[Donaldson Tan]

as chemists and chemical engineers, we often deal with rate constants.

Chemists often deal with a close system, but chemical engineers often deal with open system. The rate equation for both cases are: rate = k[A]ⁿ (considering a single=step reaction).

I am not sure if the rate constants for both open and close system should be the same, because there is mass exchange in an open system.

Arrhenius equation gives the rate constant as k = Ae^-E/RT, where A is a function of the probability that the colliding reactant molecules possess sufficient kinetic energy and colliding at the required orientation.

I don't think A would be the same in an open system versus the close system, because there is prevailing direction of mass transfer in an open system (especially in the case of chemical reactors, where forced convection is dominant). There is no prevailing direction of mass transfer in a close system.

Please advice.

[GCT]

Not too quite sure about this one, as far as I know rate laws can become quite complicating especially if you want to be very specific and are usually derived experimentally, while those thought to the the theoretical rate law from mathematical calculations are merely speculated to describe the rate dynamics at specific times.  

For closed systemsfor the mixing of ideal gases or solutions, I think that the enthalpy of mixing is zero.  In the cases where you speak of convection and for bimolecular reactions which have enthalpy values associated with them the hemholtz energy equilibrium constant and rate constants may be more adequate, such as in referring to explosions where the rate increases dramatically while the reaction progresses especially at higher pressures.

You may also want to consider this in perspective of activities of solution analytes in a closed system versus that of an open system, for which the agents have vapor pressures associated with them.

[Juan R.]

as chemists and chemical engineers, we often deal with rate constants.

Chemists often deal with a close system, but chemical engineers often deal with open system. The rate equation for both cases are: rate = k[A]ⁿ (considering a single=step reaction).

I am not sure if the rate constants for both open and close system should be the same, because there is mass exchange in an open system.

Arrhenius equation gives the rate constant as k = Ae^-E/RT, where A is a function of the probability that the colliding reactant molecules possess sufficient kinetic energy and colliding at the required orientation.

I don't think A would be the same in an open system versus the close system, because there is prevailing direction of mass transfer in an open system (especially in the case of chemical reactors, where forced convection is dominant). There is no prevailing direction of mass transfer in a close system.

Please advice.

Well, the question is that nature is multihierarchical and and each level of molecular description you need different variables. Imagine that you needs variables n = (n1, n2) for description of state of your system. If variable n1 is related to processes with a characteristic time T1 and n2 with processes with T2, you can ignore one of those variables when one of the characteristic times is much more grater than the other. This is called contraction of the description.

Example, you have three processes in your chemical reactor: mixing, external flows, and the single chemical reaction for A <--> B. In principle you would need the vectorial variable n = (a1, a2, a3,..., b1, b2, b3,...) where each a (or b) measures the number of moles (or concentration) of chemical A (or B) in each point of your reactor. Call tR, tM and tF the characteristic times of reaction, mixing and flows respectively. There is several posibilities. Take the next example.

Imagine that flows of A and B are "fast" and mixing is "perfect" (that is, tR >> tM, tF), then one can mathematically prove that all variables 'collapse' except two and your description collapse to n = (a, b) = (a1, b1) = (a2, b2) = (a3, b3) = ... for any time of the order of tR.

This can be mathematically and rigorously proven but intuitivally understood. If mixing is perfect, this means that any inhomogeneity into the reactor is diluted in a tM time and therefore to greater times you can substitute the local concentration (e.g. a3) by a single concentration (for instance a) valid for the overall reactor.

In fact, we usually exploit time scales in our laboratories. For instance, when you work in the picosecond scale, electrons are in different configurations and you may use them. In a femtosecond scale, however, electron motion can already be ignored and you can work with stationary electronic wavefunctions (because nuclear motion is more slow that electronic one). Still motion of nucleus is perceptible in that scale and there is not molecules in the usual chemical sense just atomic-molecular systems.

Below the femtosecond, nuclear motion can be ignored and you can asume that different molecular configurations are equilibrated and, therefore, you work with chemical species; such as benzene. In fact, benzene is a collective denomination for different molecular species all formed by 6C and 6H. In nanoseconds, benzene can suffer chemical reactions but in a cosmological temporal scale, for instance, almost all reactions "happen" and one would even ignore the composition. In fact, in cosmological studies it is really unnecesary talk of chemical composition and just we talk of mass of the universe. That is description is again contracted.

Take your chemical example of rate constant as k = Ae^-E/RT.

This is already asuming a slow time scale. So slow, that system has thermalized and you can define a local equilibrium temperature T. If external flows are fast, there is no time for thermalization inside the reactor and there is not T! Then one may use the concept of nonequilibrium temperature, usually denoted by theta.

