Solutions and preparation
The label on a bottle of concentrated acid never gives the molarity. It gives the percentage purity by mass and the density. To know how much substance is really dissolved in each litre, you have to translate those factory figures into the language of the laboratory: moles per litre.
Molarity and other ways to express concentration
Dissolve 5 g of salt in one glass of water and 5 g of sugar in another. The masses are the same, but the numbers of particles are very different because the molar masses are different. How do we measure concentration so that we know how many particles will actually react in each litre?
A solution is a homogeneous mixture of a solute (the substance that dissolves) in a solvent (the substance it dissolves in, usually water). At a given temperature, a solution can be:
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unsaturated: it can still dissolve more solute;
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saturated: it holds the maximum amount of solute possible;
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supersaturated: it holds more than the saturation limit (unstable, and the extra solute can crystallise out suddenly).
A standard solution is one whose concentration is known accurately.
Other units are used too, and you need to be able to convert between them. All of them describe the same thing: how much solute is in how much solution.
| Unit | Meaning | Typical use |
|---|---|---|
| g/L | mass of solute per litre of solution | lab work |
| mol/L | moles of solute per litre of solution | chemical calculations |
| %w/w | g of solute per 100 g of solution | concentrated acids |
| %w/v | g of solute per 100 mL of solution | saline, medicines |
| %v/v | mL of solute per 100 mL of solution | mixtures of liquids |
| p.p.m. | mg of solute per litre (or per kg) of dilute aqueous solution | trace amounts, water quality |
In the laboratory, an aqueous solution is to be prepared by dissolving exactly $4.00\,\text{g}$ of pure sodium hydroxide, $\mathrm{NaOH}$, in deionised water to a total…
In the laboratory, an aqueous solution is to be prepared by dissolving exactly $4.00\,\text{g}$ of pure sodium hydroxide, $\mathrm{NaOH}$, in deionised water to a total volume of $250.0\,\text{mL}$ at $20\,^{\circ}\text{C}$. Atomic mass data: $M(\mathrm{Na}) = 22.99\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{O}) = 16.00\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{H}) = 1.01\,\text{g}\times\text{mol}^{-1}$.
- a) Calculate the molar concentration (molarity) of the resulting solution.
- b) State, with reasoning, which precision volumetric glassware must necessarily be used to set this final volume.
Purity and density of commercial reagents
Density is the bridge that lets you go from %w/w to g/L and then to mol/L. Use it only for that.
A commercial bottle of concentrated nitric acid, $\mathrm{HNO}3$, states on its label a purity of $65.0\,\%$ by mass and a density of…
A commercial bottle of concentrated nitric acid, $\mathrm{HNO}_3$, states on its label a purity of $65.0\,\%$ by mass and a density of $1.40\,\text{g}\times\text{cm}^{-3}$ at $20\,^{\circ}\text{C}$. Atomic mass data: $M(\mathrm{H}) = 1.01\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{N}) = 14.01\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{O}) = 16.00\,\text{g}\times\text{mol}^{-1}$.
- a) Calculate the mass of pure nitric acid present in $1.00\,\text{L}$ of this commercial reagent.
- b) Determine the molar concentration of this commercial solution.
$500.0\,\text{mL}$ of an aqueous solution of sulfuric acid, $\mathrm{H}2\mathrm{SO}4$, $0.250\,\text{mol}\times\text{L}^{-1}$ is to be prepared from a commercial acid of…
$500.0\,\text{mL}$ of an aqueous solution of sulfuric acid, $\mathrm{H}_2\mathrm{SO}_4$, $0.250\,\text{mol}\times\text{L}^{-1}$ is to be prepared from a commercial acid of $96.0\,\%$ purity by mass and density $1.84\,\text{g}\times\text{cm}^{-3}$. Atomic masses: $M(\mathrm{H}) = 1.01\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{S}) = 32.06\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{O}) = 16.00\,\text{g}\times\text{mol}^{-1}$.
- a) Calculate the volume of concentrated commercial acid that must be taken.
- b) Set out the experimental sequence for adding the reagents, justifying in detail the specific safety rule for avoiding serious thermal accidents.
Dilution: the moles stay the same
If you top up half a glass of very strong blackcurrant cordial with water until the glass is full, the amount of cordial has not changed but the taste is weaker. What exact relationship links the concentration to the total volume after you add solvent?
Serial dilutions
A serial dilution is a series of dilutions, each made from the one before. It is the quickest way to get a range of weak solutions, for example to make a calibration set.
A concentrated solution of hydrochloric acid, $\mathrm{HCl}$, with concentration $2.00\,\text{mol}\times\text{L}^{-1}$, is available in the laboratory. Exactly…
A concentrated solution of hydrochloric acid, $\mathrm{HCl}$, with concentration $2.00\,\text{mol}\times\text{L}^{-1}$, is available in the laboratory. Exactly $500.0\,\text{mL}$ of a dilute solution of $\mathrm{HCl}$ with concentration $0.050\,\text{mol}\times\text{L}^{-1}$ is required.
