3  Reaction Progress

As noted in Chapter 2.1.4, the composition of a reacting system changes over time. However, the change in the amount of one reagent relative to the change in the amount of another reagent must be consistent with the stoichiometry of the reactions taking place. Since the changes in the amounts of the reagents are not independent, it is useful to define reaction progress variables that measure how many times each of the reactions have occurred. Given the initial amounts of all of the reagents, the amounts at any later time can then be calculated using these reaction progress variables.

3.1 Describing Single-Reaction Systems

Consider a closed system that initially contains a mixture of reagents. Let \(N_{i,0}\) represent the number of molecules of reagent \(i\) in that mixture. Suppose that there is only one reaction, \(j\), that can occur, and further suppose that in the reaction expression, the stoichiometric coefficients, \(\nu _{i,j}\), equal the number of \(i\) molecules that participate each time molecular reaction event \(j\) occurs. If \(N_{j,net}\) is the net number of times molecular reaction event \(j\) has occurred, then the change in the number of molecules of \(i\) is equal to \(\nu _{i,j} N_{j,net}\). (Remember, if \(i\) is a reactant, its stoichiometric coefficient will be negative, so the change in the number of molecules will be negative, and the number of \(i\) molecules will decrease, as it should. Similarly, if \(i\) is a product, the stoichiometric coefficient will be positive and the number of \(i\) molecules will increase.)

\[ N_i - N_{i,0} = \nu _{i,j} N_{j,net} \]

In real-world systems, the number of molecules is huge, and it is more convenient to work with moles. Dividing both sides of the equation by Avogadro’s number, \(N_{Av}\), and rearranging, gives an expression for the moles of \(i\) at any later time, given the number of moles of \(i\) at the start of the process.

\[ n_i = n_{i,0} + \nu _{i,j} \frac{N_{j,net}}{N_{Av}} \]

The net number of reaction events divided by Avogadro’s number is called the true extent of reaction j, \(\Xi_j\), Equation 3.1. Substitution in the preceding equation yields an expression for the moles of \(i\), Equation 3.2.

\[ \Xi_j = \frac{N_{j,net}}{N_{Av}} \tag{3.1}\]

\[ n_i = n_{i,0} + \nu _{i,j} \Xi_j \tag{3.2}\]

That was for a closed system. For an open, steady-state system, the change occurs between the point where the moles enter the process and the point where they leave. When talking about material entering and leaving, it makes more sense to use molar flow rates instead of moles. Doing so leads to Equation 3.3 for an open, steady-state system. It turns out that these equations are accurate no matter how the reaction expression is written, as long as it is balanced. That is, the stoichiometric coefficients in the reaction expression do not have to equal the number of molecules that participate in the molecular reaction event, but the reaction expression does have to be properly balanced. In that case, it is preferable to define the apparent extent of reaction, \(\xi_j\), as the net number of times the reaction, as written, occurs and to use the apparent extent of reaction in Equation 3.1 and Equation 3.2 instead of the true extent of reaction. Uses of the apparent extent of reaction are illustrated in Examples 3.4.1, 3.4.2, 3.4.3, and 3.4.4.

\[ \dot n_i = \dot n_{i,0} + \nu _i \dot \Xi_j \tag{3.3}\]

It’s also important to know that the extent of reaction is an extensive quantity. For a closed system it has units of moles, and for an open, steady-state system it has units of moles per time. In the latter case, the extent is related to the net number of times that the molecular reaction event occurs per unit time. Notice that all of the variable symbols in Equation 3.3 except the stoichiometric coefficient have dots over them, indicating they are flow quantities.

There’s a second quantity that is very useful for describing how far a reaction has progressed. It’s called the conversion. It can be either a fraction or a percentage, and it only applies to reactants. Quite simply, the fractional conversion of reactant \(i\), denoted as \(f_i\), is just the fraction of the starting amount of \(i\) that has reacted. It is an intensive quantity, and it does not have any units. The defining equations for closed and open, steady state systems are shown in Equation 3.4 and Equation 3.5. It doesn’t make sense to talk about how much product was converted, so again, conversion only applies to reactants.

\[ f_i=\frac{n_{i,0}-n_i}{n_{i,0}} \tag{3.4}\]

\[ f_i=\frac{\dot{n}_{i,in}-\dot{n}_i}{\dot{n}_{i,in}} \tag{3.5}\]

If the starting composition is stoichiometric, then the fractional conversion of every reactant will be the same. However, if one reactant is present in excess, its fractional conversion will be smaller than other reactants and will never reach 100%. For this reason, when the starting composition is not stoichiometric, it is preferred (but not required) that the conversion be specified for the limiting reactant because then the conversion will range from zero to 100%, assuming the reaction is irreversible. Of course, if the reaction is reversible, then even the limiting reactant will have a maximum conversion that is less than 100% because the system will reach equilibrium with measurable amounts of the reactants still remaining. The use of fractional conversion is illustrated in Example 3.4.1 and Example 3.4.4.