Do Theta = T + F(flow), where F is some known function of flows. For times >> characteristic time of flows F--> 0 and, therefore, Theta --> T. In the more general case, you may use Theta.

Imagine that F is a small correction to T then you can use the serie

(1/Theta) = (1/T)(1 - ...) =(1/T) + X.

Perhaps you would write some like

k = Ae^-XE/Re^-E/RT = A'^-E/RT

with A' not equal to A.

This topic is extensive and i cannot deal here in detail. Note that not only one needs modification of usual kinetics knowledge of textbooks (usually focused to chemicals in homogeneous bulk). Other macroscopic disciplines as usual hidrodynamics are not valid in presence of rapid flows in open OR closed systems.

A book i read in the topic of rapid processes some time ago is

http://www.amazon.com/gp/product/0387983732/002-6900299-7958415?v=glance&n=283155

The book is devoted to rational extended thermodynamics. There is other approaches like Extended irreversible thermodynamics. I do not like none of those and prefer use own methods (i have not still published anything in this point but i will do when i have got time). For example, i do not agree with definition of nonequilibrium heat defined in above monograph.

The book have a chapter in reactive mixture that could be of interest for you but i do not remember now much about it. I think that they finalized the chapter saying "this is a very difficult topic and we have done no significant advance" or similar.

If you are interested, the next wednesday i will go to library and will revise the chapter for you. But please say me tomorrow via posting or personal message.

[Donaldson Tan]

Non-equilibrium thermodynamics is indeed insightful. It allows theoretical investigation of rate constants in chemical reactors. The theta function from non-equilibrium thermodynamics extends the Arrhenius equation by providing a modified constant (A')

Given the turbulent fluid conditions inside industrial chemical reactors, I would agree with Juan.R that  there is not enough time for thermalisation to occur within a control volume inside the reactor (PFR and CSTR).

Perhaps the function F(flow) would be characterised by the various dimensionless variables such as Prandtl Number (Pr, ratio of momentum diffusion to thermal diffusion), Reynolds Number (Re, describes the extent of forced convection) and Grashoff Number (Gr, describes the extent of natural convection). It is just speculation, but perhaps it might be in the form of F(Re,Pr,Gr) = a.Re^bPr^cGr^d I am curious if there are such correlation available (in chemical literature) for theta.

The Maclaurin Expanion of 1/(x+1) can be used to find X in the Juan.R's modified Arrhenius Equation. That assumes F/T is small.

Juan.R's proposal also highlights something - what is temperature?

According to the kinetic theory of gas, the microscopic equivalent of macroscopic quantity temperature is the average velocity of the gas molecules. In real gases, energy from intramolecular vibration and intermolecular attraction must be taken in account to describe temperature. Prandtl number takes in account of all these because it is a function of heat capacity (Cp).

Perhaps temperature is best described as a response of randomness since for reversible conditions,
dQ = TdS => T = dQ/dS

Juan.R, it will be really great if you could summarise the chapter on reactive mixture. I believe this topic would be of interest to many undergraduate/graduate chemists and chemical engineers.

[Juan R.]

Perhaps temperature is best described as a response of randomness since for reversible conditions,
dQ = TdS => T = dQ/dS

Seeing your above interest in open systems, let me say that dQ = TdS is only valid for closed systems.

For open (reversible) systems the equation is

dS = dQ/T + d_matterS

where d_matterS denotes the flow of entropy with environment due to interchange of mass

Juan.R, it will be really great if you could summarise the chapter on reactive mixture. I believe this topic would be of interest to many undergraduate/graduate chemists and chemical engineers.

Rational extended thermodynamics of reacting mixtures (introduction)

Linear non-equilibrium thermodynamics (usually named TIP) is based in the supposition of small gradients (e.g. temperature) and slow processes.

There is not accepted extension of TIP to more general situations and one finds several approaches on literature. One of them is rational extended thermodynamics. As stated in my previous message, I do not like that kind of approach, but next I present a short resume of the main ideas and chemically interesting formulas of the chapter on reacting mixtures of Muller and Ruggeri’s monograph http://www.amazon.com/gp/product/0387983732/002-6900299-7958415?v=glance&n=283155

I assume that reader knows the basics of standard equilibrium thermodynamics (at level of MTE formulation), has a basic knowledge of linear non-equilibrium thermodynamics (TIP) and an elementary understanding of special relativity (including three-notations and sum conventions). Moreover, I use a LaTeX-like notation for formulas but without the required command \

Extended thermodynamics (EIT) is based in the idea that the thermodynamics state of any system may be represented by an extended collection of state variables. For example, take a simple system at constant volume and composition. The remaining variable according to equilibrium thermodynamics is internal energy U. Therefore, S = S(U) (S is entropy) and from this basic fundamental equation one can derive any thermal property of that system. In classical extended thermodynamics, the fundamental equation for the above system looks like S = S(U, q) where q is the heat flow. Therefore, one is introducing an extended description of the state; therein, the name “extended thermodynamics”.

