Problem 63 - Technical Mathematics with Calculus

Question

A solution is made by mixing 75.0 liters of alcohol with 125 liters of water. Find the percent concentration of alcohol.

Step-by-Step Solution

Answer

The percent concentration of alcohol is \( \frac{75.0}{75.0 + 125} \times 100 \approx 37.5\% \)

Step 1: Determine the total volume of the solution

To find the percent concentration of alcohol in the solution, first calculate the total volume of the solution by adding the volume of alcohol to the volume of water.

Step 2: Calculate the percent concentration of alcohol

Use the formula for percent concentration which is \( \frac{\text{volume of alcohol}}{\text{total volume of solution}} \times 100 \) to find the percent by volume of alcohol in the solution.

Key Concepts

Alcohol Solution Problem

Understanding alcohol solution problems is key to mastering the concept of percent concentration in chemistry. When we talk about such a problem, we're often dealing with a mixture of alcohol and water. The goal is generally to calculate the percentage of alcohol by volume, which indicates how much alcohol is present in the solution compared to the total amount of the mixture.

In simpler terms, the percent concentration tells us how 'strong' the solution is. For instance, in beverages, this concept is used widely to specify how much alcohol (ethanol) there is in the liquid we drink. When solving these problems, it's important not to confuse the volume of alcohol with the volume of water or the total volume of the mixture, as each plays a distinct role in the calculations. By keeping a clear distinction, you'll be less likely to make mistakes and more likely to grasp the essence of these solutions.

Additionally, understanding how to approach these types of problems prepares you to deal with real-world scenarios such as diluting antiseptic solutions or preparing specific alcohol concentrations for scientific experiments.

Volume Calculation

Volume calculation is a fundamental aspect of solving percent concentration problems. It is a straightforward process but requires attention to detail. To solve for volume, you'll want to ensure you're measuring all liquid components in the same units, such as liters, to maintain consistency in your calculations. In our original exercise, we combine 75.0 liters of alcohol with 125 liters of water, so we're already set with consistent units.

It's essential to ascertain the combined volume of the mixture, which is the sum of the volumes of the individual components. To achieve this, simply add the volume of alcohol to the volume of water. Knowing the total volume is critical because it serves as the denominator in our percent concentration formula. Missing this step or miscalculating can lead to incorrect results, highlighting the importance of accurate volume calculation in chemistry and in various real-life applications like cooking or filling up a pool.

Exercise Improvement Advice

A practical method to improve comprehension in such exercises is by relating the concept to daily tasks, such as mixing drinks or preparing recipes that require precise volumes of ingredients to achieve the desired taste or outcome.

Mathematical Formulas

Mathematical formulas are indispensable when addressing percent concentration problems. These formulas provide a reliable and consistent method for calculating values such as percent concentration, density, molarity, and more. The formula used in our exercise \( \frac{\text{volume of alcohol}}{\text{total volume of solution}} \times 100 \) is a straightforward expression to find the percent by volume of alcohol within the solution.

Percent means 'per hundred', so multiplying the volume ratio by 100 translates our fraction into a percentage. It's crucial to note that the percent concentration formula only works when the numerator (top number) is part of the denominator (bottom number); in other words, the substance we're interested in (alcohol, in this case) must be a component of the total volume.

Formulas in mathematics provide a universal language that transcends simple arithmetic, allowing us to express and solve complex problems. Gaining proficiency in the application of these formulas will ensure confidence in dealing with more advanced scientific and mathematical tasks.