If it is desired to have a conversion that always ranges from 0 to 100%, the fraction of equilibrium conversion, Equation 3.6 for a closed system or Equation 3.7 for an open, steady-state system, can be used. The fraction of equilibrium conversion, \(g_i\) is just the actual conversion relative to the conversion at equilibrium, so it, too, is an intensive variable.

\[ g_i = \frac{f_i}{f_{i,eq}}= \frac{\frac{n_{i,0} - n_i}{n_{i,0}}}{\frac{n_{i,0} - n_{i,eq}}{n_{i,0}}} = \frac{n_{i,0} - n_i}{n_{i,0} - n_{i,eq}} \tag{3.6}\]

\[ g_i = \frac{\dot n_{i,in} - \dot n_i}{\dot n_{i,in} - \dot n_{i,eq}} \tag{3.7}\]

The fraction of equilibrium conversion may not be convenient to use. For one thing, in order to use it, you have to do additional calculations to determine what the conversion is at equilibrium. In addition, the conversion at equilibrium changes with temperature which means a change in temperature would change the fraction of equilibrium conversion even if no reaction took place.

3.2 Describing Multiple-Reaction Systems

The preceding section considered systems where only one reaction was taking place. When multiple reactions are taking place in the system, additional or different variables are needed to fully describe the reacting system. Before considering such variables it is necessary to review what is meant by “mathematically independent” equations, or in this case, mathematically independent reactions.

To understand why, consider Reactions 3.8, 3.9 and 3.10. Only two of those three reactions are mathematically independent. The third is a linear combination of the others. This can be seen by noting, for example, that the third reaction is the sum of the first two. Put differently, the third reaction is equal to the net effect of the first two reactions occurring sequentially.

\[ CO + 3 H_2 \rightarrow CH_4 + H_2O \tag{3.8}\]

\[ CO + H_2O \rightarrow CO_2 + H_2 \tag{3.9}\]

\[ 2 CO + 2 H_2 \rightarrow CH_4 + CO_2 \tag{3.10}\]

If a system initially contained a mixture of CO and H2, and then at some later time it was found to contain CO, H2, CH4, H2O and CO2, it would not be possible to know how much of the CH4 was produced by Reaction 3.8 and how much was produced by Reaction 3.10. Similarly, it would not be possible to know how much of the CO2 had been produced by Reaction 3.9 and how much was produced by Reaction 3.10. It is impossible to calculate the true extents of all three reactions if only the initial and final compositions of the system are known.

In general, when some of the reactions occurring in a system are linear combinations of other reactions, then a complete, mathematically independent subset of the reactions captures all of the stoichiometry. A complete mathematically independent subset is a group of reactions taken from all possible reactions where (1) none of the reactions in the subset are a linear combination of other reactions in the subset, and (2) each reaction that is not in the subset is a linear combination of the reactions that are in the subset. Consequently, for a closed system, the change in the number of moles of any reagent, \(i\), is equal to the sum of the apparent extents, \(\xi _{j^\prime}\) of the \(j^{\prime}\) mathematically independent reactions multiplied by the stoichiometric coefficient, \(\nu_{i,j^\prime}\), of reagent \(i\) in independent reaction \(j^{\prime}\). This leads to Equation 3.11 for a closed system or to Equation 3.12 for an open, steady-state system. In those equations \(j^{\prime}\) indexes the reactions in a complete, mathematically independent subset of the reactions taking place in the system. Note that Equation 3.11 and Equation 3.12 also apply to a system where only one reaction is taking place. In that case, the summation reduces to a single term. Example 3.4.3 illustrates the identification of a complete mathematically independent subset of the reactions taking place in a system, and Example 3.4.4 illustrates the use of the apparent extents of those reactions in the calculation of system composition.

\[ \Delta n_i = n_i - n_{i,0} = \sum_{j^\prime} \nu_{i,j^\prime} \xi_{j^\prime} \]

\[ n_i = n_{i,0} + \sum_{j^\prime} \nu_{i,j^\prime} \xi_{j^\prime} \tag{3.11}\]

\[ \dot n_i = \dot n_{i,in} + \sum_{j^\prime} \nu_{i,j^\prime} \dot \xi_{j^\prime} \tag{3.12}\]

It is important to recognize that the extent of reaction, \(\Xi_j\), has physical significance. It is the number times that molecular reaction event \(j\) has occurred, expressed in moles. In contrast, the apparent extent of reaction, \(\xi_{j^\prime}\), does not have any physical significance. It is a mathematical construct that is useful for characterizing the composition of a reacting system. Indeed, for a given multiple reaction system, there are typically more than one complete, mathematically independent subsets, and any one of those subsets can be used in Equation 3.11 or Equation 3.12.

In order to use Equation 3.11 or Equation 3.12 it is necessary to determine the number of mathematically independent reactions and then to find at least one such subset. Appendix A presents a very brief and succinct description of how to do this, and Example 3.4.3 illustrates doing so.

If the number of mathematically independent reactions is \(N\), then \(N\) reaction progress variables are needed to fully characterize the system, given its initial composition. Thus, the apparent extents of a complete, mathematically independent subset of all possible reactions are sufficient to calculate the composition of a system, given its initial composition.

The conversion can be used when describing reaction progress of multiple reaction systems, but it must be used in conjunction with additional measures of reaction progress to fully specify the system composition. When using the conversion as a reaction progress variable in a multiple reaction system, it is necessary to specify the reactant it applies to. Furthermore, that reactant must be initially present in the system. As an example, again consider Reactions 3.8, 3.9 and 3.10. H2O is a reactant in Reaction 3.9, but it might not initially be present in the system. Initially the system might only contain CO and H2. H2O would be generated by Reaction 3.8 and then could participate as a reactant in Reaction 3.9. If one tried to calculate the conversion of H2O using Equation 3.4, the denominator would equal zero, leading to an infinite conversion. Therefore when multiple reactions are taking place, conversion is only defined for reactants that are initially present.

