## Extent of Reaction

The extent of reaction approach to solving multiple reaction systems is a general formulation that works for many reaction systems. It can be used whenever you know:

1. the complete composition of either the inlet or the exit stream from a reactor, and
2. one constraint (conversion, selectivity, second composition, equilibrium constant, etc.) for each reaction
The good thing about this approach is that you apply it exactly the same way to every problem where it is appropriate. In order to do this, we need to develop a generic framework and notation. This generality can be confusing for some, so we'll start out by solving a reactor problem and then use that to build up the extent approach.

Consider a reactor where nitrogen pentoxide decomposes to or is made from nitrogen dioxide:

Let's indicate the moles of each component with an n and a subscript. We'll tack a 0 on to the feed stream to keep it separate from the products.

We can write a component balance on each of the three species:

The total material balance can be obtained by summing the moles in each stream and summing the component balances:

We thus have a system of 3 independent equations. There are 9 variables in this problem, so f = V-E = 6, and we have to specify the value of 6 of the variables in order to solve the problem. It seems plausible that we might know one stream entirely (either nP, nN, and nO OR nP0, nN0, and nO0), but even then we still must specify 3 terms.

From experience, we know that the amounts of material that are produced and consumed in the reaction are related, so we can use the reaction stoichiometry to develop the needed specifications or equations. That is what we'll tackle next.

First, consider that 2 moles of pentoxide react. Then, from the stoichiometric ratios, we know that:

All of these expressions are calculated by multiplying the basis (2 moles pentoxide) by the stoichiometric coefficient of a component by a common factor of 1/(2 moles pentoxide). Let's rename the common factor X:

Put that result aside, and consider another case. This time let's say that we MAKE 2 moles of nitrogen dioxide, and applying the stoichiometry gives:

Again, there is a common factor (this time 1/4). We'll name it Y:

The factors are exactly the same as in the first case. It turns out that the same factors will occur for any reaction -- they are a consequence of the stoichiometry. What do X and Y represent? The define "how much reacts". Depending on the amounts present, the numbers we plug in may change, but the meaning doesn't.

The single variable that quantifies how much "stuff" reacts, is called the the "extent of reaction" and is commonly given the symbol . For the reaction we're studying, we can always say that

What does this do? It lets us write the production and consumption terms in our balances in terms of only one unknown, instead of three:

This system has only 7 unknowns. Accounting for the 3 equations, and the 3 specified stream compositions, we get f = 7-3-3 = 1, meaning we only need specify one additional parameter. This will likely be something that tells us about the reaction, either a second composition for one of the components or the conversion (which can also be written in terms of the extent):
and the problem is fully specified and can be solved for the 3 unknown compositions and the extent.

NOW ... let's complicate things. What if there are TWO reactions? Let's add a side reaction where the nitrogen dioxide reacts to make tetroxide:

We could go through the whole ratio bit again, but suffice it to say that because there are two reactions we will need two extent terms. The balances will become:

This system has 10 variables and 4 equations. Again, we'll assume that we have one stream completely known, so that f = 10-4-4 = 2. We'll have to specify two parameters. As before, the conversion is good, and for the second reaction, we'll add a yield or selectivity:

and the problem can be readily solved for the 4 unknown compositions and the extent.

Summmarizing the steps above:

1. Write the component balance equations using the extent
2. Write one constraint for each reaction (in terms of the extent)
3. Solve the system of equations for the extent
4. Plug the extent into the balances and constraints as needed.
Often the equation that must be solved to get the extent is quadratic or higher order. If you obtain multiple solutions, you must decide which is correct. To do this recall that the extent is always positive, and needs to "work" with all the component balances -- values which lead to negative amounts can be tossed.

### General Formulation of Extents of Reaction

The method for using extents can be generalized using variables, summations, etc.

Any set of reactions can be written in terms of its stoichiometric coefficients as:

To distinguish the stoichiometric relationships between reactants and products, we define a sign convention:

Our component balances can be written using this notation as well. The production consumption terms will be in terms of the extent, and the coefficient on is always the stoichiometric coefficient and the sign convention. The mole balance for any component now has the general form:

to give moles in the process outlet stream (or at the ending time for a batch process) in terms of the stoichiometric coefficients, the extent of reaction, and the inlet moles.

If there is more than one reaction going on, there will be a single extent term for each, so for i components and j reactions:

If you know the feed quantities (ni0) and the extent, or you know the feed and one constraint on the reaction (which will let you get the extent by balance) this formulation lets you calculate all the other quantities as well as conversion, etc., in a nicely organized fashion.

In problems dealing with conversion, selectivity, equilibrium, etc., it is often useful to set up the entire problem in terms of the extent of reaction. If you do this, you reduce all your production and consumption terms to a single unknown variable -- the extent. You can then solve for the extent and use it to get the desired answers.

Example: Multiple reaction system

This problem can be solved if we know either the inlet or outlet stream and one constraint for each reaction.

References:

1. Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2nd Edition, John Wiley, 1986, pp. 121-22, 127-29.
2. Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2005 3rd Edition, 2005, p. 119, 130-35.

R.M. Price
Original: 6/15/94
Modified: 9/26/95, 10/7/97; 1/7/2005