Reacting Systems

Why, for example, should a group of simple, stable compounds of carbon, hydrogen, oxygen and nitrogen struggle for billions of years to organise themselves into a professor of chemistry? What’s the motive?

—Robert M. Pirsig

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Let’s briefly revisit our guidelines for solving material balance problems.

Material balances: With species formation or consumption

Reaction stoichiometry is not given

Recall our material balances equations for multiple species.

In particular, at steady state, we can employ species mass balances with terms accounting for formation and/or consumption:

\[\begin{split}\sum_{\substack{\text{input}\\\text{streams}}} \dot m_{A_{in}} + R_{\text{formation, A}} = \sum_{\substack{\text{output}\\\text{streams}}} \dot m_{A_{out}} + R_{\text{consumption, A}}\end{split}\]

In this type of problem, we need some specification of the rate of formation or consumption. This could be through a chemical kinetic (rate) equation or some other auxiliary information.

Exercise: Material balance with consumption

Consider a device (operating as a continuous process) designed to remove 98% (by mass) of urea from blood.

Blood containing \(\SI{15}{mgmol/l}\) of urea enters the device at a flow rate of \(\SI{500}{ml/min}\).

The molecular weight of urea is approximately \(\SI{60}{g/gmol}\).

Questions and tasks:

• Draw a block diagram of the device, labeling all knowns and unknowns.

• What is the rate of removal of urea in units of \(\si{mg/min}\).

Reaction stoichiometry is given

So far we have had some experience using mass balances.

However, in problems involving a chemical reaction and where the stoichiometry of the reaction is known, it is usually more convenient to use mole balances to solve the problem.

To construct these balances, the following definitions are useful:

\(r_{\text{formation, A}} =\) rate that species \(A\) is formed, in units of \(\si{moles/time}\)

\(r_{\text{consumption, A}} =\) rate that species \(A\) is consumed, in units of \(\si{moles/time}\)

It follows that

\[\begin{split}\begin{align*} R_{\text{formation, A}} &= MW_{A} \cdot r_{\text{formation, A}} \\ R_{\text{consumption, A}} &= MW_{A} \cdot r_{\text{consumption, A}} \end{align*}\end{split}\]

Species mole balances

Keep in mind that these are actually mole-rate balances.

\[\begin{split}\sum_{\substack{\text{input}\\\text{streams}}} \dot n_{in} + r_{\text{formation, A}}= \sum_{\substack{\text{output}\\\text{streams}}} \dot n_{out} + r_{\text{consumption, A}}\end{split}\]

or

\[\begin{split}\sum_{\substack{\text{input}\\\text{streams}}} (y_{A} \, \dot n)_{in} + r_{\text{formation, A}}= \sum_{\substack{\text{output}\\\text{streams}}} \dot n_{out} + r_{\text{consumption, A}}\end{split}\]

or

\[\begin{split}\sum_{\substack{\text{input}\\\text{streams}}} (c_{A} \, \dot V)_{in} + r_{\text{formation, A}}= \sum_{\substack{\text{output}\\\text{streams}}} (y_{A} \, \dot n)_{out} + r_{\text{consumption, A}}\end{split}\]

or …

How do we relate the relative rates of formation/consumption for the species?

From the stoichiometry.

Assuming that the reaction stoichiometry is

\[\ce{\nu_{A} \, A + \nu_{B} \, B -> \nu_{C} \, C + \nu_{D} \, D}\]

it follows that

\[\frac{r_{\text{consumption, B}}}{r_{\text{consumption, A}}} = \frac{\nu_{B}}{\nu_{A}} \; ; \; \frac{r_{\text{formation, C}}}{r_{\text{consumption, A}}} = \frac{\nu_{C}}{\nu_{A}} \; ; \; \frac{r_{\text{formation, D}}}{r_{\text{consumption, A}}} = \frac{\nu_{D}}{\nu_{A}}\]

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Let’s start with a relatively simple example.

Exercise: Reacting system where the stoichiometry is known

Suppose we have a process shown below to produce chemical \(C\) according to the reaction stoichiometry \(\ce{2A + 3B -> C}\).

In addition to reactants or product, each of the streams contains some non-reacting (NR) chemicals.

Givens:

• The total mass flow rate of stream 1

• The volumetric flow rate and density of stream 2

• The mass fractions of \(A\) and \(B\) in stream 1 and 2

• Molecular weights of all relevant species

Questions and tasks:

• Create a diagram with the knowns and unknowns.

• What is the limiting reactant?

• Write out the various balances and other equations you would need to determine the molar flow rates of each species throughout the process.

• What equations can be derived from the given stoichiometry?

• Is the information provided sufficient? How many equations and unknowns do you have?

Solution