Yield is a reaction progress variable that specifies the amount of one particular product relative to the starting amount of one particular reactant. There are a few different ways to define yield, making it important to know which definition is being used whenever a yield is specified. When multiple reactions are occurring, it is very often true that some of the products are more desirable or valuable than other products. In some such systems, the desired product, \(D\), is produced through a combination of reaction steps where it isn’t possible to identify a stoichiometric relationship between \(D\) and any one reactant, \(i\). Under these circumstances it is common to define the yield as shown in Equation 3.13 for a closed system and in Equation 3.14 for an open, steady-state process. Example 3.4.4 illustrates the use of yield as a reaction progress variable.

\[ Y_{D/i} = \frac{n_D}{n_{i,0}} \tag{3.13}\]

\[ Y_{D/i} = \frac{\dot n_D}{\dot n_{i,in}} \tag{3.14}\]

In other reaction systems, the desired product, \(D\), is produced from reactant, \(i\), in only one reaction, \(j\). When this is true, it may be preferable to account for the stoichiometry as shown in Equation 3.15 and Equation 3.16. By doing so, the resulting yield can be expressed as a percentage that ranges from 0 to 100% (assuming reaction \(j\) is irreversible).

\[ Y^\prime_{D/i} = \frac{\nu_{i,j}}{\nu_{D,j}}\frac{n_D}{n_{i,in}} \tag{3.15}\]

\[ Y^\prime_{D/i} = \frac{\nu_{i,j}}{\nu_{D,j}}\frac{\dot n_D}{\dot n_{i,in}} \tag{3.16}\]

While yield relates the amount of a product to the amount of a reactant, selectivity is a reaction progress variable that relates the amount of one product or group of products to a different product or group of products. If there is one desirable product, \(D\), and one undesirable product, \(U\), the selectivity for \(D\) relative to \(U\) can be defined as shown in Equation 3.17 and Equation 3.18. In other instances the selectivity of interest might involve groups of products. Examples include parafins vs. olefins, napthenes vs. aromatics or hydrocarbons with 6 or more carbon atoms (sometimes denotes as C6+) vs. hydrocarbons with 5 or fewer carbon atoms (sometimes denoted as C5-). Example 3.4.1 and Example 3.4.4 illustrate the use of selectivity.

\[ S_{D/U} = \frac{n_D}{n_U} \tag{3.17}\]

\[ S_{D/U} = \frac{\dot n_D}{\dot n_U} \tag{3.18}\]

Both yield and selectivity can be used as overall measures or as instantaneous measures. The overall yield or selectivity is the value at the end of a closed process or at the outlet from an open, steady-state process, as defined in Equations 3.17 and 3.18. In a closed process, the yield or selectivity will typically change over time, and in an open, steady-state process it will vary with position within the reactor. An instantaneous yield or selectivity can be defined for that time or position within the reactor. The instantaneous yield or selectivity is related to the rates at which the amounts of the reagents are changing. Reaction rates are defined in Chapter 4, so defining equations for the instantaneous yield or selectivity will be presented in that chapter.

3.3 Mole Tables

A mole table is a convenient tool for summarizing the stoichiometric relationships among the chemical species present in a system. Mole tables can be particularly helpful for those just learning reaction engineering. They are useful when calculating the composition of a system, given its initial composition, and also when converting from one composition variable (concentration, mole fraction, partial pressure, moles, etc.) to another.

A mole table has three columns, as can be seen in Table 3.1. The first column lists each of the chemical species that are present in the system, including any inert reagents that do not participate in any chemical reactions. The second column lists, for each species, the initial moles (for a closed system) or inlet molar flow rate (for an open, steady-state system). The third column lists expressions for the moles or molar flow rates at any later time or at any point beyond the inlet to an open system in terms of the apparent extents of reaction. In other words, each row in the third column contains either Equation 3.11 or Equation 3.12 for that row’s species, with the summation expanded and the values of the stoichiometric coefficients substituted. A final row in the table is for total moles or molar flow rate. It lists the sum of the initial values in the second column and the sum of the expressions for moles at any later time in the third column. The construction and use of a mole table is illustrated in Examples 3.4.1, 3.4.3, and 3.4.4.

3.4 Examples

3.4.1 Calculation of Final Composition in a Closed System using a Mole Table

A new catalyst is being developed with the hope of generating ethylene by the oxidative dehydrogenation of ethane, reaction (1). Reaction (2) also takes place at the operating conditions of 1 atm and 150 °C. The process starts with 90% ethane and 10% air (you may take air to contain 79% N2 and 21% O2) and ends with a 50% conversion of O2 and a selectivity of 3 C2H4/CO2. Construct a mole table for this system and use it to calculate the final mole fraction of CO2.

\[ 2 C_2H_6 + O_2 \leftrightarrows 2 C_2H_4 + 2 H_2O \tag{1} \]

\[ 2 C_2H_6 + 7 O_2 \leftrightarrows 4 CO_2 + 6 H_2O \tag{2} \]

What this problem asks me to do is often one step in the completion of a reaction engineering assigment. It entails calculating the composition of a system given information about its initial composition together with some measure of how much reaction has taken place. I often refer to this as a reaction progress task.

Completion of a reaction progress task entails four general sub-tasks: 1) calculate the initial molar amount of every reagent, 2) calculate the apparent extents of reaction, 3) calculate the final molar amounts of every reagent, and 4) calculate the quantities requested in the assignment.

