Ammonia, with its four atoms, is an incredibly simple molecule:

Despite this simplicity, its large-scale industrial production from its constituent elements — nitrogen and hydrogen — was, and remains, a great scientific challenge. The balanced equation of the synthesis, again, deceptively simple, can be written as:

N2 + 3H2 → 2NH3

Hydrogen reacts with nitrogen to form ammonia. Hydrogen gas is readily available from fossil fuels, or from electrolysis of water. As for nitrogen, it is readily available… well, all around us. In fact, the very air you are breathing right now contains four nitrogen molecules out of every five!

So, what’s the problem? Why were not one, but *two* Nobel Prizes awarded for ammonia synthesis? As always in science, the devil is in the details. It turns out that nitrogen, one of our reactants, is an incredibly stable molecule. It is frequently used in the laboratory as an inert gas.

Its resistance to reaction can be attributed to its structure:

N≡N

The triple bond is extremely strong, so breaking it requires very high temperatures (reactions of N_{2} have high activation energy). If you have been following the Chemical Kinetics series, you might be able to guess the solution: you use a catalyst to lower the activation energy (we discussed biological catalysts in the last post of this series).

The synthesis of ammonia is known as the Haber or Haber-Bosch process, which is named in honor of the chemists who developed the first effective catalysts. The production catalyst, which is made of iron and iron oxide with various other additional compounds, was first used on bulk scale in 1913 and is still commonly used today. This is an active and cheap solution that allows the process to be performed directly at an appreciable rate.

It is telling that significant research on ammonia synthesis was performed during the years leading up to World War I. A short digression on why we would want to produce ammonia will inform you about the industrial importance of the process.

In a word: explosives. Ammonia is a precursor compound in the manufacturing of a large number of explosive substances that have been in use for centuries.

Before the era of industrial ammonia synthesis, these precursors were usually imported from countries rich in *guano* (bird and bat droppings). Guano was abundant in Latin America and had a geopolitical importance comparable to petroleum in the present era.

Wars were waged over the control of guano-rich regions. When the sea routes to the German Empire were sealed off by the Entente Powers, it effectively put a limit to the duration of World War I. Without guano, the production of explosives would have had to come to an end. The industrial, large-scale production of ammonia in Germany led to a prolongation of fighting and destruction in Europe.

All is not dark in the world of ammonia production, however. There is another — lighter — side to the story.

Ammonia is also used for another vital purpose: fertilization. Nitrogen compounds are required for the growth of plants and animals. Remember that proteins consist of amino acids. The “amino” part of the name is due to the nitrogen-containing group.

Until less than a century ago, agricultural output was limited by nitrogen-rich, “naturally occurring”, fertilizer, which can be viewed as the mammalian equivalent of guano. In the past, there was a very real fear of global famine because there was just not enough fertilizer available to ensure the necessary agricultural output for global food security.

The synthesis of ammonia changed all of this. Our world can, at present, sustain a much larger number of people thanks to industrially-synthesized fertilizers. It is now estimated that nearly 80% of the nitrogen atoms found in human tissue have passed through the industrial production of ammonia.

From here, we will ignore the good and bad of producing ammonia, and move straight to considering the chemical engineering involved by simulating ammonia synthesis with COMSOL Multiphysics.

It comes as no surprise that ammonia production is one of the best understood chemical processes. Countless scientific publications and textbooks have been written on the subject. Considering a global production that exceeds 200 million tons per year, even seemingly minor process improvements will yield great economic benefits.

Ammonia synthesis is an *exothermic* (heat-releasing) reaction because the combined bond strength of the *six* N-H bonds, in the products, is greater than the combined bond strength of the one N≡N and three H-H bonds in the reactants. Exothermic reactions are, thermodynamically, less-favored at higher temperatures. However, we need high temperatures, even with a catalyst, to overcome the significant activation energy. In fact, ammonia synthesis is a perfect example of a delicate balance frequently encountered in chemistry: the *kinetics* (rate) of the reaction would profit from higher temperatures, but the *thermodynamics* (yield of product at equilibrium) would be more advantageous at lower temperatures.

These subtleties suggest that modeling may help us find optimal conditions. Since the process is well understood, a high-fidelity of modeling is demanded by the chemical industry, and so the equations describing ammonia process design are very detailed and highly non-ideal (and as a consequence, extremely nonlinear). At the pressures and temperatures typically used to produce ammonia (around 400°C and 200 bar), chemicals behave quite differently from the way that the mass action and ideal gas laws would have us expect.