- a) Calculate the volume of concentrated solution that must be measured out to carry out this dilution.
- b) From the point of view of good laboratory practice and safety, explain which auxiliary instrument must necessarily be attached to the pipette to take the aliquot and which risky practice is strictly forbidden.
In the laboratory, a controlled dilution of a stock solution of copper(II) sulfate, $\mathrm{CuSO}4$, with initial concentration $0.800\,\text{mol}\times\text{L}^{-1}$,…
In the laboratory, a controlled dilution of a stock solution of copper(II) sulfate, $\mathrm{CuSO}_4$, with initial concentration $0.800\,\text{mol}\times\text{L}^{-1}$, is carried out. Using a volumetric pipette, an aliquot of $20.0\,\text{mL}$ is transferred to a volumetric flask of $250.0\,\text{mL}$, which is then made up with deionised water to $20\,^{\circ}\text{C}$. The equipment used and the detail of the final meniscus setting are illustrated below:
- a) Calculate the final molar concentration of the dilute solution obtained in the volumetric flask.
- b) A student adds water until the bottom of the meniscus is above the graduation mark when viewed at the level of the mark. Explain whether the real concentration will be higher or lower than the calculated one, and why removing some of that solution down to the mark does not correct its concentration.
Mixing solutions
When you mix two solutions of the same solute that do not react, the moles add up. If we assume the volumes are additive, the volumes add up too.
Evaporating water works the other way round: the moles of solute stay the same and the volume falls, so $c_1V_1 = c_2V_2$ still applies, with $V2 < V_1$.
When a problem gives you several sources of the same solute (solutions of different concentrations, or a solid of known purity), work out the moles from each source and add them up. Convert %w/w to moles using density and molar mass, and use the percentage purity to find how much of a solid is really solute.
Two aqueous solutions of potassium chloride, $\mathrm{KCl}$, are mixed in the laboratory, with no chemical reaction taking place. Solution A has a volume of…
Two aqueous solutions of potassium chloride, $\mathrm{KCl}$, are mixed in the laboratory, with no chemical reaction taking place. Solution A has a volume of $150.0\,\text{mL}$ and a concentration of $0.500\,\text{mol}\times\text{L}^{-1}$. Solution B has a volume of $350.0\,\text{mL}$ and a concentration of $1.200\,\text{mol}\times\text{L}^{-1}$. Assume ideal behaviour with strictly additive volumes at $20\,^{\circ}\text{C}$.
- a) Calculate the final molar concentration of chloride ions, $\mathrm{Cl}^-$, in the resulting mixture.
- b) Calculate the volume of pure water that would have to be evaporated from the resulting mixture so that the final concentration of $\mathrm{KCl}$ reaches exactly $1.500\,\text{mol}\times\text{L}^{-1}$. In b), compare the volumes at $20\,^{\circ}\text{C}$ and assume that only water is lost, with no precipitation or change in the additivity of volumes.
Exactly $1.000\,\text{L}$ of an aqueous solution of sodium chloride, $\mathrm{NaCl}$, with a final concentration of $0.750\,\text{mol}\times\text{L}^{-1}$ at…
Exactly $1.000\,\text{L}$ of an aqueous solution of sodium chloride, $\mathrm{NaCl}$, with a final concentration of $0.750\,\text{mol}\times\text{L}^{-1}$ at $20\,^{\circ}\text{C}$ must be prepared. Three sources of the reagent are available in the laboratory: solution 1 ($200.0\,\text{mL}$ of $\mathrm{NaCl}$ $0.400\,\text{mol}\times\text{L}^{-1}$), solution 2 ($300.0\,\text{mL}$ of $\mathrm{NaCl}$ at $4.50\,\%$ by mass with density $1.03\,\text{g}\times\text{cm}^{-3}$) and a jar of $\mathrm{NaCl}$ solid anhydrous of $98.0\,\%$ purity by mass. Atomic masses: $M(\mathrm{Na}) = 22.99\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{Cl}) = 35.45\,\text{g}\times\text{mol}^{-1}$.
- a) Calculate the amount of substance (moles) of $\mathrm{NaCl}$ contributed jointly by solutions 1 and 2.
- b) Calculate the mass of impure solid reagent that must be weighed to make up the necessary moles, placing it in the $1.000\,\text{L}$ volumetric flask before finally making up to the mark with deionised water.
Preparing a standard solution in the laboratory
When you look at water in the narrow neck of a flask, the surface is not flat but curved. Where exactly on that curve should the mark sit if the concentration is not to be wrong?
Primary standards
A primary standard is a substance used to make a standard solution directly from a weighed mass. To be suitable it must be:
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pure (known, very high purity);
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stable: it does not react with the air or absorb water or carbon dioxide, so its mass does not change while you weigh it;
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soluble in the solvent;
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of high molar mass, so that the mass you weigh is large and the weighing error is small in proportion.
