Lesson Video: Percentage Concentration Chemistry

In this video, we will learn how to express and calculate the percentage concentration of a solution by its volume or mass ratio.

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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.

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