal liquid solutions, however, further correction is necessary involving activity coefficients. See Smith and Van Ness for more details. 


#14
Jun613, 12:11 PM

P: 343

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. 


#15
Jun1013, 07:41 AM

P: 343

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. 


#16
Jun1013, 08:30 AM

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Chet 


#17
Jun1113, 06:18 AM

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#18
Jun1113, 11:09 AM

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