One of the many known kinetic rate expressions, for instance, is written as:

(1)

r_\mathrm{NH_3}=2k\left[K_a^2a_\mathrm{N_2}\left(\frac{a^3_\mathrm{H_2}}{a^2_\mathrm{NH_3}}\right)^\alpha-\left(\frac{a^2_\mathrm{NH_3}}{a^3_\mathrm{H_2}}\right)^{1-\alpha}\right]

Here, k refers to the Arrhenius rate constant, K_a to the equilibrium constant, a_i to the activity of species i, and \alpha to an empirical constant that is usually between 0.5 and 0.75 in literature.

For the equilibrium constant, K_a, the following expression has been empirically derived as a function of temperature, T, in units K:

(2)

$\begin{array}{r}

{\log _{10}}{K_a} = – 2.691122\;\log T – 5.051925 \cdot {10^{ – 5}}{\mkern 1mu} T\\

+ 1.848863 \cdot {10^{ – 7}}{\mkern 1mu} {T^2} + \frac{{2001.6}}{T} + 2.689

\end{array}$

{\log _{10}}{K_a} = – 2.691122\;\log T – 5.051925 \cdot {10^{ – 5}}{\mkern 1mu} T\\

+ 1.848863 \cdot {10^{ – 7}}{\mkern 1mu} {T^2} + \frac{{2001.6}}{T} + 2.689

\end{array}$

The activities can be determined from the fugacity coefficients, \phi, via:

(3)

a_i=x_i\phi_iP

where x_i is the component mole fraction and P is the system pressure.

The fugacities themselves also depend on empirical equations. These expressions quickly become extremely unwieldy. Take, for instance, the expression for hydrogen:

(4)

\begin{align}{r}

\phi_\mathrm{H_2} &=\exp \left\{e^{-3.8402\,T^{0.125}+0.541}P-e^{-0.1263\,T^{0.5}-15.98}P^2\right.\nonumber\\

&\qquad \left. {}+300\left[e^{-0.011901\,T-5.941}\right]e^{-\frac{P}{300}}\right\}\nonumber

\end{align}

\phi_\mathrm{H_2} &=\exp \left\{e^{-3.8402\,T^{0.125}+0.541}P-e^{-0.1263\,T^{0.5}-15.98}P^2\right.\nonumber\\

&\qquad \left. {}+300\left[e^{-0.011901\,T-5.941}\right]e^{-\frac{P}{300}}\right\}\nonumber

\end{align}

Besides violating any aesthetic sentiment, you can see that we have no hope of solving these equations by hand. An expression like this is extremely nonlinear and may cause difficulty in the computation. The difficulty doesn’t stop there, however. The rate expression must also be multiplied with a catalyst-dependent efficiency factor, \eta, to account for the mass transport to the interior of the catalyst:

(5)

r_\mathrm{eff}=\eta\cdot r_\mathrm{NH_3}

The efficiency factor is computed from the empirical correlation, below (with Z as the conversion):

(6)

\eta=b_0+b_1T+b_2Z+b_3T^2+b_4Z^2+b_5T^3+b_6Z^3

The b coefficients are known at three different pressure values and must be interpolated between these.

As if the problem wasn’t already complex enough, the significant heat released by the exothermic reaction means that the energy balance must be solved together with the mass balance. The reaction enthalpy also depends on the pressure and temperature — as do the heat capacities for all involved species.

In short, we are presented with a relevant problem involving several equations, complex nonlinearities, and couplings between heat and mass conservation. In other words, an ideal playground for COMSOL Multiphysics.

Fortunately for us, heat capacity and reaction enthalpy data can both be readily obtained from the National Institute for Standards and Technology (NIST) WebBook in a very useful, exportable, format. Even more fortunate, this exported data, in a simple tabulated format, can be imported directly into COMSOL Multiphysics.

Take, for example, the heat capacity data for ammonia. One can either choose the NIST option or input an empirical function derived in the 1980s:

C_\mathrm{p,NH_3}=4.184\cdot{6.5846-0.61251\cdot 10^{-2}\,T+0.23663\cdot 10^{-5}\,T^2-1.5981\cdot 10^-9\,T^3}

{+96.1678-0.067571\,P+(0.2225+1.6847\cdot 10^{-4}\,P)T}

(7)

{+(1.289\cdot 10^{-4}-1.0095\cdot 10^{-7}\,P)T^2}

where the molar heat capacity is expressed in J/mol/K.

