# Video: Molar Concentrations

In this video, we will learn how to calculate the molar concentration of a solution when you’re given the amount of solvent and the mass or number of moles of dissolved solute.

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### Video Transcript

In this video, we will learn how to calculate the molar concentration of a solution when you’re given the amount of solvent and the mass or number of moles of the dissolved solute. Before we learn how to carry out these molar concentration calculations, we should probably understand why we’re doing this and also some of the key terms. The first key term to understand is concentration. Concentration is the amount of solute which is dissolved in a certain amount of solution. In order to understand what we mean by solute and solution or solvent, let’s look at making a cup of instant coffee.

The solute is the substance which you dissolve, in this case, instant coffee. The solvent is then the liquid which you dissolve your solute in, in this case, hot water. So the concentration of our cup of coffee is how much instant coffee or solute we’ve dissolved into our whole cup of coffee, which is our solution. Another term you might hear when discussing concentration is molarity. This is often written as a capital 𝑀. Molarity is defined as the number of moles in one liter of solution. We should also consider what happens to the concentration when we change the amount of solute or solvent. Again, our cup of coffee can help with this.

If we increase the amount of solute in our coffee, which is the instant coffee granules, what happens to the concentration of our whole cup of coffee? Of course, it gets stronger, which in chemistry terms means it becomes more concentrated. But what happens if we increase the amount of solvent instead? If we add more hot water to the same amount of coffee granules, then of course our cup of coffee is going to be weaker, as in a lower concentration. You can always return to our coffee cup analogy if you ever get stuck thinking what should be happening to the concentration if we change either the solute amount or the solvent.

One final term to discuss is molality. Generally, molarity is the most common way of talking about concentration. But if we have a situation where the temperature is changing, molarity doesn’t work. And this is where we use molality. The molality of a solution is the number of moles of solute divided by kilograms of solvent. And the symbol for molality is usually a lowercase italic 𝑚. But for this video, we’re going to use molarity.

The equation we need for calculating molar concentration is 𝑛 equals 𝑐𝑣, where 𝑛 is the number of moles, 𝑐 is the concentration, and 𝑣 is the volume. It’s also important to pay attention to the units of each of these, where 𝑛 is in moles, 𝑐 is in moles per liter, and 𝑣 is in liters. Let’s try using this on an example.

How many moles of copper sulfate in 2.5 liters of 0.15 molar solution?

So 𝑣 is 2.5 liters, and 𝑐 is 0.15 molar. Molar is simply another way to write moles per liter. And we’re being asked to find 𝑛, the number of moles. So using 𝑛 equals 𝑐𝑣, we get 0.15 molar multiplied by 2.5 liters, giving us an answer of 0.375 moles.

Now let’s look at an example where we need to rearrange our equation.

Calculate the volume in liters of 0.125 molar aqueous nitric acid containing 0.400 moles.

Here, we’re given the concentration 𝑐, remembering that molar means moles per liter. And we’re given 𝑛, the number of moles. We’re being asked to find 𝑣, the volume. So here we’re going to need to rearrange our equation, which gives us 𝑣 equals 𝑛 divided by 𝑐. Now we can substitute in our values, giving us 0.400 moles divided by 0.125 molar, giving us 3.2 liters. We can double-check our answer by doing some unit analysis.

In our working, we’ve done moles divided by moles per liter. The moles are going to cancel, and dividing by per liter is going to leave us with liters. And liters, of course, are the correct unit for our volume. So it looks like we’ve done this correctly.

Now let’s look at what we need to do if we’re not explicitly given the number of moles.

Calculate the concentration of 25 grams of sodium chloride dissolved in 0.34 liters of water.

Here, we’re being asked to calculate 𝑐, the concentration. And we’re given 𝑣, but we’re not actually given 𝑛, the number of moles. Instead, we’re given a mass, so we need to convert our mass into the number of moles. To do this, we’ll need another equation. The number of moles is equal to the mass divided by our molar mass, where our units are moles, grams, and grams per mole. So to work out the number of moles, we do 25 grams divided by 22.99 plus 35.45 grams per mole. We get these bottom numbers from our periodic table. They are the masses of sodium and chlorine, giving us the molar mass of sodium chloride. This gives us 25 grams divided by 58.44 grams per mole, which equals 0.4278 moles.

Now that we have 𝑛, we can use our 𝑛 equals 𝑐𝑣 equation. But let’s rearrange it in terms of 𝑐, giving us 𝑐 equals 𝑛 divided by 𝑣. Substituting in our values, we get 0.4278 moles from what we’ve just worked out and 0.34 liters given in the question. This works out as 1.258 moles per liter. Of course, we can round our answer to two significant figures so that it matches the level of accuracy in our question. And moles per liter is the same as molar, so we could give our answer as 1.3 molar.

Now that we know how to use our key equation, let’s have a look at what happens if we’re given different units. You may come across units which are not ideal for the calculation you’re performing. So let’s look at some common unit conversions that you might need. When talking about volume, you may come across the units of decimeters cubed. So far, we’ve been using liters when measuring volume. So how do we convert decimeters cubed into liters? As it turns out, it’s actually a really easy conversion. Decimeters cubed are equivalent to liters, so any volume given in decimeters cubed is the same as it is in liters.

