Mass Balance across Linked Steps

Introduction

Sequential mixing, evaporation and transfer problems are easy to mishandle if only the total solution mass is tracked. A component table keeps solute and solvent separate at every step. The principle is conservation: material does not appear or disappear when a solution becomes more concentrated. The total concentration can change because water leaves, because a portion is removed or because more solvent is added.

Core explanation

Choose a clear system boundary and time interval. For each component, write initial amount + inputs − outputs = final amount, assuming no reaction creates or consumes that component. For a salt solution, a table might have columns salt, water and total. If 10 g salt dissolves in 90 g water, the initial solution has 10 g salt, 90 g water and 100 g total. Its salt mass fraction is 0.10. The salt and water amounts should be kept separately rather than compressing the state immediately to “10% solution.”

Suppose 20 g water evaporates and salt is nonvolatile under the conditions. The new inventory is 10 g salt, 70 g water and 80 g total; salt mass percentage becomes 10/80 × 100 = 12.5%. A claim that 20% of the salt evaporated because 20 g of solution mass disappeared would contradict the stated process. The total mass decreases, but the conserved component's mass remains fixed inside the chosen vessel.

Now remove half of a well-mixed 80 g solution. A 40 g withdrawn portion contains half of each dissolved component: 5 g salt and 35 g water. The remaining vessel also contains 5 g salt and 35 g water. If the sample were not homogeneous or salt had precipitated, this proportional split would need qualification. A solution that is truly uniform supports the proportional component balance; a suspension or crystal-containing mixture may not.

Finally, add 10 g pure water to the withdrawn 40 g portion. It now contains 5 g salt and 45 g water, total 50 g, so its salt mass percentage is 10%. The original concentration has been recovered numerically, but not because the exact original material amounts returned. The inventory makes this distinction clear: 5 g salt in 50 g solution has the same percentage as 10 g salt in 100 g solution.

Reaction problems add stoichiometric source and sink terms. If a salt reacts, its moles decrease according to a balanced equation and products increase. A single total-mass balance still holds in a closed system, but a named-component balance includes chemical conversion. Gas escape changes the mass inside an open vessel even though matter is conserved across a larger boundary including the surroundings. State whether gas products are retained or released before comparing input and final vessel mass.

Volume balance is not automatically mass balance. Mixing 50 mL of one liquid and 50 mL of another does not always make 100 mL final solution because volumes can contract or expand. Masses add if no material is lost, so a mass table is robust. A final volume measurement or density may then be used for molarity. Confusing additive mass with additive liquid volume can introduce a systematic concentration error.

Step-by-step reasoning

1. Draw a separate row for each retained chemical component and a total row. 2. Enter starting amounts, then apply each step as an input, output or reaction change. 3. Update component amounts before calculating any new percentage or concentration. 4. For sample withdrawal, use a proportional split only when the mixture is homogeneous. 5. Check that each step's component balance and overall mass balance are satisfied for the chosen boundary.

Visual explanation

Draw a four-column table labelled “start,” “after evaporation,” “withdrawn half” and “after adding water.” Fill salt row 10, 10, 5, 5 g; water row 90, 70, 35, 45 g; total row 100, 80, 40, 50 g. Underneath place percentages 10%, 12.5%, 12.5%, 10%. The salt row's constancy during evaporation and increase of water only in the final step make the logic visible.

Real-world analogy

Tracking a household budget requires separate categories for income, food spending and savings. Knowing only the bank's total balance at each date does not reveal which category changed. A solution's total mass likewise does not identify whether salt, water or both changed; component rows do.

Real-world example

A chemical plant concentrates an aqueous product by evaporating solvent, then transfers a measured fraction to another tank for dilution. Material accounting records product and water masses in each stream. This permits checking product recovery and detecting a leak or unexpected precipitation, rather than inferring everything from changing total tank mass.

Why?

Why can concentration rise while the solute mass stays constant? Evaporation removes solvent from the denominator of mass fraction or molality. The numerator may remain unchanged while the denominator decreases, increasing the ratio.

Common misconception

“A 20 g decrease in solution mass means 20 g of every component was lost.” The removed stream has a composition determined by the process. Evaporated pure water removes water, whereas withdrawing well-mixed solution removes both salt and water in their current proportions.

Worked example

Prepare 100 g solution from 10 g salt and 90 g water. Evaporate 20 g water: 10 g salt + 70 g water = 80 g, or 12.5% salt. Withdraw 40 g of the homogeneous mixture: 5 g salt + 35 g water. Add 10 g water: 5 g salt + 45 g water = 50 g, or 10% salt. At each step, total equals the sum of component masses. The final 10% is a ratio, not evidence that the original 10 g salt returned to the withdrawn portion.

Quick check

1. If only water evaporates, does the mass of dissolved nonvolatile salt in the vessel decrease? Answer: No. Salt mass remains constant unless salt is separately removed or reacts.

Exam focus

Use a component table through every stage. State what exits, whether a withdrawn portion is homogeneous, and whether gas or solids leave the chosen boundary. Calculate a new percentage only after updating the inventory.

Advanced insight

Industrial material balances are written for each conserved element or chemical species across process units. A reaction-network matrix adds stoichiometric source terms, while unit-operation streams add flows. The simple salt–water table is a small version of that general framework.

Summary

Sequential solution calculations require tracking each component, not only total mass or concentration. Evaporation, withdrawal and dilution change different inventory rows. Mass conservation applies across the declared boundary, while concentration changes as the component ratio changes.

Practice questions

1. A 200 g solution contains 20 g nonvolatile solute. If 50 g pure solvent evaporates, what is the new mass percentage? Answer: Solute remains 20 g, total becomes 150 g, so 20/150 × 100 ≈ 13.3%. 2. If half of that homogeneous solution is withdrawn, how much solute is in the withdrawn portion? Answer: 10 g, half of the 20 g dissolved solute. 3. Why might withdrawing half the vessel mass fail to remove half the solute from a suspension? Answer: A suspension may not be homogeneous; solids can settle and distribute unevenly.