Given the choice between typing out an equation, like the above, and importing benchmarked and tested data from NIST — a matter of just a few clicks in COMSOL Multiphysics — the import wins, hands down. Accordingly, all heat capacities were simply imported into the software and applied without any further refinement:

*Interpolated heat capacity for ammonia from the NIST WebBook.*

Once all the relevant data has been imported into COMSOL Multiphysics, setting up the reaction kinetics is a breeze.

We have chosen the Ammonia Synthesis PFR model (available in our Model Gallery), so we can compare it to an existing benchmark (Rase, H. F: *Chemical Reactor Design for Process Plants*, Volume 2: Case Studies and Design Data, 1977).

By using a steady-state plug flow reactor model, where time is utilized for stepping along the length of the reactor (called 0D in the COMSOL software) and an intermediate pressure, we can bypass the expensive CFD calculation and leave it for a more detailed analysis. We can incorporate an energy calculation into the reaction system with the “Energy Balance” node.

Here are some results for yields and reactor temperature:

*Mole fractions for hydrogen (green), nitrogen (blue), and ammonia (red) as a function of reactor extent.*

*Temperature profile within the ammonia conversion reactor.*

While the figures do serve to make a point regarding the descriptive power of the model, nothing beats numbers when comparing with existing benchmarks.

For the same process conditions as applied in this COMSOL Multiphysics model, Rase obtains a final ammonia mole fraction of 13% and an outlet temperature of 970°F (essentially the same values we obtained here at a fraction of the effort).

While this 13% outlet fraction may seem low at first glance, the balancing act between thermodynamics and kinetics (as mentioned above) is responsible for such a low-equilibrium composition. Because the separation of the product from the substrates is easily and efficiently accomplished, large recycle-loops are industrially applied. The unreacted nitrogen and hydrogen gases will be distilled from the mixture and returned to the reaction vessel, making ammonia synthesis one of the most efficient and highly-optimized chemical processes.

This concludes our Chemical Kinetics series. Today, we discussed chemical reaction engineering for a highly complex and nonlinear process. We also simulated one of the most important processes in the world — ammonia synthesis — with COMSOL Multiphysics and compared our results to an existing benchmark.

While this analysis is only meant to provide an initial overview of ammonia synthesis, it shows the great potential and power the COMSOL software can provide to relevant (and sometimes world-changing) chemical technology.

- Read the complete Chemical Kinetics series
- Read more about the Ammonia Synthesis PFR model
- Learn more about the software:
- Try COMSOL Multiphysics first-hand

To quote a favorite phrase of my former professor: “A good and benevolent fairy waved her wand and gave us the activation energy.” The truth is, these parameters (rate constant, activation energy, and pre-exponential factor) are the result of a great number of repeated experiments. Almost all of the experimental laboratory work I did during my research career involved running chemical reactions at different temperatures, sampling at regular intervals, transforming the results, and calculating rate constants.

Have a look at the structure of the chemical compound benzenediazonium chloride (BDC):

As innocuous as that little structure seems, it packs a punch. It belongs to the family of the *diazo compounds* (meaning with a carbon atom, C, adjacent to two connected nitrogen, N, atoms) a group of compounds that are dangerously explosive.

Recall from this series’ first blog entry on the Arrhenius Law that any molecule requires a particular amount of energy to overcome its inherent “laziness” or resistance to reaction — this reaction is the activation energy. Today, I will describe how a careful series of experiments (along with some calculations in COMSOL Multiphysics) can give you precisely that energy, thus allowing a statement on the safety of a particular process.

The decomposition of BDC proceeds irreversibly (as would be expected from an explosion), so we can write the fairly simple reaction equation:

BDC → Products

This corresponds to the rate equation (again, see “A General Introduction to Chemical Kinetics, Arrhenius Law” for details):

(1)

r=k\cdot c_\mathrm{BDC}

with the related ordinary differential equation (ODE):

(2)

\frac{dc_\mathrm{BDC}}{dt}=-r=-k\cdot c_\mathrm{BDC}

If we recall the Arrhenius equation

(3)

k=A\cdot e^\frac{-E_\mathrm{A}}{RT}

the problem becomes relatively clear. By performing the decomposition reaction at various temperatures and measuring BDC concentration at different times, we can fit the values of A and E_\mathrm{A} to best reflect the experimental results with the modeled ODE solution.

Of course, the above rate equation can be integrated directly and in practice we would use our knowledge of the concentration vs. time expression (we derived a similar expression in the post on the Arrhenius Law ) to determine the rate constant directly from the gradient of a straight-line plot of \ln{c_\mathrm{BDC}} against t. Then we can plot \ln k against 1/T to find the activation energy, which in practice can be seen as a graphical least square problem since the experimental points will never be “perfectly” aligned.

