### Video Transcript

In this video, we will learn how to
express and calculate the percentage concentration of a solution by its volume or
mass ratio. Let’s introduce this topic with a
simple story.

Two friends go shopping to buy
mulch for the garden and feathers to fill up some homemade pillows. They place one large bag of each in
the shopping cart, nearly filling it. One friend says, “The shopping cart
is made up of 50 percent mulch and 50 percent feathers because they each take up
half of the space in the shopping cart.” The other friend, who’s pushing the
cart, says, “I think the shopping cart is 95 percent mulch and five percent feathers
because there are many more kilos of mulch in the cart than kilos of feathers.” Which friend is correct?

Well, they both are. These friends are calculating the
percentage concentration of the shopping cart. The first friend is calculating the
percentage by volume. The second friend is calculating
the percentage concentration by mass. In chemistry, we don’t normally
deal with mulch and feathers. We usually apply percentage
concentration to the amount of solute present in a solution.

One household example of percentage
concentration can be found on a bottle of hand sanitizer. If we look on the back of a bottle
of hand sanitizer, we may see it say, “Ethyl alcohol, 70 percent v/v.” What does this mean? Well, the v/v means that this
percentage concentration is by volume. That means that out of the total
volume of hand sanitizer in the bottle, 70 percent of the volume is ethyl
alcohol.

For example, a small bottle of hand
sanitizer that you might see in a bathroom or on a desk would be about 240
milliliters of hand sanitizer. The volume of the bottle will be
indicated on the label. Since the solution is 70 percent
ethyl alcohol by volume, making this solution requires an amount of ethyl alcohol
equal to 70 percent of the total volume of the bottle. 70 percent of 240 is 168. So the company used 168 milliliters
of ethyl alcohol to make this bottle of hand sanitizer.

As a reminder, to find 70 percent
of 240, we can take the decimal form of 70 percent, 0.7, and multiply it by 240 to
get 168. The remaining 72 milliliters, the
part of the hand sanitizer that isn’t ethyl alcohol, is made up of inactive
ingredients. These include fillers, binders, and
fragrances.

Another slightly different example
of percentage concentration can be found on a tube of anti-itch cream. Anti-itch cream is used for bug
bites, rashes, and other itches. Its active ingredient is
hydrocortisone, a steroid.

If we look on the back of a tube of
anti-itch cream, we might see “Hydrocortisone, one percent m/m.” m/m means that this
percentage concentration is by mass. In other words, one percent of the
mass of the cream in the tube is hydrocortisone. A small tube of anti-itch cream,
about the size of a dry erase marker, might have a mass of 30 grams. One percent of this total mass is
hydrocortisone. In this case, one percent of 30
grams is 0.3 grams. So there’s 0.3 grams of
hydrocortisone in the tube.

To do this calculation on a
calculator, you could type in 0.01 times 30 equals 0.3. The remaining 29.7 grams are
inactive ingredients. For the most part, these inactive
ingredients give the cream its creamy texture, allowing the hydrocortisone to be
applied to the skin.

As we can see here, the math
involved in these two situations is basically the same. The main difference is whether we
use units for volume or units for mass. We will learn more about
calculations surrounding percentage concentration in a moment. But first, let’s define a couple
key terms.

Very generally, we talk about the
percentage concentration of a solute in a solution. As a simple example, let’s consider
sugar dissolved in water as we define these terms. The solute is the substance that is
dissolved. In this case, the solute is the
sugar. The solute is often a solid being
added to a liquid. But it could also be a liquid or a
gas being added as well. The solvent is the substance that
does the dissolving. In this case, water is the
solvent. And finally, the solution is the
mixture of the solute dissolved in the solvent. The mixture of the sugar dissolved
in the water forms a sugar–water solution.

When we talk about the percentage
concentration of a solute in a solution, it’s important to know what the solute is
and what the solution is. But solute and solvent can be easy
words to mix up. So here’s a handy mnemonic
device.

Let’s imagine for a moment a
burglar carrying a bag of stolen goods. As the police arrive, he may need
to hide his bag in order to escape. Lucky for him, he finds an air duct
nearby. So the robber’s loot goes in the
vent. Similarly, in a solution, the
solute goes in the solvent.

When we know the definitions of
these words, it becomes easier to discuss the calculations involved with percentage
concentration. Let’s take a look at some of the
calculations we use when talking about percentage concentration. The formula that we’ll use as a
base for our other formulas is the percentage concentration by volume equals the
solute volume divided by the solution’s volume.

We also might see the solution
volume referred to as the total volume. As a real-world example, if there
are 300 milliliters of alcohol in a 500-milliliter solution of hand sanitizer, we
can calculate an answer of 0.6, which means the hand sanitizer is 60 percent alcohol
by volume.

Note that there are three variables
in this equation: the percentage concentration, the solute volume, and the solution
volume. In this example, we used the known
values of the solute volume and solution volume to calculate the unknown percentage
concentration. But in other questions, if we know
any two of these three values, we can use them to solve for the third unknown
value. For example, if we’d like to solve
for an unknown volume of the solute in a given percent concentration and total
volume, we can use the formula solute volume equals percent concentration times
solution volume.

