10.3 Dilution, Concentration & Alligation

The Dilution Formula: C1V1 = C2V2

When you dilute a stock, the amount of active ingredient is conserved - only the volume changes. That conservation is written as:

C1 x V1 = C2 x V2

where C1 and V1 are the stock concentration and the volume of stock you must measure, and C2 and V2 are the target concentration and the final total volume. Concentrations must be in the same expression (both % w/v, or both mg/mL).

Worked Example 1: Diluting a 10% Stock to 2%

Question: How much 10% w/v stock is needed to prepare 500 mL of a 2% w/v solution?

1. Rearrange: V1 = (C2 x V2) / C1.
2. V1 = (2 x 500) / 10 = 1000 / 10 = 100 mL of stock.
3. Measure 100 mL of stock and add diluent up to a final volume of 500 mL (approximately 400 mL of diluent).

Worked Example 2: Diluting Concentrated Alcohol

Question: Prepare 1 litre of 70% v/v alcohol from 95% v/v alcohol.

1. V1 = (C2 x V2) / C1 = (70 x 1000) / 95 = 70,000 / 95 = 736.8 mL, practically 737 mL of 95% alcohol.
2. Add purified water to a final volume of 1000 mL.

Note the wording carefully: you do not "add 263 mL of water". Alcohol and water contract on mixing, so volumes are not strictly additive; compounding instructions always say make up to the final volume. The exam exploits this: an option reading "737 mL alcohol + 263 mL water" is a trap, the correct instruction is "737 mL alcohol, qs ad 1000 mL" (qs ad = a sufficient quantity to make).

Serial Dilution

A serial dilution repeats the same dilution step to reach very low concentrations accurately, because measuring 1 microlitre directly is impossible.

Worked Example 3: Three 1:10 Steps

Question: Dilute a 100 mg/mL stock to 0.1 mg/mL using serial 1:10 dilutions.

1. Step 1: 1 mL stock + 9 mL diluent = 10 mL at 100 / 10 = 10 mg/mL.
2. Step 2: 1 mL of step-1 solution + 9 mL diluent = 1 mg/mL.
3. Step 3: 1 mL of step-2 solution + 9 mL diluent = 0.1 mg/mL.

Each 1:10 step multiplies the concentration by 0.1; three steps give an overall 1:1000 dilution (100 to 0.1 mg/mL). On the exam, count the steps: n steps of 1:10 give a final dilution factor of 10^n.

Alligation Alternate: The Tic-Tac-Toe Grid

Alligation alternate finds the proportions in which two preparations of known strength must be mixed to obtain a third, intermediate strength. Draw the grid:

• Top-left box: the higher strength; bottom-left box: the lower strength; centre box: the desired strength.
• Subtract diagonally across the centre: parts of the higher strength = desired - lower; parts of the lower strength = higher - desired.

Worked Example 4: Mixing 10% and 2% to Make 5%

Question: Prepare 400 g of a 5% w/w ointment from 10% and 2% w/w ointments.

1. Grid: higher = 10 (top left), lower = 2 (bottom left), desired = 5 (centre).
2. Parts of 10% = 5 - 2 = 3 parts. Parts of 2% = 10 - 5 = 5 parts. Total = 8 parts.
3. For 400 g: 10% ointment = (3 / 8) x 400 = 150 g; 2% ointment = (5 / 8) x 400 = 250 g.
4. Check: drug contributed = (0.10 x 150) + (0.02 x 250) = 15 + 5 = 20 g; 20 g / 400 g = 5%. Correct.

The parts always attach crosswise: the difference (desired - lower) belongs to the higher-strength box. Attaching 3 parts to the 2% ointment is the standard exam error.

Trituration, Minimum Weighable Quantity and the Aliquot Method

Balances have a sensitivity requirement (SR) - the smallest weight change they can detect. Weighing at or near the SR gives unacceptable percentage error, so a minimum weighable quantity (MWQ) is defined:

MWQ = sensitivity requirement / acceptable error (as a decimal)

For a balance with SR 6 mg and an accepted error of 5%: MWQ = 6 / 0.05 = 120 mg. You may never weigh less than 120 mg on that balance.

When the required dose is below the MWQ, use the aliquot method:

Worked Example 5: Dispensing a 10 mg Dose (MWQ 120 mg)

1. The 10 mg dose is below the 120 mg MWQ, so weigh the MWQ instead: 120 mg of drug.
2. Dilute with an inert diluent (lactose) to a convenient multiple - here a 1:10 dilution: 120 mg drug + 1080 mg lactose = 1200 mg of mixture, using geometric dilution (triturate the drug with an approximately equal amount of lactose, then repeatedly add an amount of lactose equal to the growing mixture, mixing thoroughly each time).
3. Concentration: 1200 mg of mixture contains 120 mg of drug, so each 100 mg of mixture holds 10 mg of drug.
4. Weigh the aliquot: 100 mg of mixture delivers the required 10 mg dose.

A second example with the same balance: for a 15 mg dose, weigh 120 mg of drug, dilute with lactose to 960 mg (an 8-fold dilution, 12.5% drug), and take an aliquot of 120 mg of mixture: 120 x 0.125 = 15 mg of drug.

Reducing and Enlarging Formulas

To scale an extemporaneous master formula, compute the conversion factor = quantity required / quantity in the formula, then multiply every ingredient by it.

Worked Example 6: Scaling a Recipe Down

Question: A mixture formula for 100 mL contains 2 g of drug. Prepare 30 mL.

1. Factor = 30 / 100 = 0.3.
2. Drug needed = 2 g x 0.3 = 0.6 g (600 mg).
3. Multiply every other ingredient by 0.3 in the same way, then make up to 30 mL final volume.

Enlarging works identically (factor greater than 1) and is used to build in overage for transfer losses.

Common Traps

• Final volume vs added volume. "Dilute to 500 mL" is not "add 500 mL" - with alcohol the difference is real, and on the exam it is a deliberate distractor.
• Weight vs volume in alligation. Mixing ointments is w/w (weigh the parts); mixing solutions is w/v (measure or weigh, but keep units consistent across the grid).
• Units inside C1V1 = C2V2. Convert 10% w/v to 100 mg/mL before mixing with a mg/mL target, or the answer is 1000-fold out.
• Aliquot arithmetic. The aliquot is a weight of mixture, not of drug; always divide the required dose by the mixture's drug fraction.