Dilution, Concentration and Alligation

Strength and total quantity

When you dilute a preparation, you add more liquid (or more base). The amount of drug does not change, but it is spread through a larger quantity, so the strength falls.

Example. 5 g of drug in 100 mL is a 5% w/v solution. Water is added to make 200 mL.

• The drug is still 5 g.
• Strength = 5 ÷ 200 × 100 = 2.5% w/v

So strength and total quantity vary inversely: when one doubles, the other halves.

Diluting liquids from a concentrated stock

A stock solution is a concentrated solution kept to make weaker ones when they are needed. Because the amount of drug stays the same:

strength before × quantity before  =  strength after × quantity after
S₁ × V₁ = S₂ × V₂

To find the quantity of stock you need:

V₁ = (S₂ × V₂) ÷ S₁

Example 1. Make 500 mL of a 0.1% solution from a 5% stock.

• V₁ = (0.1 × 500) ÷ 5 = 10 mL of stock
• Measure 10 mL of the stock and add water to make 500 mL.

Example 2. Make 250 mL of a 4% solution from a 20% stock.

• V₁ = (4 × 250) ÷ 20 = 50 mL of stock
• Add water to make 250 mL.

Example 3: ratio strengths. Make 200 mL of a 1 in 1 000 solution from a 1 in 100 stock.

1. Change to percentages: 1 in 100 = 1%, and 1 in 1 000 = 0.1%.
2. V₁ = (0.1 × 200) ÷ 1 = 20 mL of stock.
3. Add water to make 200 mL.

The dilution factor

dilution factor = strength of stock ÷ strength wanted

The factor is also the total number of parts.

Example. Dilute a 5% solution to 1%.

• Dilution factor = 5 ÷ 1 = 5
• Total = 5 parts, made of 1 part stock + 4 parts diluent

Making up to volume

Always add water to the final volume, not the full amount of water. In Example 2, the water added is 200 mL (250 − 50), and this is only exact for liquids that mix without a change in volume. For other mixtures, measure the stock, then add water up to the final mark.

Diluting solids: trituration

Small amounts of a potent powder cannot be weighed accurately. To give a small dose accurately, the potent powder is mixed with a larger quantity of an inert diluent (such as lactose) to make a trituration, which is then weighed.

Example. Make 20 powders, each containing 5 mg of a potent drug. The smallest quantity that can be weighed accurately is 100 mg.

1. Make a 1 in 10 trituration: mix 100 mg of drug with 900 mg of diluent, giving 1 000 mg (1 g). This is 10% w/w, and 1 mg of the drug is in each 10 mg of trituration.
2. Each powder needs 5 mg of drug = 50 mg of trituration.
3. 20 powders need 20 × 50 = 1 000 mg = 1 g of trituration, which is all of it.
4. A 50 mg powder is below the 100 mg limit, so do not weigh the powders one by one at that weight. Mix the 1 g of trituration with 3 g more diluent to make 4 g, then divide it into 20 equal portions. Each portion weighs 4 ÷ 20 = 0.2 g (200 mg) and holds 100 ÷ 20 = 5 mg of drug.

Mix the potent drug into the diluent by geometric dilution: start with the potent powder, add an equal amount of diluent, mix, then add an amount equal to what is now in the mortar, and keep doubling until all the diluent has been added.

Compare this with the aliquot method for liquids, covered in the lesson on measurement accuracy.

Alligation

Alligation is a method for working with mixtures of different strengths.

Alligation medial: the strength of a mixture

This is the average strength of a mixture, weighted by the quantities.

strength of mixture = total drug ÷ total quantity × 100

Example 1. 200 g of a 5% ointment is mixed with 300 g of a 10% ointment.

• Drug in the first = 200 × 5 ÷ 100 = 10 g
• Drug in the second = 300 × 10 ÷ 100 = 30 g
• Total drug = 40 g in 500 g
• Strength = 40 ÷ 500 × 100 = 8%

Example 2. 100 mL of a 20% solution is mixed with 400 mL of a 5% solution.

• Total drug = (100 × 20 + 400 × 5) ÷ 100 = (2 000 + 2 000) ÷ 100 = 40 g in 500 mL
• Strength = 40 ÷ 500 × 100 = 8%

Alligation alternate: the proportions needed

This answers the question: in what proportions must I mix two strengths to get the strength I want?

1. Write the higher strength at the top left and the lower at the bottom left.
2. Write the wanted strength in the middle.
3. Subtract diagonally, always the smaller from the larger. Each answer goes opposite its strength, on the right.

Example 1. Make 20% from a 50% and a 10% preparation.

50 ─╲      ╱─ 20 − 10 = 10 parts of 50%
     ╲    ╱
       20
     ╱    ╲
10 ─╱      ╲─ 50 − 20 = 30 parts of 10%

• Ratio of 50% to 10% = 10 : 30 = 1 : 3
• For 1 000 g: 1/4 is 50%, which is 250 g, and 3/4 is 10%, which is 750 g.
• Check: (250 × 50 + 750 × 10) ÷ 1 000 = (12 500 + 7 500) ÷ 1 000 = 20% ✓

Example 2: diluting with water. Water has a strength of 0%. Make 1 000 mL of 70% v/v alcohol from 95% alcohol and water.

• Parts of 95% = 70 − 0 = 70
• Parts of water = 95 − 70 = 25
• Total parts = 95
• 95% alcohol = 70 ÷ 95 × 1 000 = 736.8 mL
• Water = 25 ÷ 95 × 1 000 = 263.2 mL

Alcohol and water shrink slightly when mixed, so measure the alcohol first and add water up to 1 000 mL.

Example 3: making a weaker ointment. Make 500 g of a 10% ointment from a 20% ointment and plain base (0%).

• Parts of 20% = 10 − 0 = 10
• Parts of base = 20 − 10 = 10
• Ratio = 1 : 1, so use 250 g of the 20% ointment and 250 g of base.

Remember: alligation alternate only works when the wanted strength is between the two strengths. You cannot make 60% by mixing 50% and 10%.