We could use this formula to answer
a question like how much alcohol would be in a 700-milliliter solution of our 60
percent alcohol solution. Following the formula, 60 percent
times 700 milliliters will give us our answer. That answer is 420 milliliters of
alcohol.

The last formula is solution volume
equals solute volume divided by percentage concentration. For example, if we wanted to take
1300 milliliters of alcohol and make a 65 percent solution out of those 1300
milliliters, we could take 1300 and divide by 65 percent or 0.65. Our answer is a 2000-milliliter or
a two-liter solution. Note that this 2000 milliliters is
the volume of the entire solution, including the 1300 milliliters of alcohol that we
started with.

If the question asked us, “How much
solvent do we add to the alcohol to get this solution?,” our final answer wouldn’t
be 2000. It would instead be the 700
milliliters of solvent we need to add to the 1300 milliliters of alcohol to reach
2000 milliliters in the total solution.

So to recap, we can use this
formula, arranged three different ways, to solve for the percentage concentration,
the solute volume, or the solution volume when we’re given the other two
variables.

Thankfully, the calculations by
mass are extremely similar to the calculations by volume. Instead of using the solute volume
and solution volume, we use the solute mass and solution mass. And instead of our units being in
volume, like milliliters, they’ll be mass units, like grams. For example, if we use 33 grams of
sugar to make a 330-gram lemonade solution, we can calculate an answer of 0.1 or 10
percent sugar by mass.

Replacing volume with mass once
again, we get a second formula: solute mass equals percentage concentration times
solution mass. If we know the solution mass and
the percent concentration, say, a 250-gram lemonade solution with five percent
sugar, we can calculate the solute mass. Five percent of 250 is 12.5. So our answer means that there’s
12.5 grams of sugar in this solution.

The last formula is solution mass
equals solute mass divided by percentage concentration. For this question, let’s say we
wanna take 54 grams of sugar and make a six percent lemonade solution out of it. What would be the mass of the final
solution?

We can take 54 grams of sugar and
divide it by six percent. Our answer is a 900-gram
solution. As a reminder, the 900-gram
solution includes the initial 54 grams of sugar. So to make this solution, we would
need to add 846 grams of solvent to end up with a solution that’s 900 grams in
mass.

As we can see here, the
calculations related to percentage concentration are essentially the same when
calculating by volume or by mass. We simply need to be mindful of the
labels and units we use.

Another kind of problem that we can
solve that has to do with percentage concentration is calculating the percentage
concentration of the new solution that arrives when we mix two solutions
together. For example, we could start with
two solutions of hand sanitizer: one a 250-milliliter solution that’s 70 percent
alcohol by volume and the other a 300-milliliter solution of hand sanitizer that’s
60 percent alcohol by volume.

And we may be interested in knowing
what is the percentage concentration of the mixture when we add these two solutions
together. Let’s use a variable 𝑥 to
represent the percentage concentration of the mixture. Let’s also refer to the two
solutions added to the mixture as solution A and solution B.

Using our formula to calculate
percentage concentration, we know that the percentage concentration by volume of any
solution is the solute volume divided by the solution volume. If we can find these two values, we
can calculate the percentage concentration of the mixture. What is the solution volume of the
mixture?

Well, we’re mixing a 250-milliliter
solution with a 300-milliliter solution. In the end, that will give us a
solution with a volume of 550 milliliters. For the other missing variable in
our formula, what is the solute volume of the mixture? Well, we find this one a similar
but slightly more intricate way. The volume of solute in the mixture
is simply going to be the volume of solute in each of the individual solutions added
together.

We can find the solute volume of a
solution by taking the volume and multiplying it by the percentage
concentration. For solution A, 70 percent of 250
milliliters gives us 175 milliliters of alcohol. Similarly, for solution B, 60
percent times 300 gives us 180 milliliters as a solute volume. Adding these two numbers together
gives us the solute volume for the mixture, 355 milliliters.

Since there are 355 milliliters of
alcohol in a 550-milliliter solution, we can divide 355 milliliters by 550
milliliters to get the percentage concentration. Our final answer is 64.5
percent. Note that our answer, 64.5 percent,
falls in between the two initial concentrations that we started with, 60 percent and
70 percent. If the calculated percentage
concentration of the mixture was somehow higher than the higher concentration or
lower than the lower concentration, we would know that we’d made an error in our
calculations somewhere along the way. So when mixing these two solutions,
the percentage concentration of the mixture is 64.5 percent.

Now that we’ve learned about
percentage concentration, let’s review some key points of the video. Percentage concentration indicates
the percent solute in a solution. Percentage concentration can be a
concentration by volume or by mass. For example, 10 percent v/v means
that 10 percent of the volume of the total solution is the volume of the solute. On the other hand, one percent m/m
means that one percent of the mass of the solution is the mass of the solute.

The formula to calculate percentage
concentration is percentage concentration equals solute volume divided by solution
volume if we’re calculating it by volume or percentage concentration equals solute
mass divided by solution mass to calculate it by mass. We can use these formulas or
rearranged versions of them to solve for the percentage concentration, solute volume
or mass, or solution volume or mass when given the other two. And we can calculate the percentage
concentration of a mixture of two substances by finding the total solute volume or
mass and the total solution volume or mass and then dividing them.