Ethanol-Water Solution: Volume Fraction Calculation

Introduction

Hey guys! Today, we're diving into a fun chemistry problem where we need to figure out the volume fraction of an ethanol-water solution. We've got 125 grams of a solution that's 66.0% ethanol by mass, and we need to find out how much of that volume is actually water. To do this, we'll use the specific gravity (s.g.) of water at 20°C, which is 0.9982. Buckle up, because we're about to get our calculations on!

Understanding the Problem

Before we jump into the nitty-gritty, let's make sure we understand what we're dealing with. We have a solution that's a mix of ethanol and water. The concentration is given as 66.0% mass/mass (% m/m), which means that for every 100 grams of the solution, 66 grams are ethanol. The rest, obviously, is water. Our goal is to find the volume fraction, which tells us what proportion of the total volume is water, assuming the solution behaves ideally. This "ideal" bit is important because it means we're assuming the volumes of ethanol and water are additive – no weird shrinking or expanding when they mix. Sounds good? Let's move on.

Breaking Down the Steps

To solve this, we're going to break it down into manageable steps. First, we'll calculate the mass of water in the solution. Then, we'll use the specific gravity to find the volume of the water. After that, we'll need to do the same for ethanol – calculate its mass, find its density (which we'll need to look up), and then calculate its volume. Finally, we'll add the volumes of water and ethanol together to get the total volume of the solution, and then we can calculate the volume fraction of water. Phew! Sounds like a plan? Let's get started with the first step.

Step 1: Calculate the Mass of Water

Alright, let's kick things off by figuring out how much water we have in our solution. We know we have 125 grams of the solution, and 66.0% of it is ethanol. That means the remaining percentage must be water. So, let's calculate the percentage of water first. If the solution is 66.0% ethanol, then the water percentage is:

100% - 66.0% = 34.0%

Now that we know the percentage of water, we can calculate the mass of water in the 125-gram solution. We'll do this by multiplying the total mass of the solution by the water percentage:

Mass of water = (34.0 / 100) * 125 grams = 42.5 grams

So, we've got 42.5 grams of water in our solution. Not too shabby! Now that we know the mass of water, we can move on to the next step: finding the volume of this water. This is where the specific gravity comes in handy. Let's dive into that next.

Step 2: Calculate the Volume of Water

Okay, now that we know the mass of water, we need to figure out its volume. This is where the specific gravity (s.g.) comes into play. Remember, the specific gravity of a substance is the ratio of its density to the density of a reference substance, which is usually water at a specified temperature (in our case, 20°C). The specific gravity of water at 20°C is given as 0.9982. But what does this mean for us?

Using Specific Gravity

Well, since specific gravity is a ratio of densities, we can use it to find the density of water at 20°C relative to pure water. The density of pure water is approximately 1 gram per milliliter (1 g/mL). So, the density of water in our solution is:

Density of water = s.g. * Density of pure water = 0.9982 * 1 g/mL = 0.9982 g/mL

Now that we have the density of water, we can use the formula:

Density = Mass / Volume

to find the volume. We can rearrange this formula to solve for volume:

Volume = Mass / Density

We know the mass of water is 42.5 grams, and we just calculated the density as 0.9982 g/mL. So, let's plug those values in:

Volume of water = 42.5 grams / 0.9982 g/mL ≈ 42.57 mL

Fantastic! We've found the volume of water in our solution. Now, we need to do the same for ethanol. This means we need to find the mass of ethanol (which we already know, kind of), find its density (which we'll need to look up), and then calculate its volume. Let's tackle that next.

Step 3: Calculate the Volume of Ethanol

Alright, let's shift our focus to ethanol. We need to figure out the volume of ethanol in the solution, just like we did for water. We already know the mass of ethanol in the solution because the problem told us the solution is 66.0% ethanol by mass. So, out of 125 grams of solution:

Mass of ethanol = (66.0 / 100) * 125 grams = 82.5 grams

Great! We have 82.5 grams of ethanol. Now, we need to find its density so we can calculate its volume. Unlike water, we weren't given the specific gravity of ethanol, so we'll need to look it up. A quick search tells us that the density of ethanol is approximately 0.789 g/mL at room temperature (which is close enough to our 20°C). Keep in mind that density can change with temperature, so it's always good to use the value that corresponds to the temperature in your problem.

Calculating the Volume of Ethanol

Now that we have the density of ethanol, we can use the same formula we used for water:

Volume = Mass / Density

Plugging in the values for ethanol, we get:

Volume of ethanol = 82.5 grams / 0.789 g/mL ≈ 104.56 mL

Excellent! We've calculated the volume of ethanol in the solution. Now we know the volume of water and the volume of ethanol. What's the next step? You guessed it – we need to find the total volume of the solution so we can calculate the volume fraction of water. Let's do that now.

Step 4: Calculate the Total Volume of the Solution

Okay, we're on the home stretch now! We've calculated the volume of water and the volume of ethanol. To find the total volume of the solution, we're going to make an important assumption: that the volumes are additive. This is what we mean by an "ideal" solution. In real life, when you mix liquids, the total volume might be slightly different from the sum of the individual volumes due to intermolecular interactions. But for our purposes, we're assuming these effects are negligible.

So, to find the total volume, we simply add the volume of water and the volume of ethanol:

Total volume = Volume of water + Volume of ethanol

We calculated the volume of water to be approximately 42.57 mL, and the volume of ethanol to be approximately 104.56 mL. So:

Total volume ≈ 42.57 mL + 104.56 mL ≈ 147.13 mL

There we go! We've found the total volume of the solution. Now, finally, we can calculate the volume fraction of water. Let's wrap this up in the final step.

Step 5: Calculate the Volume Fraction of Water

Alright, guys, the moment we've been working towards! We're finally going to calculate the volume fraction of water in the solution. Remember, the volume fraction is the ratio of the volume of the component (in this case, water) to the total volume of the solution. We've got all the pieces we need:

Volume fraction of water = Volume of water / Total volume

We calculated the volume of water to be approximately 42.57 mL, and the total volume of the solution to be approximately 147.13 mL. So, let's plug those numbers in:

Volume fraction of water ≈ 42.57 mL / 147.13 mL ≈ 0.289

And there you have it! The volume fraction of water in the ethanol-water solution is approximately 0.289. This means that about 28.9% of the solution's volume is water.

Wrapping It Up

We did it! We successfully calculated the volume fraction of water in the ethanol-water solution. We started by understanding the problem, broke it down into manageable steps, and tackled each step one by one. We calculated the mass of water, used the specific gravity to find the volume of water, calculated the volume of ethanol, found the total volume of the solution, and finally, calculated the volume fraction of water.

This problem highlights the importance of understanding concepts like mass percentage, specific gravity, and density, and how they relate to each other. It also shows how we can use these concepts to solve practical problems in chemistry. Great job, everyone! Keep up the awesome work, and remember to keep those calculations sharp!