So you mean log(r) v 1/T is for zero orders only, where r=k?
k = Aexp(-E/RT) and product rate = k([C]^c)([D]^d)... for however many reactant species you have.
rate = Aexp(-E/RT)[C]^c...
log(rate) = logA + log(([C]^c)([D]^d)...) - E/RT
It is only valid to plot if the concentration does not change. Otherwise, you can do a multiple linear regression using Excel and the data analysis tool to plot log rate vs 1/T and log(([C]^c)([D]^d)...) if you like.
In my case with solid catalysts you consider coverage (i.e. cov^n) instead of concentration and I use a multiple linear regression which use log(rate) as the y-input and log(cov^n) and 1/RT as the two x-inputs. I hope this is helps.
The reaction is oxidative & catalytic n-butane dehydrogenation i.e C410+CO2-->C4H8+CO+H2O
The experimental design is one such that Temperature is varied while fixing Conc. Thank you all for your insightful contributions. I really appreciate this.
Also I do not like to contradict someone who do not know the reasons which asks something. If you know the reaction and experimental data I can find the order of reaction and thus I can deduce the ratio r versus k in order to use the Arrhenius equation one way or another.
k=Ae^-(Ea/kB T) , not r, k.
r=-d[X]/dt,(differential) or for example; ("integrated" or otherwise solved) r=e^-kt or some other equation depending on the order.
In a temperature dependent series of initial rate measurements you are regarding the rate as being proportional to the rate constant so it is not necessary to actually measure it. (Essentially what has already been stated in the replies). Go for it with the initial rates.
The rate can include more than one rate constant, so generally I would not tend to plot log(k) versus 1/T.
BTW:
log(rate)= log A +c log [C]+d log[D] -E/RT
the intercept could be maybe used to determine reaction orders (if you don't know it and there is only one step, if there are more this doesn't work out).
You can also study this by the intercept as a function of (initial) concentrations, maybe even as a function of time
If this is a good method to determine the reaction order then this is a good reason to do so
Probably heterogeneous? Then you'd need to consider mass transport as well,
at least when the rate constant is not much smaller than the mass transport coefficient.
Dear colleagues,
This is Gregory Yablonsky from Saint Louis University.
You can always plot a dependence "observed rate" (not rate constant ) regarding "1/T"
Then, you can determine "an apparent activation energy"
Then, you have to decode this apparent activation energy.
I completely agree with Dr. Roussell that this decoding has to be done based on what kind of data we have. For example, for the CSTR reactor in which the heterogeneous "gas-solid" reaction is occured, the apparent activation energy is a function of coverages which are, in their turn, are functions of the temperature and composition. For two-step catalytic reaction
(1) A + Z goes to AZ (irreversible); (2) AZ + B = AB +Z
the apparent activation energy is
E = (E1 )x (dimensionless concentration of Z) + (E2) x (dimensionless concentration of AZ), where E1 and E2 are activation energies of reactions 1 and 2, respectively
See in detail book Marin, G; Yablonsky, G, "Kinetic of complex reactions;Decoding complexity", J-VCH, 2011
In a reaction with 0-order in reactants and therefore constant rate, plotting log(rate of product formation) vs 1/T is absolutely fine. Otherwise it is only valid using the initial reaction rates. If you have a well-defined order in reactants though I would make sure you determine the k values for each reaction using the usual linear graphs: [reactant] vs time for 0-order, log[reactant] vs time for 1st-order; 1/[reactant] vs time for 2nd order. And don't forget to convert your rates and k values into international units for your activation energy plot!
I agree with the colleague Arrowsmith. If you obsvere a linear kinetic you can simply use the rate instead of the rate constant. However there will certainly be more than one thermal activated processes in your reaction I suggest. Therfore you will monitor the process with the largest activation energy (sse the comment of Mr. Yablonsky).
Yeah, for gas there is no real meaningful mass transport limitations either for heterogeneous processes (usually), is there?
Generalising, for a simple reaction where rate = k[reactant1][reactant2]... ln [initial rate] = ln k + ln[reactant1(t=0)] + log[reactant 2(t=0)].... So as long as you keep the same initial concentrations of reactants for each temperature, it's OK to plot log(initial rate) vs 1/T. Dimensions of initial rate don't matter if you only want the activation energy, which you get from the slope (slope = E(act)/R), but the intercept, from which you get A and from that deltaS, depends on the units you use. Units of E(act) just depend on the units of R.
Short answer is yes. This gives an "apparent activation energy". Generally speaking, detailed chemical kinetics are often unknown and there may be no single "k(T)" (for example, formation can be by serial reactions and/or transport processes as others have mentioned.