To begin, I will summarize the information provided in the assignment narrative. Upon reading through the assignment narrative I can see that all of the quantities it provides are intensive. That means I can assume one extensive quantity as a basis for the calculations and add it to my summary. As a basis, I will set the initial total molar amount to 1 mol.

3.4.1.1 Assignment Summary

Reactions:

\[ 2 C_2H_6 + O_2 \leftrightarrows 2 C_2H_4 + 2 H_2O \tag{1} \]

\[ 2 C_2H_6 + 7 O_2 \leftrightarrows 4 CO_2 + 6 H_2O \tag{2} \]

Quantity of Interest: \(y_{CO_2,f}\)

Given and Known Constants: \(P\) = 1 atm, \(T\) = 150 °C, \(y_{C_2H_6,0}\) = 0.9, \(y_{\text{air},0}\) = 0.1, \(f_{O_2}\) = 0.5, and \(S_{C_2H_4/CO_2}\) = 3.

Basis: \(n_{\text{total},0} = 1 \text{ mol}\).

A mole table can be helpful when completing reaction progress tasks, and this assignment asks me to generate one, so next I will generate one.

The first column in a mole table lists the reagents in the system, which in this example are C2H6, O2, C2H4, H2O, CO2, and N2. The second column lists the corresponding initial molar amounts of the reagents, and the third column lists expressions for the molar amounts at any later time, in terms of the apparent extents of reaction. For example, the stoichiometric coefficient of C2H6 in reaction (1) is -2, and in reaction (2) it is also -2, leading to the following expression for the molar amount: \(n_{C_2H_6,0} - 2 \xi_1 - 2 \xi_2\). Analogous expressions can be written for the other reagents. A final row can be added to the mole table containing the initial and later total molar amounts.

3.4.1.2 Mathematical Formulation of the Analysis

A mole table for the system is presented in Table 3.1.

Table 3.1: Mole Table for Example 3.4.1.
Species Initial Amount Later Amount
C2H6 \(n_{C_2H_6,0}\) \(n_{C_2H_6,0} - 2 \xi_1 - 2 \xi_2\)
O2 \(n_{O_2,0}\) \(n_{O_2,0} - \xi_1 - 7 \xi_2\)
C2H4 \(n_{C_2H_4,0}\) \(n_{C_2H_4,0} + 2 \xi_1\)
H2O \(n_{H_2O,0}\) \(n_{H_2O,0} + 2 \xi_1 + 6 \xi_2\)
CO2 \(n_{CO_2,0}\) \(n_{CO_2,0} + 4 \xi_2\)
N2 \(n_{N_2,0}\) \(n_{N_2,0}\)
Total \(n_{\text{total},0}\) \(n_{\text{total},0} + \xi_1 + \xi_2\)

I need to calculate the apparent extents of reaction, and I’m given two bits of information about the final composition, namely the conversion of O2 and the selectivity for C2H4 over CO2. If I write the defining equations for those two quantities and substitute the expressions from the mole table, I’ll have two equations that can be used to calculate the apparent extents of reaction.

The conversion of O2 and the selectivity for C2H4 over CO2 can be related to the apparent extents of reaction as shown in equations (3) and (4).

\[ f_{O_2} = \frac{n_{O_2,0} - n_{O_2}}{n_{O_2,0}} = \frac{n_{O_2,0} - \left(n_{O_2,0} - \xi_1 - 7 \xi_2\right)}{n_{O_2,0}} = \frac{\xi_1 + 7 \xi_2}{n_{O_2,0}} \tag{3} \]

\[ S_{C_2H_4/CO_2} = \frac{n_{C_2H_4}}{n_{CO_2}} = \frac{\cancelto{0}{n_{C_2H_4,0}} + 2 \xi_1}{\cancelto{0}{n_{CO_2,0}} + 4 \xi_2} \tag{4} \]

The initial amount of O2 appearing in equation (3) can be calculated using equation (5), while the initial amounts of C2H4 and CO2 appearing in equation (4) are both equal to zero.

\[ \begin{align} n_{O_2,0} &= 0.21 n_{\text{air},0} \\&= 0.21 y_{\text{air},0} n_{\text{total},0} \\&= 0.21 \left(0.1\right)\left(1.0 \text{ mol}\right) \\&= 0.021 \text{ mol} \end{align} \tag{5} \]

After substitution of \(f_{O_2}\), \(S_{C_2H_4/CO_2}\), and \(n_{O_2,0}\), equations (3) and (4) can be solved for \(\xi_1\) and \(\xi_2\).

The defining equation for the final mole fraction of CO2 and the expressions in the mole table can be used to relate \(y_{CO_2,f}\) to \(\xi_1\) and \(\xi_2\). The resulting equation can then be used to calculate the final mole fraction of CO2.

The final mole fraction of CO2 is related to the apparent extents of reaction as shown in equation (6). Noting that the initial molar amount of CO2 is zero, the final mole fraction of CO2 can be calculated.

\[ y_{CO_2,f} = \frac{n_{CO_2,f}}{n_{\text{total},f}} = \frac{\cancelto{0}{n_{CO_2,0}} + 4 \xi_2}{n_{\text{total},0} + \xi_1 + \xi_2} \tag{6} \]

3.4.1.3 Results, Analysis and Discussion

The calculations were performed as described above. The apparent extents of reactions (1) and (2) were found to equal 0.00485 mol and 8.08 x 10-4 mol, respectively. These are extensive quantities that will change depending upon the basis selected for the calculations.

