Mixing Acid Solutions How To Calculate The Right Concentration

In this comprehensive guide, we will delve into a classic mixture problem involving acid solutions. Specifically, we will address the challenge faced by Eli, who aims to create a 15% acid solution by combining a 0.5-gallon 10% acid solution with a 35% acid solution. Mixture problems like this are common in chemistry and mathematics, requiring a careful balance of concentrations and volumes. This article will not only provide a step-by-step solution to Eli's problem but also offer a broader understanding of the concepts involved in solving mixture problems. Understanding how to solve these problems is crucial for various applications, from chemistry labs to everyday scenarios where solutions need to be mixed to achieve a desired concentration. We will break down the problem, explain the underlying principles, and provide a clear, concise solution. By the end of this guide, you will be equipped with the knowledge and skills to tackle similar mixture problems with confidence.

The key to solving mixture problems lies in understanding the relationships between the volumes and concentrations of the solutions being mixed. Eli has a 0.5-gallon solution that is 10% acid, meaning that 10% of the 0.5 gallons is pure acid. Eli wants to add a certain amount of a 35% acid solution to increase the overall concentration to 15%. The core challenge is to determine the exact volume of the 35% acid solution needed to achieve this target concentration. This involves setting up an equation that represents the total amount of acid in the final mixture. This amount of acid is the sum of the acid in the initial 10% solution and the acid added from the 35% solution. This total amount of acid must then equal 15% of the total volume of the final mixture, which includes the initial 0.5 gallons plus the added volume of the 35% solution. By carefully defining these relationships, we can create a mathematical equation that accurately models the mixing process. Solving this equation will provide us with the exact volume of the 35% solution required to achieve the desired 15% concentration. This process requires a clear understanding of the conservation of mass principle, specifically how the mass of the acid components is conserved during the mixing process. The total mass of the acid before mixing must equal the total mass of the acid after mixing.

To solve this problem, we need to translate the word problem into a mathematical equation. Let's define 'g' as the amount (in gallons) of the 35% acid solution that Eli needs to add. The amount of acid in the 0.5-gallon 10% solution is (0.10)(0.5). The amount of acid in the 'g' gallons of the 35% solution is 0.35g. When these two solutions are combined, the total amount of acid in the mixture is (0.10)(0.5) + 0.35g. The total volume of the mixture is the sum of the initial 0.5 gallons and the 'g' gallons of the 35% solution, which is (g + 0.5) gallons. Eli wants the final solution to be 15% acid, so the amount of acid in the mixture should also be 0.15(g + 0.5). Now, we can set up the equation: (0.10)(0.5) + 0.35g = 0.15(g + 0.5). This equation represents the conservation of acid in the mixture. The left side of the equation represents the total amount of acid from the two solutions being mixed. The right side of the equation represents the amount of acid in the final 15% solution. By setting up the equation in this way, we ensure that we are accurately accounting for all the acid in the system. This equation is a linear equation in one variable (g), which means it can be solved using basic algebraic techniques. The next step is to solve this equation to find the value of 'g', which represents the volume of the 35% acid solution that Eli needs to add. The equation provides a clear and concise mathematical representation of the mixing problem, allowing us to find the solution through algebraic manipulation.

Now that we have the equation (0.10)(0.5) + 0.35g = 0.15(g + 0.5), let's solve it step by step. First, simplify both sides of the equation. On the left side, (0.10)(0.5) equals 0.05, so the equation becomes 0.05 + 0.35g = 0.15(g + 0.5). Next, distribute the 0.15 on the right side of the equation: 0.15(g + 0.5) becomes 0.15g + 0.075. So, the equation now looks like this: 0.05 + 0.35g = 0.15g + 0.075. To isolate the terms with 'g', subtract 0.15g from both sides: 0.05 + 0.35g - 0.15g = 0.15g + 0.075 - 0.15g, which simplifies to 0.05 + 0.20g = 0.075. Next, subtract 0.05 from both sides to isolate the term with 'g': 0.05 + 0.20g - 0.05 = 0.075 - 0.05, which simplifies to 0.20g = 0.025. Finally, divide both sides by 0.20 to solve for 'g': 0.20g / 0.20 = 0.025 / 0.20, which gives g = 0.125. Therefore, Eli needs to add 0.125 gallons of the 35% acid solution to achieve a 15% acid solution. This step-by-step solution demonstrates the process of algebraic manipulation necessary to solve linear equations. Each step is designed to isolate the variable 'g', ultimately leading to the solution. This method can be applied to a wide range of similar problems, making it a valuable skill in both mathematical and scientific contexts. The solution, 0.125 gallons, represents the precise amount of the 35% acid solution needed to balance the mixture and achieve the desired 15% concentration.

