1. Dilution, Concentration and Alligation
Dr. Osama A. A. Ahmed

2. Dilution, Concentration and Alligation
There are occasions in which the dilution or concentration of a pharmaceutical preparation is either desirable or required.
1st quantity X 1st concentration = 2nd quantity X 2nd concentration
Q1 X C1 = Q2 X C2
Example(1): if 500 ml of 15% v/v solution are diluted to 1500 ml, what will be the percentage strength (v/v)
500(ml) X 15(%) = 1500(ml) X x(%) X = 5%
Example(2): if 50 ml of 1: 20 w/v solution are diluted to 1000 ml, what is the ratio strength (w/v)?
1 : 20 = 5%
50(ml) X 5(%) = 1000(ml) X x(%) X = 0.25% = 1: 400
Dr. Osama A. A. Ahmed

3. Dilution, Concentration and Alligation
Stock solutions: are concentrated solutions of active (e.g. drug) or inactive (e.g. colorant) substances and are used by pharmacists as a convenience to prepare solutions of lesser concentrations.
Example(1): How many milliliters of a 1:400 w/v stock solution should be used to make 4 liters of a 1 : 2000 w/v solution?
1: = % and : 2000 = 0.05% and liters = 4000 ml
Q1 X C1 = Q2 X C2
4000(ml) X 0.05(%) = X (ml) X 0.25(%) X = 800 ml
Example(2): How many milliliters of a 1:50 stock solution of phenylephrine hydrochloride should be used in compounding the following prescription.
R/ phenylephrine HCl %
Rose water ad 30 ml
Sig. For the nose
1 : = 2 %
x X 2% = X % X = ml
Dr. Osama A. A. Ahmed

4. Dilution, Concentration and Alligation
Some calculations used in pharmacy in which the strength of a diluted portion of a solution is defined, but the strength of the concentrated stock solution used to prepare it must be determined.
e.g. when the pharmacist need to prepare and dispense a concentrated solution and direct the patient to use small volume (5 ml, teaspoonful) and dilute it in a specified volume of water.
Example(1): How much drug should be used in preparing 50 ml of a solution such that 5 ml diluted to 500 ml will yield a 1:1000 solution?
1 : 1000 solution g : 500 ml
0.5 g drug in 500 ml is the same amount found in 5 ml
X X = 5 g
Example(2): How many milliliters of water should be added to 300 ml of a 1:750 w/v solution of benzalkonium chloride to make a 1:2500 w/v solution? (Page 242)
X X = 0.4 g
X X = ml – 300 = 700 ml
Dr. Osama A. A. Ahmed

5. Dilution, Concentration and Alligation
Dilution of alcohol: when water and alcohol are mixed there is a physical contraction in the resultant volume.
Example: How much water should be mixed with 5000 ml 85% v/v alcohol to make 50% v/v alcohol?
Q1 X C1 = Q2 X C2
5000(ml) X 85(%) = X (ml) X 50(%) X = ml
so enough water is added to make 8500 ml.
Dilution of acids: the strength of concentrated acids (official) are expressed as % w/w, while diluted acids are expressed as w/v. Consider the specific gravity of conc. Acids in calculations
Example: How many mls of 37% w/w HCl having a specific gravity of 1.20 are required to make 1000 ml of dil HCl 10% w/v?
1000 X 0.1 = 100 gm of HCl 100% in 1000 ml of 10% w/v
100g X 100(%) = X (ml) X 37(%) X = 270 g of 37% acid
270/1.2 = 225 ml
Dr. Osama A. A. Ahmed

6. Dilution, Concentration and Alligation
Dilution and concentration of solids and semisolids:
Example: How many grams of opium containing 15 w/w of morphine and how many grams of lactose should be used to prepare 150 g of opium containing 10% w/w of morphine?
Q1 X C1 = Q2 X C2
X g X 15(%) = 150g X 10(%) X = 100 g of 15% opium
150 g g = 50 g of lactose
Example: How many grams of 20% benzocaine ointment and how many grams of ointment base (diluent) should be used in preparing 5 lb. of 2.5 % benzocaine ointment?
5 lb = 454 g X 5 = 2270 g
2270g X 2.5(%) = X g X 20(%) X = or 284 g
2270 – 284 = 1986 g of ointment base
Dr. Osama A. A. Ahmed

7. Dilution, Concentration and Alligation
Arithmatical method of solving problems that involves the mixing of solutions or mixtures of solids possessing different percentage strengths.
Alligation medial
A method by which the weighted average percentage strength of a mixture of two or more substances of known quantity and concentration may be easily calculated.
Example: what is the percentage (v/v) of alcohol in a mixture of 300 ml 40% (v/v) alcohol and 100 ml 60% (v/v) alcohol ?
Q1C1 + Q2C2 + Q3C3 +…….. = QtCt
Qt = = 400 ml
300 X X 60 = 400 X Ct
Ct = 18000/400 = 45 %
Dr. Osama A. A. Ahmed

8. Dilution, Concentration and Alligation
Alligation alternate
A method by which we may calculate the number of parts of two or more components of a given strength when they are mixed to prepare a mixture of desired strength. A final proportion permits us to translate relative parts to any specific denomination.
Example: in what proportion should alcohol 95% and 55% strengths be mixed to make 70% alcohol?
Available strength desired strength number of parts
95% parts (95%)
70%
55% parts ( 55%)
minus
gives
from
Dr. Osama A. A. Ahmed