Video Transcript
What volume of water must be added to a 125-milliliter solution of ethanol to change its percentage volume from 40 percent volume by volume to 35 percent volume by volume? Give your answer to the nearest whole number.
Let’s draw some diagrams to help us understand this question. We have 125 milliliters of a solution that is 40 percent volume by volume ethanol. A solution consists of two components: a solute, the substance being dissolved, and a solvent, the substance doing the dissolving. In this example, the ethanol is the solute and the water is the solvent. We want to add more water to this solution in order to decrease the percentage volume of ethanol, but we’re not sure how much water to add. It would be helpful to know how much ethanol and water are present in this solution already.
For this, we can use the percent concentration by volume equation, where percent concentration by volume is equal to the volume of the solute divided by the volume of the solution times 100 percent. We know the percent concentration of the solution and the volume of solution. But we don’t know the volume of solute. We can call this variable 𝑣 for volume. We divide both sides of the equation by 100 percent then multiply both sides by 125 milliliters. This gives us a solute volume of 50 milliliters. To find the volume of solvent, recognize that the volume of solute plus the volume of solvent must equal the volume of the solution. If 50 milliliters of the 125-milliliter solution are ethanol, then the additional 75 milliliters must be water.
Now let’s think about the solution after the additional water has been added. We didn’t add more ethanol, so the volume of ethanol in the solution should still be 50 milliliters. The original solution contained 75 milliliters of water. To that, we added an additional amount of water. Let’s represent this additional amount with the variable 𝑤. This means that the new solution will have 75 plus 𝑤 milliliters of water. If the volume of the solution is the volume of the solute plus the volume of the solvent, then the volume of our new solution is 125 milliliters plus 𝑤. We also know from the question that adding this additional amount of water decrease the percentage volume of ethanol to 35 percent.
To determine 𝑤, we can return to the percent concentration by volume equation. We can substitute the percent concentration, volume of solute, and volume of solution into the equation. We divide both sides of the equation by 100 percent. Then we multiply both sides of the equation by 125 milliliters plus 𝑤, distributing the 0.35 inside of the parentheses. We subtract 43.75 milliliters from both sides of the equation. And finally, we divide both sides of the equation by 0.35 to determine that 𝑤 is equal to 17.857 milliliters. According to the question, we should give our answer to the nearest whole number. Rounding appropriately, we have determined that we must add 18 milliliters of water to the original solution.