The final mole fraction of CO2 was found to be 0.00321. Being an intensive quantity, this value is unaffected by the basis chosen for the calculations.

Note

This analysis could have been performed a number of different ways. For example, the total volume of the system could have been chosen as the basis. The initial total moles could then be calculated using the ideal gas law, after which the analysis presented here could have been used.

One important aspect of the results is that any extensive quantity that is calculated is only valid for the chosen basis, whereas

3.4.2 Calculation of Outlet Composition from an Open System using the Conversion and Extent of Reaction

Suppose reaction (1) was studied in an open, steady-state, isothermal, isobaric system. The flow into the system contained 0.5 mol min−1 of N2O5 and 0.5 mol min−1 of N2 at 600 K and 5 MPa. Generate an expression for the concentration of NO2 in terms of the molar flow rate of N2O5, the inlet molar flows, the temperature, and the pressure. If the outlet flow contains 30% N2, calculate the outlet concentration of NO2.

\[ 2 N_2O_5 \leftrightarrows 4 NO_2 + O_2 \tag{1} \]

This is a reaction progress task. In the first part, I’m asked to express the amount of one reagent in terms of the amount of another reagent. For this type of task, the general approach is to express each of the two amounts in terms of the apparent extent of reaction, and then eliminate the apparent extent of reaction between the two equations.

I will start by summarizing the information profided in the problem statement. The problem only mentions N2O5 and N2 as flowing into the system, so I’ll assume the inlet flows of NO2 and O2 are zero.

3.4.2.1 Assignment Summary

Reaction:

\[ 2 N_2O_5 \leftrightarrows 4 NO_2 + O_2 \tag{1} \]

Quantities of Interest: Expression for \(C_{NO_2}\), and its value.

Given and Known Constants: \(T\) = 600 K, \(P\) = 5 MPa, \(y_{N_2,out}\) = 0.3, \(\dot n_{N_2O_5,in}\) = 0.5 mol min-1, \(\dot n_{N_2,in}\) = 0.5 mol min-1, \(\dot n_{NO_2,in}\) = 0, and \(\dot n_{O_2,in}\) = 0.

I could write a mole table, but with only one reaction taking place, I think I can get by without one. The inlet molar flow rates are all given, so I’ll start by using Equation 3.12 to write expressions for the outlet molar flow rates, and add them together to get an expression for the total molar flow rate.

Then I can use those expressions to generate expressions for the concentration of NO2 and the outlet mole fraction of N2.

3.4.2.2 Mathematical Formulation of the Analysis

The molar flow rates of the reagents are expressed in terms of the apparent extent of reaction in equations (2) through (6).

\[ \dot n_{N_2O_5} = \dot n_{N_2O_5,in} - 2 \dot \xi \tag{2} \]

\[ \dot n_{NO_2} = \cancelto{0}{\dot n_{NO_2,in}} + 4 \dot \xi \tag{3} \]

\[ \dot n_{O_2} = \cancelto{0}{\dot n_{O_2,in}} + \dot \xi \tag{4} \]

\[ \dot n_{N_2} = \dot n_{N_2,in} \tag{5} \]

\[ \dot n_{\text{total}} = \dot n_{N_2O_5,in} + \dot n_{N_2,in} + 3 \dot \xi \tag{6} \]

The ideal gas law, together with those expressions can be used to write an expression for the concentration of NO2, equation (7), and the definition of a mole fraction can be used to write an expression for the outlet mole fraction of N2, equation (8).

\[ C_{NO_2} = \frac{\dot{n}_{NO_2}}{\dot{V}} = \frac{P\dot n_{NO_2}}{\dot n_{total}RT} = \frac{4P\dot \xi}{RT\left( \dot n_{\text{total},in} + 3 \dot \xi \right)} \tag{7} \]

\[ y_{N_2,\text{out}} = \frac{\dot n_{N_2}}{\dot n_{total}} = \frac{\dot n_{N_2,in}}{\dot n_{total,in} + 3 \dot \xi} \tag{8} \]

The first part of the assignment asks for an expression relating the concentration of NO2 to the molar flow rate of N2O5. Solving equation (2) for the apparent extent of reaction and substituting the result into equation (7) will yield the desired expression.

The second part of the assignment asks for the NO2 concentration if the outlet mole fraction of N2 is 0.3. Knowing the mole fraction of N2 and the inlet molar flows, equation (8) can be solved to find the apparent extent of reaction. The resulting value can then be substituted in equation (7) to calculate the concentration of NO2.

Solving equation (2) for the apparent extent of reaction yields equation (9). Then substitution of equation (9) into equation (7) yields the requested expression for the concentration of NO2 in terms of the molar flow rate of N2O5, equation (10).

\[ \dot \xi = \frac{\dot n_{N_2O_5,in} - \dot n_{N_2O_5}}{2} \tag{9} \]

\[ \begin{align} C_{NO_2} &= \frac{4P\dot \xi}{RT\left( \dot n_{\text{total},in} + 3 \dot \xi \right)} \\ &= \frac{4P}{RT} \frac{\frac{\dot n_{N_2O_5,in} - \dot n_{N_2O_5}}{2}}{\dot n_{N_2O_5,in} + \dot n_{N_2,in} + 3\frac{\dot n_{N_2O_5,in} - \dot n_{N_2O_5}}{2}} \\ &= \frac{4P}{RT} \frac{\left(\dot n_{N_2O_5,in} - \dot n_{N_2O_5}\right)}{\left(5\dot n_{N_2O_5,in} + 2\dot n_{N_2,in} - 3 \dot n_{N_2O_5} \right)} \end{align} \tag{10} \]

Equation (8) can be solved to find the apparent extent of reaction when the outlet mole fraction of N2 is 0.3. Substitution of the result in equation (7) then yields the corresponding concentration of NO2.

