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# Formulas for Solutions

Suppose that someone has already worked out the details, so all that you have to do is read a formula and make a solution. We can usually assume that a solution is to be aqueous unless stated otherwise. What about the concentration of the substance to be added? Common ways of describing the concentrations of solutions are weight-in-weight, weight-in-volume, volume-in-volume, and molarity. Less commonly used descriptions include normality and molality. These formulas all have one thing in common. A quantity of solute is measured out, mixed with solvent, and the volume is brought to some final quantity after the solute is completely dissolved. That is, solutions are typically prepared volumetrically. Because solutes add volume to a quantity of solvent, this method of preparation of solutions is necessary to ensure that an exact desired concentration is obtained.

There are exceptions, of course. For example, culture media for bacteria are typically made up by adding a measured amount of powdered medium to a measured volume of water. In such cases it isn't critical that a precise concentration be obtained, thus a weight-to-volume method is appropriate, instead of weight-in-volume.

## Weight/weight (w/w) solutions

Perhaps the easiest way to describe a solution is in terms of weight-in-weight (w/w). The weight of the solute relative to the weight of the final solution is described as a percentage. For example, suppose you have a dye that is soluble in alcohol. Rather than write the instructions, “take 3 grams dye and mix with 97 grams absolute alcohol,” you can describe the solutions simply as 3% dye in absolute alcohol. The formula applies to any volume of solution that might be required. Three grams dye plus 97 grams alcohol will have final weight of 100 grams, so the dye winds up being 3% of the final weight. Note that the final weight is not necessarily equal to the final volume.

Aqueous weight-in-weight solutions are the easiest to prepare. Since 1 milliliter of water weighs one gram, we can measure a volume instead of weighing the solvent. A very common use of w/w formulas is with media for the culture of bacteria. Such media come in granular or powdered form, often contain agar, and often require heat in order to dissolve the components. Microbiological media, especially when they contain agar, are difficult to transfer from one vessel to another without leaving material behind. They coat the surfaces of glassware, making quite a mess. Using a w/w formula the media and water can be mixed, heated, then sterilized, all in a single container. For example, tryptic soy agar, a very rich medium used for growing a variety of bacterial species, comes with instructions to simply mix 40 grams agar with one liter (equivalent to 1 kilogram) of deionized water, without adjusting the final volume. Very little material is wasted and there is less of a mess.

## Weight-in-volume (w/v) solutions

When we describe a concentration as a percentage without specifying the type of formula, we imply that the solution is to be made using the weight-in-volume (w/v) method. As with w/w, weight-in-volume is a simple type of formula for describing the preparation of a solution of solid material in a liquid solvent. This method can be used to describe any solution, but is commonly used for simple saline solutions and when the formula weight of the solute is unknown, variable, or irrelevant, which is often the case with complex dyes, enzymes or other proteins. Solutions that require materials from natural sources are often prepared w/v because the molecular formula of the substance is unknown and/or because the substance cannot be described by a single formula.

A one percent solution is defined as 1 gram of solute per 100 milliliters final volume. For example, 1 gram of sodium chloride, brought to a final volume of 100 ml with distilled water, is a 1% NaCl solution. To help recall the definition of a 1% solution, remember that one gram is the mass of one milliliter of water. The mass of a solute that is needed in order to make a 1% solution is 1% of the mass of pure water of the desired final volume. Examples of 100% solutions are 1000 grams in 1000 milliliters or 1 gram in 1 milliliter.

## Volume/volume (v/v) solutions

Volume-in-volume is another rather simple way of describing a solution. We simply describe the percent total volume contributed by the liquid solute. As with the other types of formulas used in biology, we assume that the solvent is water unless some other solvent is specified.

V/v is often used to describe alcohol solutions that are used for histology or for working with proteins and nucleic acids. For example, 70% ethanol is simply 70 parts pure ethanol mixed with water to make 100 parts total. To make a liter of such a solution we would start with 0.7 L absolute ethanol and bring the final volume to 1 liter with water. More often we might find ourselves with 95% alcohol. To make a 70% solution from a 95% stock solution requires a little more calculation. We will talk about that in a bit, when we discuss how to make dilutions.

Destaining of protein gels refers to the soaking of a stained gel in acidified alcohol so as to remove all dye that is not bound to proteins, revealing the bands. A useful destaining solution consists of 7% methanol, 10% acetic acid. This means using, per liter of final solution, 100 ml pure (or “glacial”) acetic acid and 70 ml methanol.