Therefore, Eli should add 0.125 gallons of the 35% acid solution. This solution is not just a numerical answer; it has practical implications. Adding 0.125 gallons (which is equivalent to half of a quart) of the 35% solution to the 0.5 gallons of the 10% solution will result in a total volume of 0.625 gallons of a 15% acid solution. This means Eli has successfully mixed the solutions to achieve the desired concentration. In practical applications, such precision is often crucial. For example, in chemical experiments, the correct concentration of solutions is essential for the experiment to yield accurate results. In industrial processes, precise mixing ensures the quality and consistency of products. Even in everyday situations, such as gardening or cleaning, the right concentration of a solution can make a significant difference in its effectiveness and safety. Understanding the implications of the solution helps to contextualize the mathematical process. It’s not just about getting the right number; it’s about understanding what that number means in the real world. This problem demonstrates the power of mathematics to solve practical problems and the importance of careful calculation in achieving desired outcomes. The solution highlights the interplay between volume and concentration in mixtures, a concept that is fundamental to many scientific and practical applications.

Solving mixture problems like Eli's requires a systematic approach. Here are some general tips to help you tackle these types of problems effectively. First, always define your variables clearly. In this case, we defined 'g' as the amount of the 35% solution to be added. Clearly defining variables helps prevent confusion and ensures that your equation accurately represents the problem. Second, set up an equation that represents the total amount of the substance being mixed (in this case, acid). This often involves summing the amounts from each solution and setting that equal to the amount in the final mixture. The equation should reflect the conservation principle – that the total amount of the substance remains the same before and after mixing. Third, make sure to express all quantities in the same units. For example, if volumes are given in gallons, use gallons consistently throughout the problem. This avoids errors in calculation. Fourth, carefully check your work. After solving the equation, plug the solution back into the original equation to verify that it holds true. Also, think about whether the solution makes sense in the context of the problem. For example, if you calculated a negative volume, you would know there was an error in your setup or calculations. Fifth, practice makes perfect. The more mixture problems you solve, the more comfortable you will become with the process. Look for variations in the types of problems, such as those involving different concentrations, volumes, or even multiple solutions being mixed. By following these tips, you can approach mixture problems with confidence and increase your chances of finding the correct solution. These tips emphasize the importance of clarity, consistency, and logical thinking in problem-solving.

In conclusion, we have successfully solved Eli's acid solution mixing problem by setting up and solving a linear equation. Eli needs to add 0.125 gallons of the 35% acid solution to the 0.5 gallons of the 10% acid solution to create a 15% acid solution. This problem highlights the practical application of mathematical concepts in real-world scenarios. Understanding how to solve mixture problems is a valuable skill that can be applied in various fields, from chemistry to everyday tasks. The step-by-step approach we used, including defining variables, setting up the equation, and solving it algebraically, provides a framework for tackling similar problems. Remember the importance of checking your work and ensuring that the solution makes sense in the context of the problem. By following these principles, you can confidently solve a wide range of mixture problems. Moreover, the ability to translate a word problem into a mathematical equation is a critical skill in problem-solving. This requires careful reading, identifying the key information, and understanding the relationships between the different quantities. The process of solving the equation then involves applying algebraic techniques to isolate the unknown variable. The solution, once obtained, should always be interpreted in the context of the original problem to ensure it is meaningful and accurate. This comprehensive approach to problem-solving is not only useful for mixture problems but also for many other types of mathematical and scientific challenges.