3.3 Reactive systems

Reactive vs Non-reactive Systems

In non-reactive systems, the species going in are the same species coming out. You're just mixing, separating, or moving things around. Reactive systems are fundamentally different: chemical reactions create new species and destroy others, which means your material balance has to track those changes.

Differences in Material Balances

• In non-reactive systems, each chemical species is conserved individually. If 10 mol/s of ethanol enters, 10 mol/s of ethanol must leave (at steady state).
• In reactive systems, molar flow rates change because reactants get consumed and products get formed. You can't write a simple "in = out" balance on each species anymore.
• Reactive balances require stoichiometric coefficients and the extent of reaction to relate how much of each species appears or disappears. Non-reactive systems don't need these.
• The general balance equation for reactive systems includes generation and consumption terms that are zero in non-reactive balances.

One thing worth noting: even in reactive systems, total mass is still conserved. Individual moles of each species change, but if you convert everything to mass, what goes in must come out (at steady state). That's why an overall mass balance can serve as a useful check on your work.

Impact on Process Modeling

Reactive systems are harder to model because you need to account for reaction stoichiometry on top of the usual mass flows. A few specific complications:

• Reactions can be exothermic (release heat) or endothermic (absorb heat), which couples the material balance to the energy balance.
• Multiple phases may be involved (gas-phase reactants producing liquid products, for example), requiring phase equilibrium calculations.
• Reaction kinetics and equilibrium constraints add equations and unknowns to your system.

Non-reactive systems, by contrast, can usually be solved with straightforward mass balances and don't require you to think about reaction rates or heat of reaction.

Balancing Chemical Equations

Writing Chemical Equations

A chemical equation shows what goes in (reactants, left side of the arrow) and what comes out (products, right side). The law of conservation of mass requires that atoms aren't created or destroyed, so every atom on the left must appear on the right.

• Use correct chemical formulas for all species ($H_2O$ for water, $CH_4$ for methane, etc.).
• Physical states are indicated in parentheses after the formula: (s) for solid, (l) for liquid, (g) for gas, (aq) for aqueous solution.

Balancing Techniques

The goal is to find the smallest set of whole-number stoichiometric coefficients that gives equal atom counts on both sides.

1. Write out the unbalanced equation with correct formulas.
2. Start with the most complex molecule, or an element that appears in only one compound on each side.
3. Balance elements that show up in multiple compounds on one side after you've handled the simpler ones.
4. Save oxygen and hydrogen for last, since they tend to appear in many compounds (especially in combustion reactions).
5. Adjust coefficients to use the smallest whole numbers. If you end up with fractions, multiply everything through by the common denominator.
6. Double-check by counting every element on both sides.

For example, combustion of methane:

$CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O$

Carbon: 1 on each side. Hydrogen: 4 on each side. Oxygen: 4 on each side. Balanced.

When a system has multiple reactions, each one must be balanced independently, and the overall material balance accounts for contributions from all of them.

Extent of Reaction and Limiting Reactant

Extent of Reaction

The extent of reaction ($\xi$, pronounced "ksi") is a single variable that tracks how far a reaction has progressed, measured in moles (or mol/s for continuous processes). It ties together the changes in every species through stoichiometry.

For a general reaction $aA + bB \rightarrow cC + dD$:

$\Delta n_A = -a\xi, \quad \Delta n_B = -b\xi, \quad \Delta n_C = c\xi, \quad \Delta n_D = d\xi$

Reactants get a negative sign (they're consumed), and products get a positive sign (they're formed). The extent $\xi$ is always positive.

This is powerful because instead of solving for the change in each species separately, you solve for one unknown ($\xi$) and get all the others from stoichiometry. For example, if you know 3 mol of $CO_2$ were produced in the methane combustion reaction above (where the coefficient on $CO_2$ is 1), then $\xi = 3$ mol, and you immediately know 6 mol of $H_2O$ were produced and 6 mol of $O_2$ were consumed.

Limiting Reactant

The limiting reactant is the reactant that runs out first and caps how far the reaction can go.

To identify it:

1. For each reactant, calculate the maximum possible extent of reaction by dividing its initial moles by its stoichiometric coefficient: $\xi_{max,i} = \frac{n_{i,0}}{|\nu_i|}$
2. The reactant that gives the smallest $\xi_{max}$ is the limiting reactant.
3. That smallest value is the actual maximum extent of reaction for the system.

Any reactant that isn't limiting is in excess and will have leftover moles after the reaction goes to completion. The fractional excess of a reactant is defined as:

$\text{fractional excess} = \frac{n_{fed} - n_{stoichiometric}}{n_{stoichiometric}}$

where $n_{stoichiometric}$ is the exact amount needed to react with the limiting reactant. In industrial practice, a slight excess of one reactant is often used deliberately to ensure the more valuable or harder-to-recover reactant is fully consumed.

Material Balances with Reactions

General Material Balance Equation

For reactive systems, the species balance takes the form:

$\text{Input} + \text{Generation} - \text{Output} - \text{Consumption} = \text{Accumulation}$

The generation and consumption terms come directly from the reaction stoichiometry and extent of reaction. At steady state, accumulation is zero, so:

$\text{Input} + \text{Generation} = \text{Output} + \text{Consumption}$

In practice, you'll often combine the generation and consumption into a single net term using the signed stoichiometric coefficient $\nu_i$ (negative for reactants, positive for products), giving you:

$F_{i,out} = F_{i,in} + \nu_i \xi$

This compact form is what you'll use most when solving problems.

Solving Material Balance Problems

Here's a systematic approach for single-reaction, steady-state problems:

1. Draw and label a flow diagram. Identify all inlet and outlet streams with known and unknown flow rates and compositions.
2. Write and balance the chemical equation.
3. Choose a basis (e.g., 100 mol of feed, or a known flow rate).
4. Express all unknown molar flows in terms of $\xi$. For each species: $F_{i,out} = F_{i,in} + \nu_i \xi$ where $\nu_i$ is positive for products and negative for reactants.
5. Identify the limiting reactant if not already specified.
6. Write material balance equations for each species or element, and solve for $\xi$ and any remaining unknowns.
7. Check your answer: verify that total mass in equals total mass out, and that no species has a negative molar flow. A negative molar flow means you've exceeded the physical limit of the reaction.

Common mistake: forgetting to include inert species (species that don't participate in the reaction) in your flow diagram. Inerts pass straight through, so $F_{inert,out} = F_{inert,in}$, but they still matter for calculating compositions and total flow rates.

When multiple reactions occur, you'll need a separate extent of reaction for each independent reaction ($\xi_1, \xi_2, \ldots$), and the change in moles of each species becomes a sum of contributions:

$\Delta n_i = \sum_j \nu_{i,j} \, \xi_j$

where $\nu_{i,j}$ is the stoichiometric coefficient of species $i$ in reaction $j$. This gives you more unknowns, so you'll need more independent equations or specifications (like conversion, selectivity, or yield data) to close the system.