Even if you had knowledge of what rate equations are involved, log(product formation) vs 1/T can reveal shifts in rate limiting steps and other effects. For an example of a breakpoint I recall offhand please see Fig. 1 in Ibbotson et. al. J. Appl. Phys. 58, 2939 (1984) as well as Fig. 1 in Ibbotson et al. J. Phys. Lett. 44, 1129 (1984).
You may plot it as the graph will not find any change in it's linearity. Only you should take care of the fact that the intercept in the graph will depend on the concentration of the reactant and the overall order of the reaction rather being an inherent constant. Evaluation of activation energy will not get disturbed as far I believe. Have a good day.
Yes. But this will result in the "apparent activation energy of the product formation rate". Which is equal to the apparent activation energy of the rate constant if concentrations of the reactants were kept constant when temperature was varied.
Yes, because both rate and rate constant expressing the change of concentration with time, which the main difference is concentration independency of rate constant. So, as mentioned above you could do this job only at fixed reactants concentration. In this special case you have to disregard the frequency factor A (the Intercept) because is quite meaningless in contrast to the usual determination of activation energy using rate constant.
Thank you all for taking out time to help a rookie researcher..why do i feel so blessed?
Dear colleagues
If the reactant concentration is fixed this rule can be true.
Generally No! Taking "shortcuts" doesn not pay, and in most cases it is WRONG!
It really depends which activation energy you want to measure. Remember that the activation energy Ea is a parameter in the T-representation of k in the form Aexp(-Ea/RT)).
Regarding products and reactants one has to be careful. For a rate first order in reactants (d(A)/dt=-k(A)): lnk vs. 1/T measures the activation energy for reactant disappearance, regardless of the time the measurement was taken. If you switch to observation of products in the reaction A==>B, k must be determined from: (B(t)) = (A0)(1-exp(-kt)). Plotting k vs. 1/T from observation of B only leads to the activation energy for disappearance of A (I assume that that is what you want) if there is a 1:1 correspondence between loss of A and formation of B, i.e. for a simple (non-complex) reaction. If you have parallel first-order reaction one must make sure that the branching ratio remains independent of T.
I encourage you to also determine the preexponential A (is not the concentration A above) as it is as important as the activation energy because it gives you information on what type of reaction you have.
Only in the case of a k zero order in (A) one may plot ln of reactant loss or product formation rate vs. 1/T to obtain Ea, and then only in the case if one does not have competing zero-order reactions whose branching ratio may be T-dependent.
For a first- or pseudofirst-order reaction dln(B)/d(1/T) = dln(-k(A))d(1/T) does NOT lead to Ea! Sorry!
If and only if, the conditions of concentrations of your reactants are identical at all different temperature runs. Why not just plot the rate constant?
Yes! you can use it as long as you are using the same concentrations in all your experiments. Also, keep in mind for scale - up, concentrations must be the same for plant, as in laboratory,
In the other hand, typical equation ln k = -Ea/R (1/T) + C, does not take into account the concentration, just the specific rate constant k. That's the primary reason it's general and more used for simple reactions.
You will obtain activation Gibbs free energy, nor the typical Arrhenius activation energy, Be careful.
i think natural log of k vs I/T is the right way to plot a graph as arrived from the formula k=Ae(-Ea/RT)
If the energies of the reactants and products are the same ,then we get same activation energy for the forward reaction and reverse reaction.If so,your idea (I think(mathematically)) would be correct.
Plotting ln(k) vs. (1/T) gives you Ea. When one doesn't have a kinetic model of the process and cannot calculate k, using data on W is possible. But you should be aware that plotting ln(W) vs. (1/T) gives you an estimate of the apparent Ea only. At that you have to be sure, that you work at the same gas composition while measuring different points. I mean not only feed composition, but conversion extent and product composition as well (so called, isoconversional series). The use of differential reactor is preferrable, since these apparent Ea values depend on the conversion extent. See, e.g. Ozawa-Flynn-Wall isoconversional method for model-free kinetic analysis of thermal analysis data (you may find it as ASTM E698).
This is only correct provided that (i) the reactant concentrations remain (quasi-)constant in time and are exactly the same at all experimental temperatures, AND that (ii) the reaction is an elementary one with a single product channel (as branching ratios for parallel reactions are mostly T-dependent). Since condition (i) implies that one knows and can monitor the reactant concentration(s) as a function of time, the procedure is correct only if one can determine k, i.e. the rate of product formation divided by the concentration (product). Simple conclusion: always derive activation energies from lnk vs 1/T.
It is appliable if you get only one product . Or this product is the final product and it depends on reaction rate of the sample.
O.Temkin.The Answer of A. Khassin is not correct enough, because the right part of kinetic equations can have complex structure and include other kinetic and thermodynamic konstants.