## Molarity

A disadvantage of describing formulas as w/v (%) is that the description says nothing about the actual concentration of molecules in solution. What if we want equal amounts of two chemicals to be mixed together, so that for each molecule of substance #1 there is a single molecule of substance #2? The same amount in grams will likely not contain the same number of molecules of each substance. Another disadvantage of the w/v method is that the same chemical can come in many forms, so that the same amount in grams of one form of the chemical contains a different amount of it than another form. For example, you may work with a chemical that can be in one of several forms of hydration. Calcium chloride can be purchased as a dry chemical in anhydrous form, so that what you weigh out is nearly all pure calcium chloride. On the other hand you may have a stock of dry chemical that is hydrated with seven water molecules per molecule of calcium chloride. The same mass of this chemical will contain fewer molecules of calcium chloride.

When we are interested in the actual concentration of molecules of a chemical in solution, it is better to have a universal measurement that works regardless of how the chemical is supplied. As long as the molecular weight (sometimes called formula weight) is known, we can describe a solution in the form of moles per liter, or simply molar (M).

## Working with formula weights

As with w/v solutions, we weigh out a specific amount of chemical when making a molar solution. Unlike w/v solutions, the amount to weigh depends on the molecular weight (m.w.) of the substance in grams per mole (g/mol). In order to calculate the desired mass of solute you will need to know the formula weight. Formula weights are usually printed on the label and identified by the abbreviation f.w. Formula weight is the mass of material in grams that contains one mole of substance, and may include inert materials and/or the mass of water molecules in the case of hydrated compounds. For pure compounds the formula weight is the molecular weight of the substance and may be identified as such.

For example, the molecular weight of calcium chloride is 111.0 grams per mole (g/mol), which is the same as the formula weight if the material is anhydrous. Calcium chloride dihydrate (CaCl2•2H2O) is 147.0 g/mol. For CaCl2•6H2O (hexahydrate) the formula weight is 219.1 g/mol.

A hydrated compound is a compound that is surrounded by water molecules that are held in place by hydrogen bonds. The water molecules in a hydrated compound become part of the solution when the material is dissolved. Thus, 111.0 grams of anhydrous CaCl2, 147.0 grams of dihydrated CaCl2, or 219.1 grams of CaCl2 hexahydrate in one liter final volume all give a 1 mole per liter solution, abbreviated 1M.

Suppose that you need one liter of a solution of 10 mM calcium chloride (10 millimolar, or 0.01 moles per liter), and suppose that you have only CaCl2 dihydrate. To make your 10 mM solution you weigh out 1/100 of the formula weight for dihydrated CaCl2, which is 0.01 x 147.0 = 1.47 grams and bring it to one liter.

## Complications With Formula Weights

Perhaps you cannot find a formula weight on a label or perhaps you are planning a protocol and do not have the actual chemicals on hand. You can calculate molecular weight from the chemical formula with the aid of a periodic table. You must keep in mind that when you purchase the chemical the formula weight may not be identical to the molecular weight. Suppose that you have already determined how much to weigh out based on the molecular weight, but the formula weight is greater due to hydration or the presence of inert material. Your remedy is simply to multiply your calculated mass by the ratio of formula weight to molecular weight (or simply recalculate the weight needed).

For example, suppose that you need 10 grams of pure CaCl2 (m.w. 111.0 g/mol), then discovered that all you have is the hexahydrated form (CaCl2•6H2O, f.w. 219.1 g/mol). Take 219.1 divided by 111.0 and multiply by 10. You need 19.7 grams of CaCl2•6H2O.

Materials are not always available in 100% pure form. The description on the label might indicate that the chemical is >99% pure. Such is often the case with enzymes or other proteins that must be purified from natural sources. Most of us do not worry about purity if it is above 99%. Greater precision might be important to analytical chemist, for example, but is seldom needed in biological applications. If there are significant impurities or if you insist on being as precise as you can, then calculate the amount of material you need and divide by the fraction representing purity of the substance. For example, if you need 10 grams of pure substance A but what you have is 95% pure, then divide 10 grams by 0.95 to get 10.5 gram (note that the result has been rounded to a reasonable level of precision).

Most chemicals tend to absorb water unless they are kept desiccated, that is to some extent they are hydroscopic. This problem should not be confused with the state of hydration of a substances, which refers to the direct association of water molecules with molecules of the substance through hydorgen bonding. Magnesium chloride is commonly used in biological buffers, and is notoriously hydroscopic. The formula weight does not include the added mass of water that is absorbed from the atmosphere, in fact the amount of contamination depends on how long and under what conditions the chemical has been shelved, especially with respect to humidity. It is usually not practical to worry about water content, since it is so difficult to control. If precision is critical, then chemicals should be maintained under desiccating conditions or used immediately before they can absorb a significant amount of water. 