Dilution, Concentration and Alligation

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)

Q1 X C1 = Q2 X C2

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%

Q1 X C1 = Q2 X C2

50(ml) X 5(%) = 1000(ml) X x(%) X = 0.25% = 1: 400

Altering Product Strength

•  If a mixture of a given strength is diluted to twice its original quantity, its strength will be reduced by one half because the active ingredient will be contained in twice as many parts of the whole

•  5 grams in 100 mL = 5% •  5 grams in 200 mL = ?%

Dilution and Concentration

•  If a mixture is concentrated to one half its original quantity, the active ingredient will be contained in one-half as many parts of the whole and the strength will be doubled.

•  5 grams in 100 mL = 5% •  5 grams in 50 mL = ?% •  The percentage or ratio strength decreases as the quantity of

solution increases (active ingredient remains constant) •  The converse is also true

•  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?

•  1:400 = 0.25% and 1: 2000 = 0.05% and 4 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 0.25%

•  Rose water ad 30 ml

•  Sig. For the nose

•  1 : 50 = 2 %

•  x X 2% = 30 X 0.25% X = 3.75 ml

Dilutions and Concentrations

•  Pharmaceutical Preparations may be diluted: –  by addition of diluent/solvent –  by admixture with solutions of lower strength

•  Pharmaceutical Preparations may be concentrated: –  By the addition of active ingredient or –  By admixture with solutions of greater strength or –  By evaporation of the diluent/solvent

•  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 0.5 g : 500 ml •  0.5 g drug in 500 ml is the same amount found in 5 ml •  0.5 5 •  X 50 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? •  1 750 •  X 300 X = 0.4 g •  1 2500 •  0.4 X X = 1000 ml 1000 – 300 = 700 ml

Problem1

•  If a 0.067% (w/v) methylbenethonium chloride lotion is diluted with an equal volume of water, what will be the ratio strength (w/v) of the dilution?

Problem 2

•  What is the strength of a sodium chloride solution obtained by evaporating 800 g of a 10% (w/w) solution to 250 g?

Problem 3

•  How many milliliters of water for injection must be added to 10 liters of a 50%(w/v) dextrose injection to reduce the concentration to 30% (w/v)

Problem 4 •  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?

Problem 5

•  How many milliliters of a 1:50 (w/v) stock solution of a chemical should be used to prepare 1 liter of a 1:4000 (w/v) solution?

Problem 6

•  If a syrup containing 65% (w/v) of sucrose is evaporated to 85% of its volume, what percent (w/v) of sucrose will it contain?

How much silver nitrate should be used in preparing 50 mL of a solution such that 5 mL diluted to 500 mL will yield a 1:1000 solution?

Problem 7

•  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 = 8500 ml

so enough water is added to make 8500 ml.

Dilution of Acids

•  The strength of an official concentrated acid is expressed as percentage weight-in-weight, e.g., HCl, NF contains 36.5% w/w

•  The strength of an official diluted acid is expressed as percentage weight-in-volume, e.g., Dil HCl, NF contains 9.5% w/v

•  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

Q1 X C1 = Q2 X C2

100g X 100(%) = X (ml) X 37(%) X = 270 g of 37% acid

270/1.2 = 225 ml

Problem 8

•  How many milliliters of 37% (w/w) hydrochloric acid having a specific gravity of 1.20 are required to make 1000 mL of diluted hydrochloric acid 10% (w/v)?

•  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 - 100g = 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

Q1 X C1 = Q2 X C2

2270g X 2.5(%) = X g X 20(%) X = 283.75 or 284 g

2270 – 284 = 1986 g of ointment base

Problem 9

•  If some moist crude drug contains 7.2% of active ingredient and 21.6% of water, what will be the percentage (w/w) of active ingredient after the drug is dried?

•  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 = 300+100 = 400 ml 300 X 40 + 100 X 60 = 400 X Ct

Ct = 18000/400 = 45 %

Dilution, Concentration and Alligation 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% 15 parts (95%)

•  70%

•  55% 25 parts ( 55%)

Problem 10

•  In what proportion should 20% benzocaine ointment be mixed with an ointment base to produce a 2.5% benzocaine ointment?

Problem 11 •  A hospital pharmacist wants to use three lots of ichthammol

ointment containing, respectively, 50%, 20%, and 5% ichthammol. In what proportion should they be mixed to prepare a 10% ichthammol ointment?

Problem 12

•  How many milliliters of water should be added to 100 mL of a 1:125 (w/v) solution to make a solution such that 25 mL diluted to 100 mL will yield a 1:4000 dilution?

Problem 13

•  The formula for a buffer solution contains 1.24% (w/v) of boric acid. How many milliliters of a 5% (w/v) boric acid solution should be used to obtain the boric acid needed in preparing 1 liter of the buffer solution?

Problem 14

•  If a pharmacist adds 3 g of hydrocortisone to 60 g of a 5% (w/w) hydrocortisone cream, what is the final percentage strength of hydrocortisone in the product?

Problem 15

•  Rx Castor Oil 5.0 mL Euresol 15.0 mL Alcohol 85% ad 240.0 mL

How many milliliters each of 95% (v/v) alcohol and water should be used in preparing the prescription?

Problem 16

•  Phenobarbital elixir contains 400 mg of phenobarbital in each 100 mL. A physician wants to increase the phenobarbital content to 30 mg per teaspoonsful and to prescribe the dose twice a day for 24 days. How many milligrams of phenobarbital should be added to the prescribed volume of the prescription to provide the amount needed for the dosage regimen?

Problem 17

•  How many grams of hydrocortisone should be used in preparing 2 lb of a ¼% hydrocortisone cream?

Problem 18

•  How many grams of silver nitrate should be used in preparing 500 mL of a solution such that 10 mL diluted to a liter will yield a 1:5000 solution?

Case #1 •  A pharmacist received a prescription for 100 mL of a cefuroxime

axetil suspension to contain 300 mg of drug in each 5 mL. The pharmacist has 100 mL of a suspension containing 250 mg/5 mL and also has 250-mg scored tablets of the drug. How many tablets should be pulverized and added to the suspension to achieve the desired strength? Assume no increase in the volume of the suspension.

A second Look •  The pharmacist observed that after adding the pulverized tablets,

the suspension measured 102 mL in volume. Calculations revealed that rather than the prescribed drug strength of 300 mg/5 mL, there were 294.1 mg/5 mL. What should the pharmacist do to bring the suspension to the desired strength?

Case #2

•  Rx Hydrocortisone 0.6% AQUAPHOR q.s. ad 15 g Sig: Apply to child’s affected area t.i.d.

The pharmacist has no hydrocortisone powder but does have a hydrocortisone cream, 1%. How many grams each of hydrocortisone

cream and Aquaphor should be used in filling the prescription?

References:

•  Pharmaceutical Calculations, 13th edition by Howard C. Ansel.

•  Dr. Mohammad Habib’s lecture notes.