Anhydrous sodium carbonate, $\mathrm{Na_2CO_3}$, and potassium hydrogen phthalate are primary standards. Sodium hydroxide is not: the pellets absorb water and carbon dioxide from the air, so the mass you weigh is not all NaOH. A NaOH solution therefore has to be standardised afterwards by titration against a primary standard.
Reading the volume
The method
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Weigh the solute on a balance, into a clean weighing boat or directly into a beaker.
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Dissolve it in a beaker with a little deionised water, stirring with a glass rod.
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Transfer the solution to the volumetric flask through a funnel. Rinse the beaker, rod and funnel into the flask.
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Add water until it is just below the mark, then add the last drops with a dropper until the bottom of the meniscus touches the mark.
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Stopper and invert the flask several times to mix.
Precision and accuracy
Precision is how close repeated measurements are to each other. Accuracy is how close a measurement is to the true value. A volumetric flask and a volumetric pipette are precise to a small tolerance, and a measuring cylinder is not. Random errors (reading, small fluctuations) reduce precision. Systematic errors (a badly calibrated balance, always reading from above) reduce accuracy.
Safe practice
Before you start, do a risk assessment: what is hazardous, how could you be harmed, and what will you do to prevent it. Wear safety glasses, a lab coat and gloves, and work in a fume cupboard with volatile or fuming reagents.
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Always use a pipette filler. Never pipette by mouth.
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When diluting a concentrated acid, add the acid slowly to the water, stirring, never the reverse. Dilution is strongly exothermic. If you pour water onto the acid, the heat released can boil the water locally and throw drops of acid out of the container.
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For a strongly exothermic dilution, let the beaker cool before making up to the mark. A hot solution has a larger volume, and the concentration will be wrong when it cools.
In a quality control laboratory, the preparation of $100.0\,\text{mL}$ of a standard solution of primary anhydrous sodium carbonate, $\mathrm{Na}2\mathrm{CO}3$, at a…
In a quality control laboratory, the preparation of $100.0\,\text{mL}$ of a standard solution of primary anhydrous sodium carbonate, $\mathrm{Na}_2\mathrm{CO}_3$, at a nominal concentration of $0.0500\,\text{mol}\times\text{L}^{-1}$ at $20\,^{\circ}\text{C}$ is standardised. An analytical balance with uncertainty $\pm 0.001\,\text{g}$ and a class A volumetric flask of $100.0\,\text{mL}$ with calibration tolerance $\pm 0.10\,\text{mL}$ are available. Atomic masses: $M(\mathrm{Na}) = 22.99\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{C}) = 12.01\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{O}) = 16.00\,\text{g}\times\text{mol}^{-1}$.
- a) Calculate the theoretical mass of solute that must be weighed to obtain exactly the specified concentration.
- b) Determine the total relative uncertainty in the molarity by propagating the maximum absolute uncertainties of weighing and volume, and express the final result with its uncertainty interval: $M \pm \Delta M$. For this estimate, use the conservative approximation $\Delta M/M=\Delta m/m+\Delta V/V$ and treat the molar mass as exact.
A technician tries to prepare $500.0\,\text{mL}$ of standard solution of $\mathrm{NaOH}$ $0.1000\,\text{mol}\times\text{L}^{-1}$ by weighing $2.000\,\text{g}$ of…
A technician tries to prepare $500.0\,\text{mL}$ of standard solution of $\mathrm{NaOH}$ $0.1000\,\text{mol}\times\text{L}^{-1}$ by weighing $2.000\,\text{g}$ of commercial pellets and making up to the mark with deionised water in a volumetric flask. When this solution is subsequently titrated against potassium hydrogen phthalate (primary standard), the real molarity obtained is found to be only $0.0885\,\text{mol}\times\text{L}^{-1}$. Atomic masses: $M(\mathrm{Na}) = 22.99\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{O}) = 16.00\,\text{g}\times\text{mol}^{-1}$; $M(\mathrm{H}) = 1.01\,\text{g}\times\text{mol}^{-1}$.
- a) Assuming that the volume was made up exactly to $500.0\,\text{mL}$, and that the impurities present in this batch are water and substances that do not react in the titration, determine the real percentage purity in $\mathrm{NaOH}$ of the commercial pellets used.
- b) Explain, with reasoning, two chemical and experimental reasons why solid $\mathrm{NaOH}$ can never be considered a direct primary standard, and justify why it is necessary to carry out a standardisation titration.
Back to the start: to know how much solute is in a bottle of reagent, take the label (percentage purity and density), work out the molarity with $c = 1000\,d\,(\%/100)/M$, and then measure the exact volume $V = n/c$ with a pipette before making up in a volumetric flask. As a check on the p.p.m. idea we met at the start, a water supply fluoridated at $0.8$ p.p.m. contains only $0.8\ \text{mg}$ of fluoride per litre.