9
Density, displacement volumes and displacement values
At 20°C, 1 mL of water weighs 1 g. Thus 1g of water occupies a volume of 1 mL and so 100 g of water occupies a volume of 100 mL and 1000 mL (1 litre) of water weighs 1000 g (1 kg). This 1 : 1 relationship is considered to hold, for most general pharmaceutical purposes, at the normal ambient temperatures. However, this 1 : 1 relationship between weight and volume is not true for other liquids, solids and semi-solids. For this reason, it is necessary to understand and be able to use the terms ‘density’, ‘displacement volume’ and ‘displacement values’ for some pharmaceutical calculations.
Density
The density of a liquid is the weight per unit volume and is expressed as grams per millilitre. In official references many liquids have their densities expressed as wt/mL, e.g. glycerol has a wt/mL of 1.255–1.26 g. Using the density value it is possible to convert a volume to a weight or a weight to a volume. The conversion to a weight or volume may be for the convenience of the formulator and enables a liquid to be measured as a volume or a weight, whichever is the easiest or most suitable. For example, in the case of a very mobile liquid it may be easier to measure by volume rather than have to weigh the liquid. Alternatively, if the liquid is very viscous, it may be easier to convert a volume to a weight and weigh the viscous liquid.
In the following example a volume is converted to a weight.
In the next example a weight is converted into a volume.
The ability to use density values is needed when a liquid (expressed as a volume) is one of the components of an ointment, cream or gel and the final quantity of the ointment, cream or gel is expressed as a weight. Consider the following example:
Displacement volumes involving solids in liquids
The definition of displacement volume is the quantity of solvent that will be displaced by a specified quantity of a solid during dissolution (i.e. the preparation of a solution). It is the space or volume of solvent that will be occupied by the solute after a solution has been produced. Knowledge of displacement volumes is required to calculate the volume of a solvent to add to a known amount of solid in order to obtain an exact volume of solution.
The following example may help to explain the concept of displacement volumes.
If we add 5 mL of water to 95 mL of water then we will have a total volume of 100 mL of water. However, if we add 5 g of salt to 95 mL of water and stir the mixture until the salt has dissolved, then we will have a total volume of solution above 95 mL, but it is unlikely to be exactly 100 mL. It may be less than 100 mL or it may be greater. We would need to measure the volume accurately to find out the exact volume. What has happened is that the 5 g of salt dissolves in the water and increases the total volume. In other words the salt will occupy a space and it will displace some of the water, such that the total volume of the solution will be more than 95 mL. This phenomenon by which the salt displaces some of the water and results in an increase in the volume of the resulting solution is important in pharmacy and has resulted in the determination and publication of displacement volumes.
An example of a drug with a known displacement volume is meticillin powder for reconstitution into an injection, which has a displacement volume of 0.7 mL/g. This means that 1 g of meticillin powder will displace (or occupy the space of) 0.7 mL, when it is dissolved in water.
The correct use of displacement volumes is essential when water for injection or another suitable diluent is added to lyophilised drug powder in order to produce an injection solution of a known concentration and volume. For example, the displacement volume of diamorphine is quoted as 0.06 mL/5 mg. This means that 5 mg of diamorphine will displace (or occupy the volume of) 0.06 mL of water. This information can be used to calculate the correct volume of water for injections to add to diamorphine powder to make a known volume of injection solution at a known concentration. This is shown in the following example.
The following example demonstrates the use of a displacement volume in the preparation of an injection.
Let us try another example, which involves determining the volume required to produce a correct concentration of the drug.
A table of displacement volumes of powder injections is published in The Pharmaceutical Codex. This table, however, gives only the displacement volume for one specific quantity of each drug. For example, in the case of diamorphine the displacement volume is that quoted above (0.06 mL/5 mg). Diamorphine is available as a powder for reconstitution in quantities greater than 5 mg. If the prescriber requires more than 5 mg, then these other quantities of powders will need to be used. In such a case it will be necessary to calculate a displacement volume for another quantity of diamorphine.
Displacement volumes may be used in calculations for different formulations. Many antibiotics are liable to hydrolysis when mixed with water. However, small children cannot take antibiotic tablets/capsules and so, rather than prescribe an antibiotic injection, prescribers may request an antibiotic mixture. Such an antibiotic mixture will have a short time span before the antibiotic is hydrolysed. Thus the pharmacist will prepare the mixture at the time the prescription is presented. Most commercial antibiotic mixture preparations consist of an antibiotic powder mix to which the pharmacist has to add a specific volume of water to produce an exact volume of mixture at a known concentration. This specified volume of water will depend on the manufacturer determining a displacement volume for the antibiotic powder mix (see Example 9.8).
Displacement values involving solids incorporated into other solids
Displacement values are used in calculations when solids are incorporated into another solid. For example, a solid may be mixed intimately with a molten solid, which on cooling forms a solid mix. In a similar way to solids dissolving in a liquid and occupying a volume, so any solid incorporated into another solid will also occupy a volume. Similar weights of different solids may occupy very dissimilar volumes. For example, 1 g of a heavy metal such as mercury will occupy much less volume than 1 g of a fluffy, crystalline powder such as benzoic acid. Thus, solids have densities and this concept of different bulk densities is used in calculations involving the preparation of suppositories and pessaries.
Calculations for suppositories and pessaries
Extemporaneously prepared suppositories and pessaries involve incorporating a drug or drugs into either a waxy or, less frequently, a glycogelatin base. The process involves the use of a mould, so that the size of the suppository is the same irrespective of which base or drug(s) is used in the preparation. The mould will contain a known volume and weight of the base (see below). In the preparation of suppositories the base is mixed with the drug. However, the base and drug may have different bulk densities. So a drug with a low bulk density will displace more of the base than a similar weight of a drug with a high bulk density. In order to ensure that each suppository contains the correct amount of the drug, the bulk densities of the drug have to be taken into account. For many drugs that would be incorporated into a suppository base, the weights of drug that would displace 1 g of the traditional suppository base (theobroma oil) have been calculated and published in The Pharmaceutical Codex. These values are referred to as displacement values.
For example, the displacement value of zinc oxide is 4.7. This means that 4.7 g of zinc oxide will displace (or occupy the space of) 1 g of theobroma oil (the suppository or pessary base). In other words, zinc oxide is much more dense than theobroma oil. Another example is menthol which has a displacement value of 0.7. This means that 0.7 g of menthol will displace 1 g of theobroma oil. In this case, menthol is less dense than the suppository base.
In order to calculate the amount of base to be added to the drug to provide the correct amount of drug in each suppository, the displacement value must be used – but note that a displacement value is used only when the drug is expressed as an actual quantity, e.g. x mg or y g. (Suppository and pessary moulds are produced in nominal sizes of 1 g, 2 g, etc. The nominal mould size refers to the quantity of theobroma oil that would occupy the mould. Before use a pharmacist should calibrate the mould by filling with theobroma oil and weighing the suppository produced. The weight of the suppository will indicate the actual size of the mould. If this is substantially different to the nominal value, then the actual value should be used in the calculations shown below.)
Waxy or oily bases
In the section above, theobroma oil was described as the traditional suppository base. Theobroma oil is a natural waxy material, which is solid at room temperature and melts at body temperature, i.e. when inserted into a body cavity. However, in recent years synthetic suppository bases have been developed that have almost replaced the use of theobroma oil. Most of the modern synthetic bases have similar density properties to theobroma oil and so the displacement values determined for drugs incorporated into theobroma oil can be used for the modern synthetic bases.
Displacement values have to be used to calculate the amount of suppository base required when a specified amount of a drug has to be incorporated into a suppository.