Here we will use the *Parameter Estimation* tool to demonstrate the corresponding numerical solution for this simple case. Automated parameter estimation becomes much more valuable when used for chemical mechanisms where we don’t know the expressions for concentration as a function of time in closed form, and so we have to proceed by modeling.

Using the Model Builder, we can select the *0D Reaction Engineering* interface with a time-dependent study from the Model Wizard. We input the reaction via BDC → Products and define an initial BDC concentration of 1 mol/L. Defining a parameter called “Tiso” (for the system temperature) and setting the temperature to this parameter is just a matter of three clicks. By right-clicking the Reaction Engineering node we can add a *Parameter Estimation* interface where we can choose our control variables and define our least square problem.

We can actually reduce the computational effort significantly if we choose *not* to optimize for A, but rather for e^{A}. This way, the solver is dealing with two exponential dependencies at the same time — rather than one linear and one exponential. Suitable initial values are a matter of experience; both activation energies of around 150 kJ/mol and prefactors of around e^{50} are reasonable.

Getting the experimental data into COMSOL Multiphysics® is as easy as right-clicking Parameter Estimation, selecting the option to load from file, and selecting the corresponding .csv-file (these files can be easily exported from Excel®). At the bottom of the Experiment settings window, we can select the experimental parameter that was varied for the experiments. The data is automatically plotted after the import, so we can ensure it makes sense:

Finally, we can select the option to use the Arrhenius expression from the Reaction settings window and type in the control variable names (“exp(Aex)” for the prefactor and “E” for the activation energy). By choosing study times between 0 and 5,000 seconds, we ensure all the experimental data is accounted for.

Choosing a solver in the study depends on the purpose, and the choice is usually done from experience. In the case of parameter estimations for reaction engineering, the Levenberg-Marquardt solver is often the best choice, as it is well established in the field of parameter estimation for reaction kinetics. While running the simulation only takes a few seconds, the results are striking:

*Modeled results with optimized reaction parameters (solid lines) together with experimental results (o)*.

The obtained values are e^{36.9}s^{-1} and 116 kJ/mol for the prefactor and the energy, respectively. Experimental work is tricky at best; it requires patience, precision, and a great deal of nerve. By making the import and interpretation of your experimental data this easy, you can focus on what’s really important.

The next blog post in this Chemical Kinetics series will look at another way of getting external data into your system — by using interpolation functions for thermodynamics. In doing so, we will come across one of the most fascinating processes in industrial chemistry. A reaction with a history as dramatic and complex as Tolstoy’s *War and Peace*! Stay tuned.

The previous blog post introduced the concept of an activation energy. This is the required energy with which molecules must “impact” in order to react. For a great number of vital reactions, this energy is quite high. Some molecules, it turns out, are just lazy and quite content to stay in their present form. This is not necessarily a bad thing — take cattle, for example.

Cattle, like any organic matter, including humans, are inherently thermodynamically unstable; besides their water content, they consist largely of hydrocarbons, usually oxygenated and nitrated. This is a potentially combustible mixture that would be significantly more stable as carbon dioxide, nitrogen, and water. The *thermodynamics* would indicate that this is a much more energetically favorable form. Thankfully, the *kinetics* of their decomposition reactions are all quite slow — the molecules simply lack the activation energy necessary to decompose.

Now consider leaving a steak on a hot grill. If you do that, you’ll see what happens when activation energy is present in abundance; first, the meat will char, and eventually, you will be left with a pile of ash (the mineral, inorganic components of meat). This brings us to my next point: how do we humans gain energy from eating a (properly cooked) piece of meat? The answer is through *catalysis*.

A *catalyst* is a substance that somehow pulls and twists a molecule in such a way that it can react according to a different, energetically lower, pathway. The effect of this is that a reaction can suddenly proceed at an appreciable rate at much lower temperatures. Catalysis is everywhere — around 95% of all industrial chemical products go through at least one catalytic step in their lifetime.

One important group of catalysts, optimized over millions of years to work at very moderate temperatures, are *enzymes*. Enzymes are large molecules (usually consisting of several thousand atoms) that allow biological systems to perform the chemical reactions necessary to sustain life. Chew a piece of bread for a while and it will turn sweet — this is due to *amylase*, an enzyme present in human saliva that can turn starch into sugar to allow for easier digestion.

