### 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.