3.2 Material Balances With Chemical Reaction

3.2 Material Balances With Chemical Reaction

Quick Answer: At steady state, total mass In = Out, but species balances include generation/consumption. Use extent of reaction (\xi) (or conversion of the limiting reactant) with a stoichiometric table, or write elemental balances that never need generation terms for ordinary chemistry.

Once nonreactive balances are automatic, the UPDA Chemical exam expects you to handle reactors, combustors, and reformer-style problems. The general balance still holds; reaction simply activates generation and consumption for molecular species.

Species vs Total Mass

If you only remember one rule under stress: mass and atoms are conserved; molecules may not be.

Extent of Reaction

For a single reaction written with stoichiometric coefficients (\nu_i) (negative for reactants, positive for products):

[ \sum_i \nu_i A_i = 0 ]

Define the extent of reaction so that the molar change of each species is proportional to (\nu_i).

Batch (amount basis):

[ n_i = n_{i0} + \nu_i ,\xi ]

Continuous flow (rate basis):

[ \dot{n}i = \dot{n}{i0} + \nu_i ,\dot{\xi} ]

Here (\xi) has units of moles of “reaction progress,” and (\dot{\xi}) is moles of reaction progress per time. One independent reaction ⇒ one independent extent.

Stoichiometric Table

Build a table with rows for each species and columns for feed, change, and outlet:

Inerts (N₂ in air, carrier gas) have (\nu = 0): they pass through unchanged in molar flow (unless the problem allows them to dissolve or react).

Conversion-Based Formulation

Fractional conversion of limiting reactant A:

[ X_A = \frac{\text{moles of A reacted}}{\text{moles of A fed}} = \frac{\dot{n}_{A0} - \dot{n}A}{\dot{n}{A0}} ]

For a single reaction with coefficient (\nu_A) (typically −1 if the reaction is written per mole of A):

[ \dot{\xi} = \frac{X_A \dot{n}_{A0}}{|\nu_A|} ]

when A is limiting and the stoichiometry is scaled so you know how extent maps to A consumed. Many textbook reactions are written as (A + \ldots \rightarrow \ldots) with (\nu_A = -1), so (\dot{\xi} = X_A \dot{n}_{A0}).

Related definitions you may see on process items:

• Limiting reactant: would be exhausted first if the reaction went to completion as written.
• Excess reactant: fed beyond stoichiometric requirement relative to the limiting reactant.
• Yield (often moles of desired product formed / moles that could form from limiting reactant reacted or fed—read the problem’s definition carefully).
• Selectivity (desired product rate / undesired product rate, or fractional forms).

On UPDA MCQs, if the stem gives conversion but not extent, convert to extent (or work entirely in “moles reacted”) before filling the stoich table.

Elemental Balances

When many species participate or you do not care about intermediate molecular identities, write balances on C, H, O, N, S, …. Because ordinary reactions rearrange atoms only:

[ \text{atoms of } E \text{ in} = \text{atoms of } E \text{ out} ]

Elemental balances are especially powerful for:

• Combustion with incomplete product specs
• Partial oxidation
• Systems where outlet analysis is given in atom % or elemental composition
• Checking a species-based solution for arithmetic errors

You still need enough independent relations (including reaction extents or conversion) if the molecular outlet composition is the unknown set.

Multiple Reactions Overview

If two independent reactions occur, introduce two extents (\dot{\xi}_1, \dot{\xi}_2):

[ \dot{n}i = \dot{n}{i0} + \nu_{i1}\dot{\xi}1 + \nu{i2}\dot{\xi}_2 ]

DOF rises: you need conversion of one reactant and selectivity, two measured outlet flows, or two independent analytical specs. Do not force a single conversion to describe two independent paths unless the problem defines an overall conversion and supplies yield/selectivity.

Worked Example: Ammonia Oxidation (Clear Stoichiometry)

Reaction (single, as written):

[ 4,\mathrm{NH_3} + 5,\mathrm{O_2} \rightarrow 4,\mathrm{NO} + 6,\mathrm{H_2O} ]

Feed (continuous): 40 mol/h NH₃ and 100 mol/h O₂ (no inerts). Conversion of NH₃ is 80%. Assume no side reactions.

Step 1 — Limiting reactant check (for complete conversion reference).

Stoichiometric O₂ for 40 mol/h NH₃: ((5/4)\times 40 = 50) mol/h O₂. Feed has 100 mol/h O₂, so NH₃ is limiting and O₂ is in excess.

Step 2 — Extent from conversion.

Moles NH₃ reacted = (0.80 \times 40 = 32) mol/h.

Per the reaction as written, 4 mol NH₃ correspond to 1 mol of “reaction event,” so:

[ \dot{\xi} = \frac{32}{4} = 8,\mathrm{mol\ reaction/h} ]

(Equivalently, if you rewrite the reaction per 1 mol NH₃: (\mathrm{NH_3} + 5/4,\mathrm{O_2} \rightarrow \mathrm{NO} + 3/2,\mathrm{H_2O}), then (\dot{\xi}' = 32) mol/h with coefficients scaled by 1/4.)

Step 3 — Outlet molar flows (using original coefficients and (\dot{\xi} = 8)).

Total moles rise because (\sum \nu_i = +1) per reaction event as written ((-4-5+4+6 = +1)).

Step 4 — Elemental check (N atoms).

• In: NH₃ contributes 40 mol N/h
• Out: NH₃ (8) + NO (32) = 40 mol N/h ✓

O atoms: In = 200 mol O/h from O₂; Out = O₂ (120 atom-mol O) + NO (32) + H₂O (48) = 200 ✓

H atoms: In = 120 from NH₃; Out = NH₃ (24) + H₂O (96) = 120 ✓

Fractional excess O₂ (common exam follow-up):

[ \text{Excess O}_2 = \frac{\text{O}_2\text{ fed} - \text{O}_2\text{ stoichiometric for feed NH}_3}{\text{O}_2\text{ stoichiometric}} = \frac{100 - 50}{50} = 100% ]

(Some stems define excess based on O₂ required for the NH₃ actually converted—read the wording.)

Exam Workflow for Reactive Balances

1. Write the balanced reaction(s) and identify limiting reactant if conversion is given for “the feed reactant.”
2. Choose mole basis for species balances (mass only after you know molecular flows, or for total mass check).
3. Build a stoichiometric table with one extent per independent reaction.
4. Insert conversion, yield, or product specs to solve extents.
5. Close with elemental balances or an unused species equation.
6. Only then compute mass fractions or mole fractions for outlet streams.

Traps That Cost Marks

• Applying In = Out to a reactive species without generation/consumption.
• Using mass fractions directly in a stoichiometric table built for moles.
• Forgetting that total moles change when (\sum \nu_i \neq 0).
• Double-counting: specifying both conversion and product rate that imply different extents.
• Treating air as pure O₂ in combustion balances (N₂ is inert but dilutes).

Reactive balances feed directly into reactor sizing (Chapter 10) and energy balances with heat of reaction (Chapter 4). For flowsheets with recycle around a reactor, you will combine this section with Section 3.3.