In extended TIP

The generalized Gibbs equation for above case is (with C a proportionality parameter)

S = (1/T) [U + C q^2]

The number of variables is still bigger in rational extended thermodynamics. Note that at equilibrium there are not flows (q = 0) and, therefore, the extended fundamental equation reduces to standard Gibbs expression postulated by equilibrium thermodynamics

S = (1/T) U

Note also that T in the equilibrium Gibbs formula is not the same that T at nonequilibrium. both coincide just when the system is at equilibrium i.e. when q=0

I said in my above post that the book had a chapter in reactive mixtures but i did not remember much about it. Unfortunately, authors of the monograph do not present us a detailed study of chemical reactions in the original geodome query, just study some of the modification of usual thermodynamics when there exists great gradients (including chemical gradients), still I will present some basic results. Authors justify their option with [page 157]:

The constitutive theory of a reacting mixture –albeit a nondiffusive one– is quite complex as we shall see.

[Juan R.]

Some main results are summarized in equations (1.5) and concern the modifications of three transport coefficients, viscosity, bulk viscosity, and thermal conductivity in function of the heat of reaction. Expressions are a bit complex (and this board has not extensive mathematical capabilities) and I will write here only if anyone is interested and solicit it (via posting or sending me a personal message).

The modifications of the coefficients are really important, since in some cases the heat of reaction can double the size of the coefficient of thermal conductivity, for example. By the coupling of processes (the famous Onsager relationships –Nobel prize for chemistry– prove that processes interfere ones with others) we know others flows (e.g. diffusion) may modify the kinetics constants, but at what extension? I have done none detailed study on this and the cited monograph is rather superficial in this topic. Next, I present a short resume of the chapter on reacting mixtures. I follow the monograph non-standard notation by commodity.

Authors present a theory of extended thermodynamics for a reacting binary mixture without diffusion. The 15 fields (i.e. the 15 variables describing the thermodynamic state) may be worked in that approach are

Number densities n_alpha
Velocity U^A
Stress deviator t^<AB>
Pressure P
Energy density e
Heat flux q^A

A and B denote the components in relativistic three-notation. For example, q^1 is the component on the x axis of the heat flow q.

For easing measurements, the above n and e fields are substituted by fugacities a_alpha for each reactant and the absolute temperature T (note it is a nonequilibrium temperature!). However, fugacities are not independent in presence of chemical reaction and thus the two fugacities (binary mixture) are replaced by an eq. mixture fugacity more the chemical affinity Delta.

a = gamma_1 m_1 a_1^E = – gamma_2 m_2 a_2^E

Superindex E denotes the value at chemical equilibrium. The gammas are the stoichometric coefficients of the reaction and the ms are the rest masses of the chemicals.

Delta = Sum_alpha gamma_alpha m_ alpha a_alpha

And, of course, Delta = 0 at equilibrium.

[Juan R.]

Using fundamental laws of state (2. and (2.10) one obtains expressions for the fugacities

a_1 = (1/gamma_1 m_1) [a + A_1 Delta]

a_2 = (1/gamma_2 m_2) [–a – A_2 Delta]

where the As are a short notation for the complex expressions summarized in (2.11)

Pressure is also decomposed into an equilibrium part more a dynamical contribution pi

P = Sum_alpha p_alpha^E + pi

In short, we may fix the new 15 fields for specifying any state of the thermodynamic system

a, Delta, T, U^A, t^<AB>, pi, q^A

which may be combined into the particle flux vector A_alpha^A and the energy momentum tensor A^{AB}

A_alpha^A = m_alpha n_{alpha(a, T Delta)}U^A

A^{AB} = t^<AB> + (Sum_alpha p_alpha^{E(a, T)} + pi) h^{AB} + (1/c^2) e(a,T, Delta)U^A U^B m_alpha n_{alpha(a, T Delta)}U^A.

Detailed functional expressions for n_alpha, and e are given in (2.12). h denotes entropy.

The relationship with reaction rate density ell (= LaTeX  \ell ) is given by balance laws type. For instance,

A_alpha^A _{,A} = gamma_alpha m_alpha ell                     (alpha = 1,2)

[the “ , ” denotes the usual noncovariant differentiation]

Next, one may introduce constitutive relations for the flows. In nonequilibrium thermodynamics, the constitutive relations correlate the functional dependence between flows (chemical, thermal, etc.) and the thermodynamic state variables for the system of interest.