3.4.2.3 Results, Analysis and Discussion

The requested expression for the concentration of NO2 is given in equation (10). Performing the calculations as described above yields a value of 0.535 mol/L for the concentration of NO2 at the time when the outlet contains 30% N2.

This example illustrated two common ways in which the apparent extent of reaction is used. The first is expressing the amount of one reagent in terms of the amount of a different reagent. Doing this is often necessary when fitting a rate expression to kinetics data analytically (see Example 19.5.5). It is not needed if the fitting is performed numerically, as is routinely done in Reaction Engineering Basics. The other common use is calculting the final amount of a reagent given the final amount of a different reagent. As an example, if the inlet to a reactor was set and the outlet amount of one reagent was measured, the apparent extent of reaction can be used to calculate the outlet amounts of other reagents.

3.4.3 Identification of a Complete, Mathematically Independent Subset of a Group of Reactions

A chemical system initially contains CO and H2. At system conditions, reactions (1) through (6) occur. Identify a complete, mathematically independent subset of those reactions and construct a mole table for the system.

\[ CO + 2 H_2 \leftrightarrows CH_3OH \tag{1} \]

\[ CO + 3 H_2 \leftrightarrows CH_4 + H_2O \tag{2} \]

\[ CH_4 + H_2O \leftrightarrows CH_3OH + H_2 \tag{3} \]

\[ CO + H_2O \leftrightarrows CO_2 + H_2 \tag{4} \]

\[ CO_2 + 4 H_2 \leftrightarrows CH_4 + 2 H_2O \tag{5} \]

\[ CO_2 + 3 H_2 \leftrightarrows CH_3OH + H_2O \tag{6} \]

I know from math courses I’ve taken that identifying mathematically independent equations involves constructing a matrix, and the same is true for identifying independent chemical reactions. (As noted in Appendix A, identifying independent reaction is prerequisite knowledge for Reaction Engineering Basics.) Each column in the matrix corresponds to one of the reagants and each row corresponds to one of the reactions. The matrix element at any one row and column is the stoichiometric coefficent of the reagent corresponding to that column in the reaction corresponding to that row.

Here the reaction matrix will have six rows, one for each reaction, and six columns, one for each reagent. I’ll let the first column corresponds to CO, the second to H2, the third to CH3OH, the fourth to CH4, the fifth to H2O, and the last to CO2. Letting the first row correspond to reaction (1), I see that the stoichiometric coefficient of CO in that reaction is -1, and so there is a -1 in the first column of row 1 (see equation (7) below). The stoichiometric coefficeint of H2 in the first reaction is -2, so there is a -2 in the second column of row 1. The stoichiometric coefficient of CH3OH in reaction (1) is +1, so there is a 1 in the third column of row 1. CH4, H2O, and CO2 do not appear in reaction (1), so there are zeros in the fourth, fifth and sixth columns of row 1. The remaining rows are filled in analogously.

The rank of this matrix will then equal the number of mathematically independent reactions. I’ll use computer software to find the rank.

3.4.3.1 Mathematical Formulation of the Analysis

Equation (7) shows a reaction matrix for this problem. The first row corresponds to reaction (1), the second row to reaction (2), and so on. The first column corresponds to CO, the second to H2, the third to CH3OH, the fourth to CH4, the fifth to H2O, and the last to CO2.

\[ \begin{bmatrix} -1 & -2 & 1 & 0 & 0 & 0 \\ -1 & -3 & 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & -1 & -1 & 0 \\ -1 & 1 & 0 & 0 & -1 & 1 \\ 0 & -4 & 0 & 1 & 2 & -1 \\ 0 & -3 & 1 & 0 & 1 & -1 \end{bmatrix} \tag{7} \]

Mathematics software can be used to find that the rank of the matrix in equation (7) is equal to 3. This means that there will be 3 equations in any complete, mathematically independent subset of reactions (1) through (6).

In the matrix shown in equation (8) the first row corresponds to reaction (1) and the second row to reaction (2). The rank of this matrix is found to equal 2, so reactions (1) and (2) are mathematically independent.

\[ \begin{bmatrix} -1 & -2 & 1 & 0 & 0 & 0 \\ -1 & -3 & 0 & 1 & 1 & 0 \end{bmatrix} \tag{8} \]

In the matrix shown in equation (9) a third row, corresponding to reaction (3), was added to the matrix in equation (8). The rank of the matrix in equation (9) is 2. This means that reaction (3) is not independent; it is a linear combinations of reactions (1) and (2).

\[ \begin{bmatrix} -1 & -2 & 1 & 0 & 0 & 0 \\ -1 & -3 & 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & -1 & -1 & 0 \end{bmatrix} \tag{9} \]

Reaction (3) was removed from the matrix and replaced with reaction (4), resulting in the matrix shown in equation (10). The rank of the matrix shown in equation (10) is 3, so the reactions corresponding to the rows of that matrix, in this case reactions (1), (2), and (4), represent a complete, mathematically independent subset of reactions (1) through (6).

\[ \begin{bmatrix} -1 & -2 & 1 & 0 & 0 & 0 \\ -1 & -3 & 0 & 1 & 1 & 0 \\ -1 & 1 & 0 & 0 & -1 & 1 \end{bmatrix} \tag{10} \]

