Working Solution Calculator

The general dilution equation carries four terms because it has to work in every direction. This page only ever needs one of those directions, from a known stock towards a target in a known final volume, so the four terms collapse into two numbers: the volume of stock to add and the volume of diluent that goes with it. The stock volume is the amount of solute the target requires, which is target times final volume, divided by what each millilitre of stock already contains. Everything else on this page is that division and the subtraction that follows it.

The general form is still worth having open in a second window, because it is the version that catches a unit slip. When the four-term page returns a number and this one returns a different number for the same preparation, the difference is nearly always a volume entered as diluent when the calculation wanted a total. The C1V1 dilution calculator also accepts all four values as a consistency check, which makes it a cheap audit of a result already computed here. Two pages answering the same question in different forms is a feature of a reference page rather than an inconsistency, provided both are right, and a disagreement between them is worth investigating rather than averaging away.

The subtraction at the end is small enough to treat as an afterthought and large enough to matter. A stock at 25 mg/mL taken to 5 mg/mL in a final volume of 40 mL needs 8 mL of stock and 32 mL of diluent. Enter the final volume as the diluent volume, which is the commonest entry error on this page, and the finished volume becomes 48 mL. The solution looks perfectly clear and perfectly normal, and it holds 200 mg in 48 mL, which is 4.17 mg/mL rather than 5.

The size of that error is not constant. It is proportional to the stock volume, which is proportional to how close the target sits to the stock. When the target is a tenth of the stock, the stock volume is a tenth of the total and the slip is small. When the target is close to the stock, the stock volume approaches the whole final volume and the slip approaches halving the concentration. A 10 mg/mL stock taken to 9 mg/mL in 100 mL needs 90 mL of stock and 10 of diluent, and treating the hundred as the diluent gives 190 mL at about 4.7 mg/mL, which is nearly a factor of two out. Rounded figures make the same class of problem, and the guide to rounding in small volumes covers that case.

A working solution is made from the stock rather than from the solid because that moves the error to the one place where it is measured once. Weighing for every preparation pays the weighing error again each time, in whatever direction the balance happened to drift. Diluting leaves only the stock concentration uncertain, and every preparation inherits that same uncertainty in the same direction. The stock solution calculator fixes that single uncertain quantity, which is why it sits upstream of this page in a workflow. Where the solid is plentiful and the weighing is easy, none of that applies, and dissolving the material directly at the target is perfectly reasonable.

Across a chain of dilutions the errors accumulate rather than average out, because each fold is slightly wrong in its own way. If three successive dilutions carry relative errors of one, two and one percent, the relative error in the finished concentration is close to the sum of them, about four percent, not the average of them. That is arithmetic rather than a flaw in the method, and it is the argument for checking a chain against a single direct dilution whenever both are possible. The serial dilution calculator lays out the chain when more than one step is genuinely needed.

There is one combination of inputs this page refuses, and refusing it is correct. If the target concentration is higher than the stock concentration, no volume of diluent reaches it, because adding solvent only ever lowers a concentration. The stock volume the formula returns is larger than the final volume that was asked for, which would require a negative volume of diluent, and a negative volume is not a preparation. A negative diluent is the arithmetic signature of a request that cannot be met by dilution at all, which makes the refusal informative rather than inconvenient. The honest answer is a stronger stock, which is a different calculation and a different page.

The near-boundary cases are worth knowing because they produce results that look like answers. A target equal to the stock gives a diluent volume of zero, which is not a dilution at all but a relabelling of the stock. A target just below the stock gives a small diluent volume and a large stock volume, which is the fragile case described above. When the target sits that close, dissolving weighed material directly at the target is usually the more robust route. Where the strength has to come from two solutions rather than one, the solution mixing calculator carries both amounts explicitly.