Moving back to our steak example, it turns out that the meat itself would be quite tough, unappetizing and difficult to digest if it weren’t for enzymes. Even before the beef hits our plate, the butcher will have hung the meat for several weeks to allow some of the naturally occurring enzymes to begin the process of protein degradation — thus tenderizing the meat and enhancing its overall flavor. Further down the line, the enzyme responsible for the degradation of protein in our digestive tract is called *pepsin*. Pepsin was in fact the very first enzyme to be discovered, so I feel it’s an apt example for this blog entry. Let’s take a closer look at how pepsin works.

In an initial step, the substrate **S** (in this case the beef protein) must associate with the enzyme **E**.

**E + S → ES**

This substrate-enzyme complex **ES** can just as well separate again into the substrate and the enzyme according to:

**ES → E + S**

The two reactions are usually written as one, using a shorthand description for reversible reactions as:

**E + S ↔ ES**

Instead of dissociating, the substrate may react while complexed to the enzyme. The enzyme then releases the product protein:

**ES → E + P**

The entire system may thus be written as:

**E + S ↔ ES → E + P**

This reaction is closely related to the example in the previous blog post on chemistry. Let us now attempt to formulate the rate equations for this system.

The production of the substrate-enzyme complex **ES** is:

(1)

r_1 = k_1\,c_\mathrm{E}c_\mathrm{S}

And its consumption:

(2)

r_2=k_{-1}c_\mathrm{ES}+k_\mathrm{cat}c_\mathrm{ES}

We now assume the so-called (*pseudo-*)*steady state approximation* (PSSA or SSA) for **ES**. This common approximation suggests that a reactive intermediate like the enzyme-substrate complex is highly reactive. The reactions that consume the product species — dissociation or reaction — are assumed to be inherently fast compared to complex formation. In this case we can approximate that the rate of change of the concentration of **ES** is negligible. We are allowed to do this even though the concentration of **ES** may change as the reaction proceeds; what matters is that on the time-scale of the reaction we are interested in, the concentration of **ES** responds very quickly to a change in the overall concentration of the reactant substrate **S**.

This approximation leads to:

(3)

k_1\,c_\mathrm{E}c_\mathrm{S}=k_{-1}c_\mathrm{ES}+k_\mathrm{cat}c_\mathrm{ES}

which is equivalent to:

(4)

\frac{k_{-1}+k_\mathrm{cat}}{k_1}c_\mathrm{ES}=c_\mathrm{E}c_\mathrm{S}

Collecting all the terms on the left and lumping them into a *Michaelis-Menten equilibrium constant* K_\mathrm{M}, we obtain

(5)

K_\mathrm{M}\,c_\mathrm{ES}=c_\mathrm{E}c_\mathrm{S}

Regarding the concentration of the enzyme present in the system, we know the initial concentration of **E**, and we already have defined a variable for **ES**, so we can write:

(6)

c_\mathrm{E}=c_\mathrm{E}^0-c_\mathrm{ES}

Inserting this into the above and solving for the concentration of **ES**, we are left with:

(7)

c_\mathrm{ES} = \frac{c_\mathrm{E}^0c_\mathrm{S}}{K_\mathrm{M}+c_\mathrm{S}}

This result can be used to determine the production rate of the protein:

(8)

\frac{dc_\mathrm{P}}{dt}=k_\mathrm{cat}\frac{c_\mathrm{E}^0c_\mathrm{S}}{K_\mathrm{M}+c_\mathrm{S}}

It is interesting to note here that under conditions of high substrate concentration (as is usually the case when the reaction has not yet proceeded far), we can approximate that c_\mathrm{S} \gg K_\mathrm{M}. Under these conditions:

(9)

\frac{dc_\mathrm{P}}{dt} \approx k_\mathrm{cat} c^0_\mathrm{E} \equiv v_\mathrm{max}

Under these conditions, the rate of substrate consumption is independent of the substrate concentration itself: it is “zeroth order”. The rate only depends on the available concentration of the enzyme, which catalyzes the substrate reaction as fast as it can. The quantities K_\mathrm{M} and v_\mathrm{max} together are used to define the kinetics of the reacting system.

This whole derivation is known as the *Michaelis-Menten mechanism*, named after Leonor Michaelis and Maud Menten. It finds wide application in a plethora of biochemical reactions. At the time Michaelis and Menten published their results 101 years ago, they had no access to a digital computer with which to quickly integrate the system of differential equations for the four species. Instead, by performing the above analysis with an intuitive, physically motivated approximation, they obtained a simple relationship that showed them what influence the enzyme and substrate concentrations had on the rate of enzymatic reaction. How good is this approximation? Thankfully, we can now calculate the full model — without the PSSA for **ES** — and compare it with the result for the simplified equation.

