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:

- the complete composition of either the inlet or the exit stream from a reactor, and
- one constraint (conversion, selectivity, second composition, equilibrium constant, etc.) for each reaction

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

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

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 n_{P}, n_{N}, and n_{O}
**OR** n_{P0}, n_{N0}, and n_{O0}),
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:

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:

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:

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:

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:

Summmarizing the steps above:

- Write the component balance equations using the extent
- Write one constraint for each reaction (in terms of the extent)
- Solve the system of equations for the extent
- Plug the extent into the balances and constraints as needed.

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:

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:

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 (n_{i0}) 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.

**References:**

- Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2nd Edition, John Wiley, 1986, pp. 121-22, 127-29.
- 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

Copyright 1997, 2005 by R.M. Price -- All Rights Reserved