In the breakdown of matter (section 1.1), a homogeneous mixture was shown to be know as a solution. It is important to know the components of a solution and the names of each component in a general sense. Although there are a myriad of various solution types, we will focus on the most important one for us which is the liquid solution. So here's a pretty good brief definition:

A solution consists of a major component known as the solvent which is the liquid in which you will dissolve the solute (the minor component).

This is a general (broad strokes) definition here. But the general rule is that you measure out a specific amount of solute (the stuff you dissolve), and put that into a specific amount of the solvent. The amount of solute that you mix into the solvent will determine the concentration of the solution (see next section with many concentration terms). Also remember, you must have the solute completely dissolve and make a homogeneous mixture (same composition throughout). The diagram below illustrates the idea of solute + solvent = solution.


When determining concentration, you must know the exact amount of solute in either grams (mass) or moles (amount). You must know the amount of solvent OR the final amount of solution. This can be mass or moles or even volume. Typically with volumes and solutions, you measure the volume of the solution which you can only do properly after you have mixed the solution.

Molarity, [A]

By far the most common concentration term for a chemist. It is the ratio of moles of solute to the overall volume of the solution in liters. It is usually depicted or symbolized as the dissolved chemical species set in square brackets, [A].

\[{\rm [A]} = {\rm \;molarity\;of\;A} = {{\rm mol\;A\;(solute)}\over{\rm L\;solution}}\]

And although IUPAC recommends that the units for molarity be explicitly shown (mol/L), it is still common practice to use the abbreviation of a capital "M" for molarity. So an example of this would be a 0.25 M HCl solution or even say a solution of 1.5 × 10-5 M NO3 ions. When you read that term you say "molar". So if you were using that 0.25 M HCl solution you would say "I'm using a 0.25 molar solution of hydrochloric acid".

Molality, mA

The solute is measured in moles (just like molarity), but the denominator is the mass of the solvent (not solution) in kg. We will show molality as a lowercase italicized m with a subscript showing the species. Molality of A is mA.

\[m_{\rm A} = {\rm molality\;of\;A} = {{\rm mol\;A\;(solute)}\over{\rm kg\;solvent}}\]

Molality is a more robust unit of concentration than molarity since the number of moles and mass don't change with conditions (temperature and pressure). Molality is used less than molarity for concentrations - although there are measurements that depend on it such as some colligative properties like boiling point elevation and freezing point depression. Also, molality might be symbolized with a lowercase b in some books - which is actually the IUPAC recommendation.

A pleasant surprise for dilute aqueous solutions: The density of water (1 g/ml) means that 1 L of water has a mass of 1 kg. Therefore, molarity and molality have the same approximate values for relatively dilute aqueous solutions (at room temperature).

Mole Fraction, XA

Another convenient conc term to use that is just the ratio of the moles of measured component (be it solute or solvent - it doesn't matter) to ALL the moles (total). We typically use an x with a subscript to denote mole fraction. So \(x_{\rm A}\) is the mole fraction of A. Some folks prefer the greek version of x and use a chi like this, \(\chi_{\rm A}\).

Consider the mole fraction of A in a mixture of A, B, and C.

\[{\rm mole\;fraction\;of\;A} = x_{\rm A} = {n_{\rm A}\over {n_{\rm B} + n_{\rm C} + n_{\rm A}}}\]

Where \(n\) is the number of moles in each component. Note that mole fraction is a fraction and is a number between 0 and 1.

Other measures of concentration

There are a wide array of other measures of concentrations of mixtures. Here are a few examples.

Percent Concentration (by mass, %m)

Also known as mass percent, percent concentration is a very common concentration type and is the number one concentration term for most "non-science" concentration terms. It is simply the mass of the solute divided by the mass of the solution, and then that fraction multiplied by 100%.

\[\%{\rm conc} = {{\rm mass\;of\;solute}\over{\rm mass\;of\;solution}}\times 100\% \]

Important Note: Always assume percentages are based on masses unless told otherwise or you are in a field of study known for other types of percentages. For example, when dealing with gases, most all percentages given are in moles of gas which means that percent is really just mole fraction times 100. If you are making consumable alcoholic beverages like a brewery, you will calculate alcohol percent by volume. So when using %vol you will use a volume unit of your solute and divide by the volume unit of your solution. That is what is on beer labels. They will have something like 5.0% ABV, which stands for alcohol by volume.

Parts-per-million (ppm)

Remember that ppm is parts-per-million and is just like percentage except that it is based on 1 million parts (not 100). So the only difference is you ratio the mass of the solute to the mass of solution but multiply the fraction by 106 instead of 100. We typically go to ppm when our percentages get too small... like instead of saying 0.053% you could say 530 ppm, a "friendlier" number to most humans.

\[{\rm ppm} = {{\rm mass\;of\;solute}\over{\rm mass\;of\;solution}}\times 10^6 \]

Also consider this: In dilute aqueous solutions, the specific gravity (or density) is approximately 1 and therefore 1 ppm = 1 mg/L. So you can CHANGE ppm to mg/L for most all our aqueous calculations. Once you have mg/L, it is much easier to convert out to other more useful concentrations like molarity (mol/L) which is needed for all aqueous equilibrium calculations.

Realize that you can keep going on the multiplier of the mass ratios. If you multiply by 109 or a billion, then you'll have parts-per-billion (ppb). If you go further to multiply by 1012 or a trillion, then that will be parts-per-trillion - although, be careful on the abbreviation for that ppt can be parts-per-trillion or parts-per-thousand. Make sure you know what the author of the unit meant. I will say though, parts-per-thousand is rare, almost nobody uses it.

Let's Get REAL here...

Please know that for ALL of the above concentration terms, there IS a limit on just how much you can dissolve into a solvent. That limit is known as the solubility of the solute. The solubility of a solute will limit the top end of just how concentrated a solution you can make of a given solute. Some solutes are extremely soluble in water - like table salt (NaCl) and even more so, sugar (sucrose, C12H22O11). Table salt has a limit of about 360g per 1000g of water which amounts to about 5.5 M or 27% concentration (those are extremely high concetrations BTW). Want another reference point? How 'bout a Coca Cola? There are 39 g of sugar (sucrose) in a 12 oz can. That amounts to about 0.32 M sugar. And that is very very sweet! So 0.3 M sugar is a very sweet drink.

Sugar and salt are more of a rarity though in terms of solubility. Most other substances have far lower solubility limits. So if you ever see really large molarities like 50 M, 100 M and beyond... they are going to be bogus. Think about water itself. One liter of water is 1000 g of water which is 55.6 moles of water. So if you count every single molecule of water itself in pure water, your looking at 55.6 M water - and that is the solvent!

As a good rule of thumb, think of 1 M as "concentrated". Think of 5% concentration as concentrated as well. A 1 M solution of hydrochloric acid has a pH of ZERO which is extremely acidic and it would definitely hurt like hell to get it on your skin and not wash it off quickly. The same goes for 5% HCl which is even stronger (1.4 M to be exact). So if 1 M is concentrated, what is like a "normal" concentration? How 'bout 0.1 M or 1 % solutions? Those are the high end of "normal" concentrations. We also call those normal concentrations "reasonable concentrations". It's the Goldilocks "just right" spot - not too high, not too low. Speaking of low, what is the low end? Well, once you reach about 0.01 M and lower, you can say it's a "dilute" solution... and to continue, once your down to like 0.0001 M you can say "very dilute" solution.

So there you go, PLEASE get an idea of the range of what we call "reasonable concentrations". It is always a good idea to know the range of reasonable values for any measured quantity.

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