Using the *Reaction Engineering* interface, typing in the two relevant reactions is simple and intuitive. By using the <=> symbols, COMSOL Multiphysics immediately understands that a reversible reaction is to be simulated, enabling the necessary input fields for both forward and reverse rate constants (as well as generating variables for **E**, **S**, and **ES**). In the same manner, once the equation for the production of **P** has been typed in, a variable for the corresponding concentration is created.

Values of k_{1} = 1700 M^{-1} s^{-1}, k_{-1} = 10^{-3} s^{-1}, and k_{cat} = 0.5 s^{-1}, as well as reasonable values for the initial concentrations of **E** and **S**, were taken from literature (Calorimetric Observation of a GroEL-Protein Binding Reaction with Little Contribution of Hydrophobic Interaction; K. Aoki et al. J Biol. Chem. vol. 272, pp. 32158-32162, 1997). As we are dealing with such significant differences in scales, the two highly concentrated species (**P** and **S**, Figure 1) are shown separately from the highly dilute species (**E** and **ES**, Figure 2). Note that this difference in scales inherently justifies our use of a steady state approximation!

*Figure 1: Concentration profiles of substrate (blue) and product (green) over time.*

*Figure 2: Concentration of enzyme (blue) and enzyme-substrate complex (green) over time.*

*Figure 3: Comparison of Michaelis-Menten kinetics (green solid line) with fully integrated reaction mechanism (blue asterisks).*

How can these results give us an idea of the error associated with the Michaelis-Menten mechanism? We can compare the results with the calculated formula for the rate of formation of product. To do this, the COMSOL variable **comp1.re.r_2**, the rate for the production of protein, and **kcat*E0*comp1.re.c_S/(Km+comp1.re.c_S)**, the derived Michaelis-Menten expression, are plotted on the same 1D Plot Group as functions of substrate concentration. This yields Figure 3, above. You can see that the agreement is very good between the approximate theory and the complete numerical approach.

Just after the 100-year anniversary of the Michaelis-Menten mechanism, we honor their trailblazing publication the best way we know how — by simulating.

*Next month, we’ll discuss chemical parameter estimation. Bookmark our Chemical Reaction blog series to stay in the loop.*

Chemistry is, quite literally, vital. Any biological system relies on chemical reactions to function. Anytime we want to influence biological systems — be it to brew beer or to cure a (possibly beer-related…) headache using painkillers — we are actually influencing chemistry. But chemistry is not merely the underlying principle of biology: it is also vital to any aspect of our modern age. From solvents, colors, fuels, and lacquers to plastics and pesticides, there are more than 70,000 industrially-produced chemical compounds. Therefore, it is no surprise that the chemical industry is one of the largest contributors to the global economy. The total value of products of the chemical industry is estimated to be almost 5 trillion USD per annum! This corresponds to around 7% of the gross world product.

Take, for example, the transformation of some substance \mathrm{A} (called a *reactant*, *substrate*, or *feedstock*) into some other, ideally more valuable, substance (called a *product*) symbolically represented by the *reaction equation*:

2\mathrm{A} \rightarrow \mathrm{B}

Even though this equation makes no statement on the *mechanism *of the reaction (that is, in what order bonds are broken and formed, and which intermediate chemical species are transiently created), it still allows us to mathematically describe the macroscopic *rate* of reaction. To better understand the concept of a macroscopic reaction rate, imagine a virtual experiment where we allow substance A to react in a beaker and intermittently take a sample with a fixed volume. We then place this sample into an analytic device that, through some means, can count the number of particles of \mathrm{A} and \mathrm{B} that are present in the sample. By measuring the number of particles and dividing it by the sample volume, we are in fact measuring a *concentration*.

Concentrations are usually described in the units of mol/L or mol/m^{3}. A *mole* is a unit of “amount of substance”, and is a shorthand for N_A atoms, molecules, electrons, or whatever we are measuring. This is convenient because the number of molecules occurring in practical samples is very large; Avogadro’s number N_A represents a very large number of molecules (roughly 6×10^{23}/mol).

