Lesson 5 · 30 min

Solution stoichiometry: concentration, dilution and titration

Molarity, C₁V₁ = C₂V₂, preparing a standard solution step by step on a virtual bench, and finding an unknown concentration from titration data.

In this lesson

Convert between concentration, volume and moles with c = n/V
Plan a dilution with C₁V₁ = C₂V₂ and explain why it holds
Describe how to prepare a solution of known molarity and the error each step can introduce
Calculate an unknown concentration from titration data and the coefficient ratio

The saline in a hospital drip is labelled 0.9% — 9.0 g of sodium chloride per litre. Chemists prefer to say it differently: $9.0 / 58.44 = 0.154$ mol of NaCl per litre, or 0.154 mol/L. Why concentration in moles? Because reactions go by numbers of particles: once the concentration is known, measuring out a volume is the same as counting out a number of solute particles, and reactions in solution can be calculated as precisely as weighing a solid.

Predict first

10.00 mL of 0.500 mol/L $\ce{CuSO4}$ is pipetted into a volumetric flask and made up to 100.0 mL with water. What is the new concentration?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

0.500 mol/L 0.0500 mol/L 0.0450 mol/L 5.00 mol/L

Molarity and dilution

c_\mathrm{B} = \frac{n_\mathrm{B}}{V} \qquad C_1 V_1 = C_2 V_2

The moles of solute are the same before and after dilution.

$c_\mathrm{B}$: molar concentration of solute B, mol/L (often written M)
$n_\mathrm{B}$: amount of solute, mol
$V$: volume of solution (not of solvent), L
$C_1V_1,\ C_2V_2$: concentration and volume before and after; any volume unit, as long as it is the same on both sides

The $V$ in $c = n/V$ is the total volume of solution. To make 1.00 mol/L you do not add 1.00 mol of solute to 1.00 L of water; you dissolve it and then make the total volume up to 1.00 L — which is exactly what a volumetric flask is for.

The standard procedure is calculate → weigh → dissolve (cool if it warms) → transfer → rinse → make up to the mark → mix. Each step has its own error: poor rinsing loses solute (concentration too low); overshooting the mark adds water you cannot remove (too low, start again); forgetting to mix leaves the neck and the bulb at different concentrations.

Calculate · weigh · transfer · make up · mix

Preparing a standard solution

Solute Volumetric flask Target concentration

1 Calculate
2 Weigh
3 Dissolve & transfer
4 Make up to the mark
5 Mix

How many grams do you need? c = 0.1000 mol/L · V = 0.1000 L · M( $\ce{CuSO4*5H2O}$ ) = 249.68 g/mol

Go through the whole procedure: calculate the mass, weigh, transfer and rinse, add water to the mark drop by drop, and mix. Then skip the rinsing or overshoot the mark on purpose and see how far the concentration moves.

Stoichiometry of titrations

In a titration a standard solution of known concentration is added drop by drop to the unknown until the two have reacted exactly in the ratio of the coefficients (the equivalence point, shown by an indicator colour change or a pH jump). There $n = cV$ holds for both solutions, and the coefficient ratio bridges them. Try one yourself on the acid–base titration bench and watch the curve steepen near equivalence.

Worked example

The concentration of sulfuric acid by titration

25.00 mL of sulfuric acid is titrated with 0.1000 mol/L NaOH; the endpoint needs 32.46 mL. The reaction is $\ce{H2SO4} + \ce{2NaOH} \mathrel{\ce{->}} \ce{Na2SO4} + \ce{2H2O}$ . Find the acid’s concentration.

Concept check

To prepare exactly 250.0 mL of a 0.1000 mol/L solution, which glassware should you make it up in?

250 mL beaker 250 mL graduated cylinder 250 mL volumetric flask 250 mL conical flask

Try the calculation

How many grams of copper(II) sulfate pentahydrate, $\ce{CuSO4*5H2O}$ , make 250.0 mL of 0.1000 mol/L $\ce{CuSO4}$ ?

Key takeaways

Molarity $c = n/V$ , with $V$ the total volume of solution; measuring a volume counts out solute particles.
Dilution conserves the moles of solute: $C_1V_1 = C_2V_2$ .
Procedure: calculate, weigh, dissolve, transfer, rinse, make up, mix; primary standards can be weighed directly.
At equivalence $c_\mathrm{A}V_\mathrm{A}/\nu_\mathrm{A} = c_\mathrm{B}V_\mathrm{B}/\nu_\mathrm{B}$ — the coefficient ratio is essential.

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Data and references

D. C. Harris, Quantitative Chemical Analysis, 9th ed., Ch. 1–2 (molarity, volumetric glassware, Table 2-3 flask tolerances) and Ch. 7 (titrations, primary standards).
人民教育出版社《普通高中教科书化学必修第一册》（2019），实验活动“配制一定物质的量浓度的溶液”。
World Health Organization, Model List of Essential Medicines — sodium chloride 0.9% injectable solution.