Determine the mass (in g) of each sucrose solution that contains \(12 \mathrm{~g}\) of sucrose. (a) \(4.1 \%\) sucrose by mass (b) \(3.2 \%\) sucrose by mass (c) \(12.5 \%\) sucrose by mass

When we talk about the concentration of a solution, one of the ways to express it is through mass percent concentration. This is a measure that tells us how much of a certain substance (the solute) is contained within a given amount of solution. It's like a recipe where you know how much of each ingredient is in the cake mix compared to the total amount of the mix.

Calculating mass percent concentration is like figuring out what percentage of a cake is flour. You look at the amount of flour compared to the overall amount of ingredients. If you have 100 grams of cake mix, and 2 grams of that is flour, then the flour is 2% of the mix. Similarly, if we have 100g of solution, and there's 4.1g of a substance like sucrose in it, then we say it's a 4.1% solution.

Let's put this concept into practice with a sucrose solution. Suppose a beverage contains sucrose (table sugar), and you're asked to calculate what total mass of the beverage is needed to get a specific amount of sucrose. The key here is in the percentage by mass we're given.

If you're told that a juice is 4.1% sucrose by mass, it means for every 100g of that juice, 4.1g is pure sucrose. To figure out how much juice you need to provide a certain amount of sucrose, like 12g, we use a proportion. Here, the 'part over whole' logic is applied. If 4.1g is to 100g of solution, then 12g of sucrose should be to how many grams of the solution? This forms the essence of our calculation.

How do we actually calculate concentration? It involves setting up and solving a simple proportion. A proportion compares two ratios or fractions and says that two ratios are equivalent. For example, if the fraction of sugar to solution is 4.1/100, and we know we want 12g of sugar, we can set up the proportion 4.1/100 = 12/x to find the unknown total mass (x) of the solution.

To solve for x, we cross-multiply and divide. Cross-multiplication gives us 4.1x = 1200. Dividing both sides by 4.1 gives us x, which represents the total mass of the solution needed to get 12g of sucrose. This cross-multiplying and dividing is the crux of how we find our answer.

Understanding the proportion of a solution is essential. It's not just about mixing stuff together; it's about mixing them in the right amounts. When we adjust the amount of one component (like sugar in our drink), it shifts the balance of the whole mixture. Think of it as you're the DJ of your mixed substances and you have to get the balance of beats (components) just right.

With sucrose solutions, adjusting concentration changes taste and other properties. So getting that proportion correct is critical not just in the kitchen but in labs and industries. That's why being able to calculate the total mass of these solutions when given a mass percent concentration is such a valuable skill.