Having identified a complete, mathematically independent subset of the reactions, I can now construct a mole table as requested in the problem statement. There are six reagents in this system: CO, H2, CH3OH, CH4, H2O, and CO2. As in Example 3.4.1, the first column of the mole table lists these reagents along with a row at the bottom for “Total.” The second column lists variables representing the initial molar amounts of each reagent and the third column lists expressions for the molar amounts of the reagents at any later time in terms of the apparent extents of the independent reactions. I need to use Equation 3.11 for the entries in the third column of of the mole table

\[ n_i = n_{i,0} + \sum_{j^\prime} \nu_{i,j^\prime} \xi_{j^\prime} \]

3.4.3.2 Results, Analysis and Discussion

While there are six reactions taking place in this system, only three reactions are mathematically independent. Therefore the mole table only uses the apparent extents of three reactions. Knowing that reactions (1), (2), and (4), represent a complete, mathematically independent subset of reactions (1) through (6), the apparent extents of those reactions were used to construct the mole table, Table 3.2.

Table 3.2: Mole Table for Example 3.4.3
Species Initial Amount Later Amount
CO \(n_{CO,0}\) \(n_{CO,0} - \xi_1 - \xi_2 - \xi_4\)
H2 \(n_{H_2,0}\) \(n_{H_2,0} - 2 \xi_1 - 3 \xi_2 + \xi_4\)
CH3OH \(n_{CH_3OH,0}\) \(n_{CH_3OH,0} + \xi_1\)
CH4 \(n_{CH_4,0}\) \(n_{CH_4,0} + \xi_2\)
H2O \(n_{H_2O,0}\) \(n_{H_2O,0} + \xi_2 - \xi_4\)
CO2 \(n_{CO_2,0}\) \(n_{CO_2,0} + \xi_4\)
Total \(n_{\text{total},0}\) \(n_{\text{total},0} - 2 \xi_1 - 2 \xi_2\)
Notes
  1. Reactions (1), (2), and (4) are not the only complete, mathematically independent subset. For example, reactions (1), (4), and (5) is also a complete, mathematically independent subset and the mole table could have been written using \(\xi_1\), \(\xi_4\), and \(\xi_5\).
  2. The solution presented here used a reaction matrix. Appendix A describes identifying independent reactions using the stoichiometric coefficient matrix where the columns correspond to the reactions and the rows to the reagents. The reaction matrix is the transpose of the stoichiometric coefficient matrix, and the rank of a matrix equals the rank of its transpose. For that reason, either the method used here or the method described in Appendix A can be used.

3.4.4 Calculation of the Final Composition of a Closed System using the Conversion, Yield, Selectivity and Apparent Extents of Reaction

A chemical system initially contains 34 mol CO and 66 mol H2. At system conditions, reactions (1) through (6) occur. At the end of the process, the CO conversion is 40%, the yield of CH3OH is 10.7% (mol CH3OH per initial mol CO), and the selectivity for methanol vs. methane is 0.38 mol CH3OH per mol CH4. Calculate the number of moles of each species at the end of the process.

\[ CO + 2 H_2 \leftrightarrows CH_3OH \tag{1} \]

\[ CO + 3 H_2 \leftrightarrows CH_4 + H_2O \tag{2} \]

\[ CH_4 + H_2O \leftrightarrows CH_3OH + H_2 \tag{3} \]

\[ CO + H_2O \leftrightarrows CO_2 + H_2 \tag{4} \]

\[ CO_2 + 4 H_2 \leftrightarrows CH_4 + 2 H_2O \tag{5} \]

\[ CO_2 + 3 H_2 \leftrightarrows CH_3OH + H_2O \tag{6} \]

This is a reaction progress task that provides three reaction progress variables and asks for the corresponding final composition. I can do that by writing expressions for the final molar amounts of the reagents in terms of the apparent extents of reaction, substituting those expressions in defining equations for the three known reaction progress variables, solving the resulting equations for the extents of reaction, and using the extents of reaction to calculate the final molar amounts.

I’ll begin by summarizing the information provided in the assignment narrative. The system described here is the same as in Example 3.4.3, so I can use the mole table from that example, Table 3.2.

3.4.4.1 Assignment Summary

Reactions:

\[ CO + 2 H_2 \leftrightarrows CH_3OH \tag{1} \]

\[ CO + 3 H_2 \leftrightarrows CH_4 + H_2O \tag{2} \]

\[ CH_4 + H_2O \leftrightarrows CH_3OH + H_2 \tag{3} \]

\[ CO + H_2O \leftrightarrows CO_2 + H_2 \tag{4} \]

\[ CO_2 + 4 H_2 \leftrightarrows CH_4 + 2 H_2O \tag{5} \]

\[ CO_2 + 3 H_2 \leftrightarrows CH_3OH + H_2O \tag{6} \]

Quantities of Interest: \(n_{CO}\), \(n_{H_2}\), \(n_{CH_3OH}\), \(n_{CH_4}\), \(n_{H_2O}\), and \(n_{CO_2}\).

Given and Known Constants: \(n_{CO,0}\) = 34 mol, \(n_{H_2,0}\) = 66 mol, \(n_{CH_3OH,0}\) = \(n_{CH_4,0}\) = \(n_{H_2O,0}\) = \(n_{CO_2,0}\) = 0 mol, \(f_{CO}\) = 0.4, \(Y_{CH_3OH}\) = 0.107, and \(S_{CH_3OH/CH_4}\) = 0.38.