Returning to our virtual experiment, we might observe behavior according to the plot below:

*Concentration profile in time: Substance A (blue) decreases as substance B (green) increases. Note that the decrease of A is twice as fast as the increase of B.*

Can we propose a mathematical expression to describe the rate of the chemical reaction? As the concentration of the reactant increases, it is likely that the rate of reaction will also increase, but with what proportionality? One suggestion is the *law of mass action*, which imagines reactions due to collisions between reactants. We propose that the rate at which \mathrm{A} reacts is proportional to the likelihood of *two* particles of \mathrm{A} meeting each other. This suggests the following fundamental *rate law* for the examined reaction:

r_1=k_1\cdot{c_\mathrm{A}^2}

where k_1 is called the *reaction rate constant* (we will qualify the use of the term “constant” below, but for now it may be assumed). This is called a “second-order” reaction in the reactant \mathrm{A} since the rate is proportional to the second power of the reactant concentration: that proportionality arises from the collisional argument above. It is important to emphasize that the *law of mass action* does not apply to all chemical reactions; as discussed above, the observed rate law does not correlate directly with the actual mechanism of the reaction.

In any rate law, the units of the rate constant k depend on the reaction order: in this case k_1 has the unit of (L/mol)/s (pronounced “per molar per second”). We can now use this expression for r_1 to determine the change in concentration for \mathrm{A} and \mathrm{B}:

\begin{split}

\frac{dc_\mathrm{A}}{dt} %26= -2{r_1}\\

\frac{dc_\mathrm{B}}{dt} %26= +r_1

\end{split}

\frac{dc_\mathrm{A}}{dt} %26= -2{r_1}\\

\frac{dc_\mathrm{B}}{dt} %26= +r_1

\end{split}

Note the multipliers -2 and +1 in the respective equations for \mathrm{A} and \mathrm{B}. This is due to the *consumption* of two particles of \mathrm{A} for the *production* of one particle of \mathrm{B}. The multipliers are called *stoichiometric coefficients* or *stoichiometric numbers*, and they are often written with the Greek letter *nu* (\nu).

The chemical rate law yields a set of two ordinary differential equations (ODEs), which are intrinsically simpler to solve than the usual space-dependent partial differential equations (PDEs) that COMSOL Multiphysics users encounter. The equation set for the reaction system described above can easily be integrated to obtain:

{c_\mathrm{A}} = \frac{c_\mathrm{A,0}}{1 + 2c_\mathrm{A,0}\cdot{k_1} {t}}

Here, c_\mathrm{A,0} represents the initial concentration of species \mathrm{A} in the reaction vessel. This expression is graphically shown below alongside the numerically derived values for the concentration:

*Numerically (asterisks) and analytically (solid line) derived concentrations of A over time.*

For this example, an initial concentration for \mathrm{A} of 1 mol/L was used, as well as a value for {k_1} of 0.001 (L/mol)/s at 20°C.

In general, the chemical rate expressions will be a combination of terms, in this form:

r_i = k_i \prod_j{c_j^{\nu_{ij}}}

for the reaction, i, of a set of chemical species, j, with respective reaction orders, \nu_{ij}. Again, the reaction order will often correspond to the stoichiometry, but for some reaction mechanisms this is not necessarily the case.

It is apparent that these expressions can very quickly yield analytically unsolvable equation sets. This is an especially important consideration because several highly relevant chemical systems (e.g. combustion, cracking of long-chain hydrocarbons, biochemical systems, etc.) may involve dozens or even hundreds of participating species, each with their corresponding rate equations. But to fully appreciate the necessity of chemical reaction engineering simulation, you need only add a single additional reaction. Going back to our system of interest, let’s further assume that our value-added product \mathrm{B} can further react to a waste product \mathrm{C}. So:

2\mathrm{A} \rightarrow \mathrm{B} \rightarrow \mathrm{C}

Rather than being only of academic interest, this sort of *reaction in series* is in fact one of the most frequently encountered chemical systems in industry. Take, for instance, oxidation reactions, one of the largest fields in industrial chemistry. A particular challenge is oxidizing the substrate to a large degree without proceeding to the *fully* oxidized product (in organic cases, this might be carbon dioxide), which is essentially without value.

