Concentration and dilution arithmetic

Most preparation errors are not technique failures. They are arithmetic done quickly, in a notebook, three weeks before anyone notices the numbers do not reconcile. The arithmetic itself is simple; the discipline is in writing down what you did.

Concentration

Concentration is mass divided by volume. With the mass stated on the vial and the volume of diluent you added:

concentration = mass in vial ÷ volume of diluent

10 mg ÷ 2 mL  = 5 mg/mL
10 mg ÷ 5 mL  = 2 mg/mL
 5 mg ÷ 2 mL  = 2.5 mg/mL

This treats the volume of diluent as the final volume, ignoring the small contribution of the dissolved solid. At milligram masses in millilitre volumes the error is negligible. Where it is not, make the solution up to a marked volume rather than adding one.

Units

Two conversions cover almost everything, and one of them is a trap worth internalising.

Relationship	Note
1 mg = 1000 µg	Straightforward.
1 mL = 1000 µL	Straightforward.
1 mg/mL = 1 µg/µL	The trap. Both terms scale by 1000, so the ratio is unchanged. A concentration in mg/mL is already in µg/µL.

So 5 mg/mL is 5 µg/µL. People who convert both halves and then divide by 1000 again arrive at a figure a thousandfold out, and it is a large enough error that it usually announces itself. The dangerous errors are the small ones.

Dilution

To dilute a stock to a lower concentration, use:

C₁V₁ = C₂V₂

C₁ = concentration of the stock
V₁ = volume of stock required
C₂ = concentration you want
V₂ = final volume you want

Rearranged for the quantity you normally need, which is how much stock to take:

V₁ = (C₂ × V₂) ÷ C₁

Worked example
  stock            5 mg/mL
  wanted           0.5 mg/mL, 1 mL of it

  V₁ = (0.5 × 1) ÷ 5 = 0.1 mL of stock
  diluent to add   1 mL − 0.1 mL = 0.9 mL

Note the final step. The diluent volume is the final volume minus the stock volume, not the final volume itself. Adding 1 mL of diluent to 0.1 mL of stock gives 1.1 mL at roughly 0.45 mg/mL — a 10% error, small enough to survive unnoticed and large enough to matter.

Serial dilution

Where a wide range of concentrations is needed, dilute in steps rather than preparing each from the stock. Transferring 0.1 mL into 0.9 mL gives a tenfold dilution; repeating it from each new tube walks the concentration down a decade at a time.

The cost is that errors compound. A 5% pipetting error at each of four steps is a 20% error at the end, all of it in the same direction if the pipette is miscalibrated rather than merely imprecise. Mix each tube before drawing from it, change tips between steps, and where accuracy at the bottom of the range matters, prepare that point directly from the stock instead.

Errors worth checking for

• Taking the vial’s stated mass as exact. Fill tolerances exist, and the certificate of analysis is the authority on content — see how to read a certificate of analysis.
• Confusing peptide content with purity. A 98% pure preparation is not 98% peptide by mass; salts and residual water are typically not represented in that figure.
• Adding the final volume of diluent instead of the difference. As above.
• Recording the volume added but not the concentration produced. Obvious on the day, guesswork later.
• Pipetting below a pipette’s reliable range. Below roughly 10% of nominal volume, accuracy falls off sharply. Dilute further and take a larger volume.

Preparation technique is covered in reconstituting lyophilised peptides.