Reconstitution Math: How to Work Out Solution Concentration

The one equation behind every solution

A concentration is an amount of substance divided by the volume of the solution that contains it. For a dry solid the amount is usually a mass, so the working equation is concentration = mass ÷ volume. Rearranged, volume = mass ÷ concentration, and mass = concentration × volume. Everything else in this guide is a refinement of those three lines.

Units are where most arithmetic slips happen, so it helps to fix a few equivalences first. One milligram per millilitre (1 mg/mL) is the same as one microgram per microlitre (1 µg/µL) and the same as one gram per litre (1 g/L). One milligram is 1,000 µg, and one millilitre is 1,000 µL.

• Concentration from mass and volume. Dissolving 5 mg of dry material in 2 mL of solvent gives 5 ÷ 2 = 2.5 mg/mL.
• Volume for a target concentration. To reach 1 mg/mL from 5 mg, the solvent volume is 5 ÷ 1 = 5 mL.
• Mass in a given volume. A solution at 2.5 mg/mL holds 2.5 × 0.2 = 0.5 mg of material in 0.2 mL.

Those figures are illustrations of the arithmetic, not recommendations about any material or any amount. The solution concentration calculator performs the same steps with numbers you type in.

Which mass goes into the equation?

The mass printed on a label is normally the mass of dry powder in the vial. That powder is not pure peptide. It also holds counter-ions left from synthesis and purification, and bound water, which together make the gross weight larger than the mass of peptide itself. The fraction of the powder that is peptide is the net peptide content.^[1]

Counter-ion content can be measured directly. Ion chromatography and capillary electrophoresis methods have been described for determining the counter-ions in synthetic peptides, which is one route to a net content figure.^[2] Amino acid analysis is another: it quantifies the peptide through its constituent amino acids.^[1]

A hypothetical example shows why this matters. If a powder had a net peptide content of 80%, then 10 mg of powder would hold 8 mg of peptide, and a solution made as if the whole 10 mg were peptide would be about 25% more concentrated on paper than it is in the vial. Unless a net content is known, a calculated concentration is a nominal one, and it is worth labelling it that way. The purity versus net peptide content guide explains the difference between this figure and chromatographic purity.

Volume: the number you add is not always the number you have

The arithmetic above assumes that the final volume equals the volume of solvent added. Dissolving a solid adds a little volume of its own, so the assumption becomes less exact as the amount of solid per millilitre rises. For dilute solutions the difference is small; for concentrated ones it is worth recording the assumption rather than leaving it implicit.

The same care applies to the diluent itself. The identity of the solvent, whether it carries a preservative and the volume actually delivered all belong in the record of how a solution was made. Bacteriostatic water versus sterile water covers the choice of diluent.

From mg/mL to molarity

Molarity counts molecules rather than mass, which is what matters in many experiments. The conversion needs the molecular weight in g/mol: molarity (mol/L) = concentration (g/L) ÷ molecular weight (g/mol). Because 1 mg/mL equals 1 g/L, a solution at 2.5 mg/mL of a compound with a molecular weight of 1,000 g/mol is 2.5 ÷ 1,000 = 0.0025 mol/L, or 2.5 mM.

Two cautions apply. First, use the molecular weight of the form you actually hold; a salt form weighs more than the free peptide, and the net content correction above still applies. Second, mass spectrometers report monoisotopic masses while calculations of solution molarity use the average mass; the two differ by an amount that grows with molecular size, because heavier molecules contain more of the naturally occurring heavier isotopes.^[3] The molarity calculator accepts any molecular weight you enter, and the peptide molecular weight estimator can add up a mass from a typed sequence.

Checking a concentration against the arithmetic

Calculated concentrations can be tested. Light absorbance follows the Beer–Lambert relationship, in which absorbance equals the molar absorption coefficient times concentration times path length, so a measured absorbance and a known coefficient give a concentration. For peptides and proteins that contain tryptophan, tyrosine or cystine, the coefficient at 280 nm can be estimated from the sequence.^{[4, 5]}

Many short peptides contain none of those residues and absorb little at 280 nm. For them, coefficients measured at 214 nm are available for the peptide bond and for each amino acid, which allows the 214 nm absorbance of a peptide to be predicted from its composition.^[6] In either case the estimate carries its own uncertainty, so agreement within the stated accuracy of the method is the realistic target. The extinction coefficient entry in the glossary defines the term.

Why a measured concentration can come out lower

If a measured concentration is lower than the calculated one, the arithmetic is only one candidate. Peptides stick to surfaces, and recoveries differ between glass, plastic and treated tubes in ways that are hard to predict for a given sequence.^[7] The effect is most visible in dilute solutions: in one study of a decapeptide, solutions at 0.2 to 4 µg/mL gave a poor and variable HPLC response because of loss to the vial surface, and the adsorption depended on both the dissolution medium and the type of vial.^[8] See adsorption in the glossary.

Incomplete dissolution and aggregation also lower the concentration of what remains in solution; those topics are covered in solubility and aggregation basics. And if the net peptide content was lower than assumed, the true concentration was lower from the start.

Dilutions and rounding

A working solution is often made from a stock by dilution, using C1 × V1 = C2 × V2: the concentration and volume of the stock taken equal the concentration and volume of the final solution. For example, taking 0.1 mL of a 2.5 mg/mL stock and making it up to 1 mL gives 2.5 × 0.1 ÷ 1 = 0.25 mg/mL. A serial dilution repeats this step, and every step carries the uncertainty of the one before it, so a long series is less certain than a single larger dilution to the same final value.

Carry full precision through the arithmetic and round only at the end, to no more significant figures than the least certain input supports. The dilution calculator lays out the transfer and solvent volumes for each tube of a series.

A record worth keeping

Most disputes about a concentration come down to a missing line in the record. These are the lines worth writing down every time:

1. The mass used. Whether it is the label mass or a balance reading, and the balance reading itself if there is one.
2. The net content assumed. A measured figure, a stated figure, or none, in which case the result is nominal.
3. The diluent. Its identity, whether it is preserved, and the volume added.
4. The calculation. The equation and the numbers, so that someone else can repeat it.
5. The result with its date. And any later check, such as an absorbance reading, against it.

The free printable concentration worksheet has blank fields for exactly these items.