Plan reaction extent when reagent supplies have bounds

Try: diagnose an impossible product proposal

The supplied equation consumes two Ag⁺ ions per carbonate ion to form one Ag₂CO₃ formula unit. The initial proposal is 10 mmol of product. Inspect the full initial/change/final ledger: which proposed remainder is negative? Equal atom totals alone cannot make that inventory possible.

Predict and convert the records

The synthetic 3.40 g sample is 80% AgNO₃ by mass. Its reactive mass is 3.40 × 0.80 = 2.72 g; divide by the stipulated 170 g/mol to obtain 0.016 mol, or 16 mmol. The solution provides 50 mL × 0.200 mmol/mL = 10 mmol carbonate. The two reaction capacities are therefore 16/2 = 8 and 10/1 = 10 mmol.

Call the reaction extent ξ, measured here in mmol. For this supplied equation it consumes 2ξ mmol silver ions and ξ mmol carbonate, and forms ξ mmol target. Zero ≤ ξ ≤ 8 is nominally feasible. This is not integer molecule counting: fractional millimoles are permitted. The interface samples half-millimole proposals for practice, not a physical quantization.

Inspect a feasible change

At ξ = 8, silver ions have zero remainder, carbonate has 2 mmol, and target is 8 mmol. Sodium and nitrate spectator inventories stay unchanged. At ξ = 4 both reactive species remain. The maximum target mass is 8 × 276/1000 = 2.208 g using the rounded teaching molar mass; this is a conditional theoretical ceiling, not a measured isolate.

Compare the halved batch and dilution record. Halving both supplies halves the ceiling. Doubling solution volume while halving concentration preserves carbonate amount. This model does not imply dilution leaves real equilibrium, kinetics or precipitation recovery unchanged.

Understand uncertainty without pretending it is probability

In the overlapping record, silver mass is 3.40 ± 0.34 g at stipulated purity 1; carbonate concentration is 0.200 ± 0.020 mol/L in 50 mL. Each supported extent lies between 9 and 11 mmol. At silver capacity 9 and carbonate 11, silver limits; at 11 and 9, carbonate limits. Nominal capacities tie, but the full allowed ranges do not establish a unique limiter.

For independent allowed intervals [A_low, A_high] and [B_low, B_high], the minimum capacity ranges from min(A_low, B_low) to min(A_high, B_high). Here “independent allowed” means every pair of endpoints/interior values is permitted, not statistical independence or a confidence level. A proposed 9 mmol extent is feasible throughout, 10 is feasible only for some inputs, and 11.5 exceeds all allowed capacities. Feasibility throughout still does not guarantee the reaction reaches that extent.

Apply and continue

Use the separated-interval record to justify a limiter that remains the same across all allowed values. Capture a contrasting overlapping case and explain what better input evidence would resolve. Complete the unfamiliar 3:2 practice. Revisit bulk limiting reactants for mass conversion or product yield for why an isolate can contain contaminants. Net ionic cancellation explains why spectators remain physically present.

Sources and assumptions

OpenStax Chemistry 2e §4.4 supplies the theoretical-yield and limiting-reagent framework. Input records, bounds and figures are original synthetic teaching material. The single-reaction model assumes inert impurity and specified ionic products; no practical measurement, equilibrium extent or recovered yield is asserted. Saved explanations require independent review.

Connect a reaction ratio to a separately specified gas state

A balanced equation supplies amounts; a gas relation connects amount with volume at stated pressure and temperature. In a separate original paper record, 2 H₂O₂ → 2 H₂O + O₂ goes to the stipulated extent, with 0.0240 mol H₂O₂ consumed. The equation gives 0.0120 mol O₂. For a dry ideal oxygen product at 300 K and 100 kPa, using R = 8.314 L kPa/(mol K), V = nRT/P = 0.299304 L. This is a supplied accounting scenario, not instructions for generating or collecting oxygen.

Changed boundary: If the same amount of dry ideal gas is instead at 200 kPa and 300 K, predict its volume. Would a wet collected volume automatically equal that dry-gas prediction?

Check amount, state and collection assumptions

The new volume is 0.149652 L: amount is unchanged while pressure doubles. A wet gas mixture also contains water vapor, and retained amount can differ from generated amount through loss or dissolution. Separate partial pressures and the collection boundary before inferring reaction extent. Neither gas volume alone nor the balanced equation establishes quantitative recovery.

Stoichiometric amount relationships and the existing wet-gas inference lesson supply the two connected frameworks; checked 31 October 2026.