Equally, a volume given in centimeters cubed is actually equivalent to the same volume in milliliters. So using centimeters cubed and decimeters cubed is fairly straightforward. But you may need to know how to convert milliliters into liters. This is, of course, the same conversion as converting centimeters cubed into decimeters cubed. There are 1000 milliliters in every liter. Milli- as a prefix means thousandth, and that tells you that there are 1000 milliliters in a liter or that one millimeter is one thousandth of a liter.

So you could think of converting milliliters to liters as dividing by 1000. You could also write this as multiplying your volume by one liter per 1000 milliliters. This gives you the same numerical value but is more useful if you’re performing unit analysis. This is because performing the calculation like this allows you to cancel the milliliters, leaving you with liters.

Another unit conversion you might need is converting milligrams to grams. Again, we have this milli- prefix, meaning thousandth, so there are 1000 milligrams in one gram. So calculating this conversion is very similar to that of milliliters to liters. So as an example, if we have 250 milligrams, we multiply by one gram per 1000 milligrams, giving us 0.25 grams. So let’s try doing some molar concentration calculations alongside unit conversions.

A 500-milligram tablet of paracetamol, C8H9NO2, was dissolved in 200 milliliters of water. What is the concentration of the resulting solution? Give your answer in units of moles per decimeter cubed and to two significant figures.

This question is asking us to work out the concentration of a solution. In order to do this, we’re going to need a key equation. Of course, the equation we need is 𝑛 equals 𝑐𝑣, where 𝑛 is the number of moles, 𝑐 is the concentration, and 𝑣 is the volume. We’re given the value of 𝑣, volume, in the question as 200 milliliters. However, the question specifically asks for our answer in units of moles per decimeter cubed. So we’re going to need to do some unit conversion. Remember that decimeters cubed are exactly the same as liters. So you can imagine giving our answer in units of moles per liter.

But the volume we’re given is in milliliters. So we’re going to need to convert our milliliters into liters. To convert milliliters to liters, we need to multiply by one liter per 1000 milliliters. You can remember that there are 1000 milliliters per liter by remembering that the prefix milli- means thousandth. Performing this conversion gives us 0.2 liters. As a side note, remember that milliliters are exactly the same as centimeters cubed. Now that we have our volume in the right units and we know that we’re working out the concentration, we just need to find the number of moles, 𝑛.

However, in the question, we’re not given the number of moles. Instead, we’re given a mass in milligrams. So we’re going to need to convert our mass into the number of moles. To do this, we need another key equation. The equation we need is moles equals the mass divided by molar mass, where our units are moles, grams, and grams per mole. So the first thing we need to do is convert our mass from milligrams into grams. Converting from milligrams to grams is very similar to converting milliliters to liters in that we multiply by one gram divided by 1000 milligrams. So converting 500 milligrams into grams, we multiply by one gram per 1000 milligrams, leaving us with 0.5 grams.

Now we can begin to work out the number of moles. From our second key equation, we know that the number of moles equals 0.5 grams divided by the molar mass of paracetamol. To work out our molar mass, we need the periodic table and the formula for paracetamol given in the question. We can see that it contains eight carbon atoms, nine hydrogen atoms, one nitrogen atom, and two oxygen atoms. So to get our molar mass, we add the masses of eight carbons, nine hydrogens, one nitrogen, and two oxygens altogether, which gives us a molar mass of 151.165 grams per mole.

We can now use this to work out the number of moles of our paracetamol. So 𝑛 equals 0.5 grams divided by 151.165 grams per mole, which gives us 0.0033076 moles. Now that we have both 𝑛 and 𝑣, we can use 𝑛 equals 𝑐𝑣 to give us the concentration. Rearranging our equation in terms of 𝑐, we get 𝑐 equals 𝑛 divided by 𝑣. By inputting our values, we get concentration equals 0.0033076 moles divided by 0.2 liters. This works out at 0.016538 moles per liter.

However, we’re not finished yet. The question asks for our answer to be given to two significant figures and with units of moles per decimeter cubed. Rounding to two significant figures gives us 0.017. And since liters are the same as decimeters cubed, we don’t need to do anything regarding units. We can simply write the units as moles per decimeter cubed. And this is our final answer.

Let’s review the key points. The key equation that we need for calculating concentration is 𝑛 equals 𝑐𝑣, where 𝑛 is the number of moles, 𝑐 is the concentration, and 𝑣 is volume ⁠— being careful to watch our units where the units should be moles, moles per liter, and liters. In some cases, unit conversions may be required, remembering that decimeters cubed are exactly equal to liters and centimeters cubed are exactly equal to milliliters.

To convert milliliters into liters, we multiply by one liter per 1000 milliliters. Similarly, to convert milligrams into grams, we multiply by one gram per 1000 milligrams. Molarity means the concentration in moles per liter, where moles per liter can also be written as a capital 𝑀, standing for molar.