The new reaction can be accounted for by writing out its rate as:

{r_2} = {k_2}\cdot c_\mathrm{B}

and the ODEs as:

\begin{split}

\frac{dc_\mathrm{A}}{dt} %26= -2{r_1}\\

\frac{dc_\mathrm{B}}{dt} %26= +{r_1}-{r_2}\\

\frac{dc_\mathrm{C}}{dt} %26= +{r_2}

\end{split}

\frac{dc_\mathrm{A}}{dt} %26= -2{r_1}\\

\frac{dc_\mathrm{B}}{dt} %26= +{r_1}-{r_2}\\

\frac{dc_\mathrm{C}}{dt} %26= +{r_2}

\end{split}

*Note that the expression for substance B now has two terms as it participates in two reactions.*

Adding just this one additional reaction to our system makes the ODE set much harder to solve by hand. Mathematically minded readers may be interested to try it by hand — the concentrations of this system can be expressed in closed form using some less familiar special functions. It is certainly a lot faster and more extensible, though, to solve the problem using chemical engineering software. Solving the equations in COMSOL Multiphysics, we observe the behavior described in the below figure (for a value of {k_2} measuring 0.0015 1/s). If we now place ourselves in the position of a chemical engineer designing this reaction process, we might have some very specific criteria that must be fulfilled.

Perhaps \mathrm{C} is a toxic substance, whose concentration may not exceed a critical value. Or perhaps the process is not economically feasible unless a particular threshold for yield of \mathrm{B} is exceeded. Let’s assume the latter; any value of \mathrm{B} above 0.15 mol/L at the end of reaction is acceptable.

*Example of a reaction in series: A (blue) is transformed to B (green), which in turn can further react to C (red).*

If our goal is to obtain a certain amount of \mathrm{B} from a particular amount of \mathrm{A}, we would have to somehow stop, or *quench*, the reaction after around seven minutes. As this is usually not feasible, we have to start looking at other methods of optimizing our yield of \mathrm{B}.

Let’s take a closer look at the idea of a reaction rate “constant” k. It turns out that two particles of \mathrm{A} meeting is not enough to lead to a reaction. They also have to impact each other with enough energy to surmount an unstable barrier to reaction (called a *transition state*) and transform into the stable reactant state. This energy that is to be surmounted is referred to as the *activation energy*. Since we are looking at so many particles at the same time (remember that a mole contains over 10^{23} molecules), we can use statistics to consider that a certain fraction of particles will collide with energies above the activation energy, and a certain fraction with energies below. Based on statistical thermodynamics and the concept of a *Maxwell-Boltzmann distribution*, the following equation for the temperature dependence of {k} can be derived:

k(T) = A(T)\;\mathrm{exp}\left(\frac{-E_A}{RT}\right)

This equation is known as the *Arrhenius* equation.

A(T) is the so-called *pre-exponential factor*, or simply *pre-factor*. It is often assumed to be temperature-independent over the temperature range of a reaction. {E_A} is the activation energy and R is the ideal gas constant (= 8.314 J/(K*mol)). Returning to our system of choice, assume that a brief literature review for reactions 1 and 2 has provided us with the necessary parameters:

\begin{split}

{A_1} %26= 1.32 \times {10^{19}}\;\mathrm{(L/mol)/s}\\

{A_2} %26= 1.09 \times {10^{13}}\;\mathrm{1/s}\\

{E_{A1}} %26= 140\;\mathrm{kJ/mol}\\

{E_{A2}} %26= 90\;\mathrm{kJ/mol}

\end{split}

{A_1} %26= 1.32 \times {10^{19}}\;\mathrm{(L/mol)/s}\\

{A_2} %26= 1.09 \times {10^{13}}\;\mathrm{1/s}\\

{E_{A1}} %26= 140\;\mathrm{kJ/mol}\\

{E_{A2}} %26= 90\;\mathrm{kJ/mol}

\end{split}

Inputting all this into COMSOL Multiphysics, we can now run an optimization for the reaction temperature, in order to obtain the ideal temperature at which to perform this reaction. As it turns out, simply reducing the reaction temperature to 10°C fulfills the desired criterion:

*The same reaction in series, now performed at 10°C.*

The definition of this problem in the COMSOL software is both simple and intuitive. The reactions can be input directly, replacing the → symbol for “=>”. After that, COMSOL Multiphysics offers the option of inputting Arrhenius parameters and initial concentrations (in this case only substance \mathrm{A} is present at the beginning). The choice of integration times depends very strongly on the kinetics one wants to study; chemical reactions can occur on any time scale between µs and years. After that, it is only a matter of clicking “Compute” and choosing the output format that delivers the most information for the application of interest.

Using COMSOL simulation software, you can study the intricacies of even the most simple reaction systems, providing you with valuable feedback when designing an industrial-scale process. The next blog post in this Chemical Kinetics series will take a look at what happens when we can no longer apply the simple power law approach to define the rates of chemical reactions, and what this may mean for your next steak dinner.

Stay tuned!

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