3.4.4.2 Mathematical Formulation of the Analysis

This reaction system was analyzed in Example 3.4.3 where it was determined that reactions (1), (2), and (4) constitute a mathematically independent subset of the reactions and Table 3.2 was generated.

Mole table (from Section 3.4.3)
Species Initial Amount Later Amount
CO \(n_{CO,0}\) \(n_{CO,0} - \xi_1 - \xi_2 - \xi_4\)
H2 \(n_{H_2,0}\) \(n_{H_2,0} - 2 \xi_1 - 3 \xi_2 + \xi_4\)
CH3OH \(n_{CH_3OH,0}\) \(n_{CH_3OH,0} + \xi_1\)
CH4 \(n_{CH_4,0}\) \(n_{CH_4,0} + \xi_2\)
H2O \(n_{H_2O,0}\) \(n_{H_2O,0} + \xi_2 - \xi_4\)
CO2 \(n_{CO_2,0}\) \(n_{CO_2,0} + \xi_4\)
Total \(n_{\text{total},0}\) \(n_{\text{total},0} - 2 \xi_1 - 2 \xi_2\)

Now I can write the defining equations for the conversion, yield and selectivity, Equations 3.4, 3.13, and 3.17, and substitute their known values and the expressions for the molar amounts from the mole table.

The CO conversion, methanol yield, and methanol to methane selectivity can be expressed in terms of the apparent extents of reaction as shown in equations (7), (8), and (9).

\[ \begin{align} f_{CO} &= \frac{n_{CO,0} - n_{CO}}{n_{CO,0}} \\ &= \frac{n_{CO,0} - \left(n_{CO,0} - \xi_1 - \xi_2 - \xi_4\right)}{n_{CO,0}} \\ &= \frac{\xi_1 + \xi_2 + \xi_4}{n_{CO,0}} \end{align} \tag{7} \]

\[ Y_{CH_3OH} = \frac{n_{CH_3OH}}{n_{CO,0}} = \frac{n_{CH_3OH,0} + \xi_1}{n_{CO,0}} \tag{8} \]

\[ S_{CH_3OH/CH_4} = \frac{n_{CH_3OH}}{n_{CH_4}} = \frac{n_{CH_3OH,0} + \xi_1}{n_{CH_4,0} + \xi_2} \tag{9} \]

Equations (7), (8), and (9) can be solved to find the apparent extents of reactions (1), (2), and (4) which can then be substituted into the expressions for the molar amounts in Table 3.2.

3.4.4.3 Results, Analysis and Discussion

The calculations were performed as described above to calculate the final molar amounts of the reagents. To check those results, they were used to calculate the CO conversion, methanol yield, and methanol to methane selectivity. The results are shown in Table 3.3 which shows that the calculated molar amounts are consistent with the conversion, yield, and selectivity given in the assignment narrative.

Table 3.3: Final Molar Amount of Each Reagent
Reagent Final Moles
CO 20.400
H2 30.400
CH3OH 3.640
CH4 9.570
H2O 9.190
CO2 0.388
CO Conversion 0.400
Methanol Yield 0.107
Methanol/Methane Selectivity 0.380

3.5 Symbols Used in Chapter 3

Symbol Meaning
\(eq\) Subscript denoting an equilibrium value.
\(f_i\) Fractional conversion of reactant \(i\).
\(g_i\) Fraction of equilibrium conversion of reactant \(i\).
\(i\) Subscript denoting a fluid phase reagent.
\(j\) Subscript denoting a reaction occurring in the system.
\(j^\prime\) Subscript denoting a reaction from a complete, mathematically independent subset of the reactions occurring in the system.
\(n\) Number of moles; a subscripted \(i\) denotes the number of moles of reagent \(i\); an additional subscripted \(0\) denotes the initial moles.
\(\dot n\) Molar flow rate; a subscripted \(i\) denotes the molar flow rate of reagent \(i\); an additional subscripted \(in\) denotes the inlet molar flow rate.
\(y_i\) Gas phase mole fraction of reagent \(i\); an additional subscripted \(0\) denotes the initial mole fraction; an additional subscripted \(in\) denotes the inlet mole fraction.
\(C_i\) Molar concentration of reagent \(i\).
\(N_{Av}\) Avogadro’s number.
\(N_i\) Number of molecules of reagent \(i\); an additional subscripted \(0\) denotes the initial number of molecules of \(i\).
\(N_{j,net}\) Net number of times reaction event \(j\) has occurred.
\(P\) Pressure.
\(R\) Ideal gas constant.
\(S_{D/U}\) Overall selectivity for product or products, \(D\), relative to product or products, \(U\).
\(T\) Temperature.
\(\dot V\) Volumetric flow rate.
\(Y_{D/i}\) Overall yield of product \(D\) from reactant \(i\).
\(Y^\prime_{D/i}\) Overall stoichiometric yield of product \(D\) from reactant \(i\).
\(\nu_{i,j}\) Stoichiometric coefficient of reagent \(i\) in reaction \(j\); if only one reaction is taking place the index, \(j\) is optional.
\(\xi_{j\text{-}ind}\) Apparent extent of reaction \(j\text{-}ind\).
\(\dot \xi_{j\text{-}ind}\) Apparent extent of reaction \(j\text{-}ind\) in an open, steady-state system.
\(\Xi_j\) True extent of reaction \(j\).
\(\dot \Xi_j\) True extent of reaction \(j\) in an open, steady-state system.