A drop falls into a volumetric flask from above, the solution in the neck slowly rises to the pink mark and settles again, as if being made up to volume, while mixing bubbles swirl in the round body. On the right a centrifuge rotor turns with four tubes, and at the bottom a marker travels along the scale up to one mole per litre.
Quantity data sheet
Non-SI · analytical chemistry sheet 1/1 · rev. 2026-09Mole per litre
Non-SI unit · quantity: amount concentration c · 1 M = 1000 mol/m³
A hot solution made up to the mark will cool stronger than the logbook says
A lab technician dissolves forty grams of caustic soda, and the flask grows noticeably warm in their hands: the alkali releases so much heat that the solution warms by ten to twenty degrees, depending on how much water is already in the flask. If the water is topped up to the mark right away in a hurry, the solution is still hot and expanded, and as it cools it shrinks below the mark, so the logbook says “exactly one mole per litre” while the flask holds a little more.
The mole per litre is convenient because it counts particles, not grams: a litre of a one-molar solution of any substance holds the same number of molecules, six hundred and two thousand billion billion, so reactions can be calculated by simple ratios of volumes. It is exactly this property that made molarity the working unit of titration, blood tests and every laboratory recipe.
But a litre is a volume, and the volume of a liquid depends on temperature, which is why every volumetric flask is etched with “20 °C”: only at that temperature does the mark mean what it says. Molarity turns out to be a quantity that changes with the room the flask stands in, and in precise work it is replaced by molality, which has kilograms in the denominator.
An upright Latin M without a full stop, separated from the number by a space: 0.1 M. The strict notation is mole per litre, mol/L or mol/dm³, and in SI the mole per cubic metre. The substance is always given alongside: c(NaCl) = 0.154 mol/L, because a concentration without a substance means nothing.
Molarity M counts moles per litre of solution, while molality b counts moles per kilogram of solvent. The first depends on temperature, because the solution expands, the second does not; in dilute water they nearly coincide, but in concentrated acid they differ by tens of percent. One more thing: the normality N of sulfuric acid is twice its molarity.
01 · Definition
weighing · flask · mark · temperatureAmount concentration is the amount of dissolved substance divided by the volume of the solution. One mole per litre means that each litre of the finished solution contains one mole of particles of that substance, about 6.022·10²³ of them.
Such a solution is prepared with a balance and a volumetric flask: a weighed portion equal to the molar mass times the required number of moles is transferred into the flask, dissolved, and topped up with water to the mark on the neck. What matters is that the denominator is the volume of the solution, not of the water: a litre of water plus the solute gives more than a litre, which is why water is added up to the mark rather than measured out in advance.
The flask is calibrated at twenty degrees. If the solution is made up to the mark at another temperature, its volume at twenty will be different: a warm solution shrinks as it cools and the concentration ends up higher than recorded, while a cold one expands as it warms and becomes weaker. For aqueous solutions the correction is a few tenths of a percent per ten degrees, which is already a lot for titration.
Dilution preserves the number of moles: as much substance as the pipette took ends up in the new flask. So the product of concentration and volume is the same before and after, and a series of calibration solutions comes from one stock solution by simple arithmetic.
On the left, the weighed portion on the balance; in the middle, the volumetric flask; on the right, a second flask into which a pipette transfers part of the solution for dilution. The colour of the solution deepens with concentration by the absorption law, the amber dotted line in the neck shows where the meniscus will be once the solution reaches twenty degrees, and yellow crystals appear at the bottom if there is more substance than the water can dissolve.
Take a finished solution and change only the room temperature. The blue line is its molarity, which falls as the solution expands; the pink dotted line is its molality, which does not depend on temperature at all; the amber dot is your temperature, and the vertical line is the twenty degrees of calibration.
02 · Conversion
units · dilution · centrifugeFrom grams to moles and from moles to the tube
The first tab converts concentration into millimoles, grams per litre and particles per millilitre for the selected substance. The second works out how much stock solution to take for a dilution series, and the third tells you what force a tube of this solution will experience in a centrifuge.
A rule to remember: grams per litre equal molarity times molar mass, and the force in a centrifuge grows as the square of the speed — double the speed and you get four times the force.
For crystal hydrates the molar mass includes the water: copper sulfate pentahydrate weighs almost 250 grams per mole, while anhydrous copper sulfate weighs only 160. Taking the wrong formula, a technician is off by a factor of one and a half in concentration.
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Nine orders of magnitude separate the permitted trace of lead in drinking water from the water itself, and all of them are written in one unit. The top of the scale is curious: a litre of pure water is fifty-five and a half moles of water, while concentrated sulfuric acid at eighteen moles per litre is almost a pure substance: it contains only about two moles of water per litre, nine times less than the acid itself.
04 · Measuring instruments
volumetric flask · burette · centrifuge · spectrophotometerGlass, drop, rotor and beam
The narrow neck makes a millimetre of height correspond to a small fraction of the volume: for a 250-millilitre flask the class A tolerance is fifteen hundredths of a millilitre. The bottom of the meniscus is set on the mark, looking at it strictly level.
A solution of known molarity is delivered drop by drop from a burette until the indicator changes colour. The volume used, multiplied by that molarity, gives the number of moles in the sample — so the molarity of one solution measures another.
A precipitate is separated from the solution by centrifugal force, and protocols specify it not in revolutions but in g-force, because it also depends on the rotor radius. Here molarity meets revolutions per minute: the same suspension needs different speeds on different centrifuges.
The absorbance of a solution is proportional to its molarity as long as it is not too strong, so an instrument calibrated with a dilution series reads concentration straight from the light. It is the same law by which the colour deepens in the flask of block 01.
05 · Writing rules
substance in brackets · M in Latin · temperature for precise workConcentration of what — always written
Molarity is written with the formula of the substance in brackets, an upright Latin M or the unit mole per litre, and in precise work the temperature at which the solution was made up to the mark is added.
The first error does not name the substance, and the number cannot be converted into grams or into normality. The second writes a Cyrillic М that search engines and databases do not recognise. The third says “per litre” twice. The fourth forgets that sulfuric acid gives up two protons and its normality is twice its molarity. The fifth changes the capital to a lower-case letter and thereby molarity into molality, and the sixth puts a litre of solvent in the denominator instead of a litre of solution.
06 · Neighbouring units
mol/kg · N · mol/m³Molarity lives among three neighbours: molality, which does not fear temperature; normality, which counts not molecules but reacting particles; and the SI unit, in which the same quantity is a thousand times larger in number.
Amount of substance per kilogram of solvent: weighing does not depend on temperature, which is why thermochemistry and cryoscopy use exactly this.
Moles of equivalents per litre; IUPAC has long advised against it, but it still lives on the bottles of old-school titrants, and it is easy to confuse with molarity.
A centrifuge turns revolutions into g-force, and a sedimentation protocol specifies it in g; on the revolution sheet there is also the stroboscope used to check the rotor speed.
07 · Historical section
archive · how the mole became a numberFrom Avogadro’s hypothesis to an exact number
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The notation “one mole per litre” joins two very different things. Since 2019 the mole has been defined by an exact number of particles and is therefore flawless, while the litre in a flask rests on glass, on a mark and on the temperature at which it was measured. All the uncertainty of molarity lives in the denominator: in the purity of the weighed portion, in the class of the flask and in whether the solution had time to cool.
That is why standard solutions for checking analytical instruments are prepared not by volume but by mass, and certificates give the concentration in moles per kilogram. Molarity remains the working unit of everyday analysis: convenient, clear and honest, as long as the flask and the thermometer show the same twenty degrees.