For the reaction rate flux,

ell = hat{ell} (n_alpha, U^M, A^{MN}) = hat{ell} (a, T, Delta, U^M, t^<MN>, q^M, pi)

where hat{ell} denotes the constitutive function.

[Juan R.]

If all constitutive functions are known explicitly, we would solve for obtaining 15 independent equations (one for each field to be determined) and every solution of that set is a thermodynamic process from which one can obtain any thermal property of the system. The constitutive functions are restricted by the three main principles of extended rational thermodynamics: the principle of relativity, the entropy inequality, and the requirement of convexity and causality.

For example, the principle of relativity implies isotropic functions with respect to spacetime translations and then authors write

ell = l_pi + l_Delta Delta

with l_pi and l_Delta certain coefficients.


[There is an error of notation in the monograph, first it introduces one notation in (3.4)_1 but after it is used another notation (via letter “l”) for both the pressure part and the chemical part. See text just below (3.4) system of equations]

Next, they exploit the entropy principle; that is, that divergence of the entropy flux was nonnegative: hA_{,A} >= 0     [the “ , ” denotes the usual noncovariant differentiation]. This introduces an additional equation (4.2) with the usual Lagrange multipliers.

Finally, authors explore the nonrelativistic limit of expressions they obtained. After of some manipulations, the reaction rate density (Section 5.2) finally is

ell = (1/D) ({Mc^2}/{kT}) [(n_1^E + n_2^E)U^A_{,A} – (1/kT) q^A_{,A}]

with D a simplification for the large (5. expression. Mc^2 denotes the heat of reaction and k the Boltzmann constant.

The above extended expression can be approximated via an iteration procedure which higher orders of the flows are ignored. For example above expression depends on heat flow q. If the flow is very small one can neglect it and solve for q = 0. Then one obtains a first approximation for ell and pi [the equation for pi (5.7) is similar to above equation for ell but I do not write here because I would needs around five times more LaTeX code than for ell]. Reintroducing the first approximation expression in the full equations and solving one obtain higher order approximations. One can repeat this procedure until obtaining the desired precision.

In the zeroth order approximation, the chemical expression reduces to

ell = (l_Delta – l_pi) [{B_1^Delta} / {B_1^pi}] Delta +

+ [{l_pi} / {B_1^pi}] (2/3D) ({Mc^2}/{kT})^2 [(n_1^E + n_2^E)U^A_{,A}

or introducing abstract coefficients L^11 and L

ell = L^11 Delta + L U^A_{,A}

which is the result obtained by the standard TIP for reacting mixtures (which is valid for slow processes and small gradients). In above extended expression, the Bs are coefficients to be determined. For example the inverse of B_1^pi has, according to the authors, the interpretation of mean time of free flight of the atoms and molecules.

[Juan R.]

If anyone needs more information, further explanation or details can post here or send me a personal message.

As stated above I do not like that kind of approach which I consider not very rigorous from the mathematical side, limited in applications, and based in dubious principles.

In fact, there are authors that state that the variables chose by rational thermodynamicists are not justified even with the appeal to kinetic theory.

For example, the final equations are always derived after of application of three main principles to the generic balance laws and constitutive relationships:

- Principle of relativity, which has no absolute validity as stated by numerous attempts to generalize it. I myself have a paper in press related to this.

- The entropy inequality. Which is just valid on absence of memory effects. In fact, the inequality is usually derived from microscopic theory (e.g. Zwanzig PO) after of applying the so-called Markovian approximation (absence of memory). In the most general case, the rate of entropy can be negative in certain situations. Note this does NOT invalidate the second law of thermodynamics, just the H-theorem. The Second law continues being valid even with a negative entropy rate.

- The convexity requirement. As proven in my work [CPS:physchem/0309002] and as proven by a number of others authors (e.g. my colleague D. H. E. Gross) small systems does not hold the convexity requirement. For example, the entropy of a metallic cluster of 1000 atoms is not convex.

I also doubt that rational thermodynamics can be applied to chaotic regimes.

If anyone needs more information or wants to know more details of the monograph chapter can post or send me a message.

[Juan R.]