For APPARENT activation energy, yes. Notwithstanding, one ought not conceive the result of any meaning in what mechanism is concerned. The apparent activation energy, for complex reaction, will most likely vary with temperature and even reactants concentration, while you will not have any understanding of the underlying elementary steps. This is a result of changes in paths by which reaction mostly occurs, parallel reactions may change preferability and, hence, selectivity.
This approach is ultimately a black-box handling of what you have at hand.
In the general case, this is inadmissible, because of the possibility of the presence of a contribution of the inverse reaction, rate inhibition by products, side reactions, etc.
Everyone seems to enjoy a linearized plot of k = k_o*exp(-E/RT) as a way to obtain its parameters. As noted, you must certainly confirm all assumptions behind the translation of log(r) to log(k). You must also confirm that your process is truly a simple Arrhenius expression.
Even in a perfect case where the Arrhenius equation applies unambiguously to your underlying process, the use of a linearized plot log(r) or log(k) versus 1/T should no longer be accepted to obtain parameters to report in a journal publication. We have far better analysis tools in ready access. The proper method is to use non-linear regression fitting to the equation directly. You thereby also obtain statistically-meaningful values for the uncertainties on activation energy and pre-factor.
The approach to solution of the problem which is in the basis of this question as applied to the heterogeneous catalysis depends on a number of concrete conditions and on the aim of calculation of the temperature dependence. It is necessary to take into account the following principal features: (1) if the reaction proceeds near the equilibrium, the factual temperature dependence depends on the degree of approaching to the equilibrium; (2) if this effect has no importance, it is necessary to take into account that the reaction rate can be dependent on the concentration of the components, and therfore, it is desirable to know the kinetics and to determine the temperature dependence of the rate constant; (3) however, on frequent occasions, the constant really represents the product of the real rate constant and the equilibrium constant of a portion of the reaction under study (this is rather frequent case if not the principal rule; this is shown in my papers) and these to constants have different temperature dependenced; therewith, the equilibrium constant may decrease with the temperature, and therefore the really measured product may decrease with the temperature. According to the oscillation theory of catalysis which is developed by us, all heterogeneous catalytic reactions at crystal catalysts proceed in such a way that one simple reaction is the rate-determining (RDS) and the residual reaction portion is the equilibrium one. According to this ideology, the best method of characterization of the temperature dependence of the reaction rate includes the revealing of the RDS and determination its temperature dependence. For example, in the case of CH3OH synthesis at the Zn/Cu/Al2O3, the real temperature dependence of the rate constant of the RDS is almost zero and the temperature dependence of the reaction rate is determined by the temperature dependence of the equilibrium constant which enters in the kinetic equation as the product kK, where k is the rate constant of the RDS. Analogeous situation is in many other reactions; the corresponding my papers are available at the ResearchGate.
This is detalization of my previous answer.
No! The rate of product formation is not at all constant.It changes with time.
Dear Feng Peng and Anbarasan Sakkarai,
I do not understand the meaning of yours "nos".
As for the "no" of Anbarasan, my notion relates, of course, to stationary processes; as for the "no" of Feng, his "no" is born by the lack of knowledge.
I address you to my numerous papers which relate to this small and related common problems of heterogeneous catalysis; these papers are available at the RewsearchGate.
Agree with the answer of Prof. Ostrovskii.
From ln(W) versus 1/T one finds APPARENT activation energy.
This APPARENT activation energy HAS SENSE and relates to thermodynamics of the reaction (when reaction is close to equilibrium) or - otherwise - to the activation energy of the RDS or to some more complicated algebraic function of Ea and dH of elementary stages of the reaction. For example, when heterogeneous catalytic reaction rate is determined by adsorption of one of the reactant, APPARENT activation energy can be negative (reaction rate decreases with temperature).
OF COURSE, to determine activation energy you should have STATIONARY process, all measurements have to be done at exactly the same reaction composition (preferably in differential reactor).
Diffusion constrains should be avoided, since they affect the apparent activation energy value. E.g. apparent activation energy of a diffusion limited first order reaction is 0.5 of the true activation energy value.
I would further disagree with all NOеs, mentioned above. With all respect to my colleages, true first order reaction it is rather rare situation. Sometimes we can't even propose the mechanism of the reaction. Full scale kinetic study is a sophisticated task and consumes a lot of time and efforts. And for many complex reactions we can't find a single kinetic constant - kinetic equation is much more complicated! Therefore measuring temperature dependence of the reaction rate is often the only feasible way for a researcher.
I am very glad to hear from my distant friend Prof. Alexander Khassin, and I take this opporttunity to send kind regards and the best wishes to him and all members of his familly. I am also very glad that our views coincide this time, although such coincidences occurr not always.
log k vs 1/T is linear plot the slope of which is used for the calculation of the activation energy. The specific reaction may be either for the formation of the product of reaction or disappearance of reactant depending upon what you measure during the progress of the reaction.
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