Geodome,

In the lecture [1], you can find an explicit expression for a modification of temperature from the point of view of extended thermodynamics (no rational). In extended thermodynamics, the nonequilibrium total entropy of the system is S = S(U, Q) with U the total internal energy and Q the total heat flux. The local expression is (eq. 19 of [1])

(1 / T) = (1 / T_0) – (1 / 2) {partial (tau/{rho lambda T^2}) / partial u} q^2

with T_0 the local equilibrium temperature, lambda the heat conductivity, tau the relaxation time for the local heat flux q, and rho is the mass density. Of course, u is the internal energy per unit mass.

Substituting the nonequilibrium component of the temperature into the Arrhenius law, one finds nonequilibrium corrections to usual equilibrium chemical constant as expressed in an above post. Extended thermodynamics also introduces modification into other usual laws transports for very rapid processes and strong gradients: transport of heat (the traditional Fourier law is not valid then), usual laws of hydrodynamics do not work, etc.

The lecture does not address the case of material flow, not study chemical reactions. But, I think that would be addressed in other bibliography of extended thermodynamics. Perhaps it is detailed in the monograph [2].

Another point of view is explored in [3]. Authors study explicit correction to the rate of chemical reaction. The paper includes some references to studies on the topic of nonequilibrium effects on chemical reactions (perhaps, you would consult them for your research).

They compute the rate for the reaction

A + B  -> Products

for a three component gas using the standard Boltzmann kinetic equation. Then they just compute the change on the reaction from a change on the velocity distribution functions of reagents. The kinetic basic expression is

v_{ch}^{(0)} = 2 n_A n_B d_{AB}^2 s_F (2 pi k_B T / m_AB)^{1/2} exp (-E* / k_B T)

where (0) denotes that is computed assuming Boltzmann/Maxwell equilibrium for reactants. In above formula n_i are the number densities of gas, d_{AB} the average diameters of colliding spheres, s_F the steric factor, m_{AB} the reduced mass for the collision pair, and E* the threshold energy (see [3] for more details in the model).

The change on chemical rate v_{ch} obtained from the introduction of a nonequilibrium T_A is important, computing modifications near the 30% (they argue that nonequilibrium corrections could even exceed 50%) for the deviation

nu = 1 – {v_{ch}}/{v_{ch}^{(0)}}

in a broad range of threshold energies (see figure 1 of [3]). I find interesting that for determined E* the modifications are bigger for higher temperatures.

At priori, one could intuitively think that at higher temperatures the molecules are more randomly distributed and the effects would “dissolve” but there exists an increasing until the lower limit E* = k_B T and thus a rapid increasing to achieve the 7%. Interesting!

The interesting is that the nonequilibrium temperature (Shizgal–Karplus temperature) is there computed from a nonequilibrium distribution function of velocities. I think that would be easier for you to adapt the formalism to a nonequilibrium distribution of velocities generated for some rapid flow in the chemical reactor. Since your original thought was about modifications caused by

In any case the modification may be important in most cases, I believe!

The article [3] finalizes with

A nonequilibrium temperature description for chemically reacting systems may be a convenient tool for making evaluations.

I wait that the typical phrase in chemical kinetics textbooks

thermodynamics is defined only for equilibrium, chemical kinetics deals with rate processes

was eliminated in future textbooks.


[1] Casas-Vázquez J. Jou, D. Dynamical evolution and fluctuations in heat conduction In Algunas reflexiones acerca de la Termodinámica. M Criado-Sancho (Ed.). UNED: Madrid, 1990.

[2] Thermodynamics of Fluids Under Flow

[3] Cukrowski, A.S.; Popielawski, J.; Stiller, W.; Schmidt, R. J. Chem. Phys. 1991, 95(8), 6192.

[Donaldson Tan]

This thread is a good read.

[sraman30]

Hi I need to know whether any online resources is there for problems in octave levenspiel book.

[Donaldson Tan]

You can post the problem and your solution here.

I love to discuss theoretical chemical reaction engineering problems.

[baleverywhere]

Hi,
I have to design a fermenter where glucose is converted to ethanol:
C6h1206 -----yeast---> 2C2H5OH + 2CO2
The thing is that I need the reaction time and to calculate that I need the reaction rate. Ive been looking at several books to get the values of activation energy and pre-exponential value to calucalte the reaction rate and then the reaction time, but I couldn't find any..So can anyone help me please?
Thankss

[eugenedakin]

as chemists and chemical engineers, we often deal with rate constants.

Chemists often deal with a close system, but chemical engineers often deal with open system. The rate equation for both cases are: rate = k[A]ⁿ (considering a single=step reaction).

Hi Geodome,

I hope that you are doing well.

I don't know if I am the exception, but I hardly deal with closed systems.  I might actually suggest that it is the other way around, chemical engineers deal with closed systems and chemists deal with open systems.   

Just my $0.02.

Sincerely,

Eugene