Acid Solution Mix: Equation For 15% Concentration

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Acid Solution Mixing Equation: How to Get a 15% Concentration

Hey guys! Let's dive into a common chemistry problem – mixing acid solutions. This is something you might encounter in a lab, or even in practical applications like cleaning or preparing solutions for various uses. The key here is understanding how to set up an equation that accurately represents the mixture and the desired concentration. We'll break down a specific scenario where Eli wants to combine different acid solutions to achieve a target concentration. Let's get started!

Understanding the Problem

The problem states that Eli wants to mix 0.5 gallons of a 10% acid solution with some amount of a 35% acid solution. The goal is to create a solution that has a 15% acid concentration. The main challenge is to figure out how many gallons of the 35% solution Eli needs to add. To solve this, we need to formulate an equation that relates the volumes and concentrations of the solutions being mixed.

Breaking Down the Components

Before we jump into the equation, let's identify the key components:

  • Volume of the 10% acid solution: 0.5 gallons
  • Concentration of the 10% acid solution: 10% (or 0.10 as a decimal)
  • Volume of the 35% acid solution: This is what we need to find, so let's call it x gallons
  • Concentration of the 35% acid solution: 35% (or 0.35 as a decimal)
  • Desired concentration of the final solution: 15% (or 0.15 as a decimal)
  • Total volume of the final solution: 0.5 + x gallons

Key Concept: Amount of Acid

The fundamental idea behind solving mixture problems like this is to focus on the amount of the solute (in this case, acid) in each solution. The total amount of acid in the final mixture will be the sum of the amounts of acid in the individual solutions. This can be expressed as:

Amount of acid in 10% solution + Amount of acid in 35% solution = Amount of acid in 15% solution

This concept forms the basis of our equation, and by understanding it thoroughly, we can easily translate the word problem into a mathematical expression. The next step is to express each of these amounts in terms of volumes and concentrations, setting us up to solve for the unknown variable.

Formulating the Equation

Now that we have all the components and the key concept, let's build the equation. Remember, the amount of acid in a solution is the product of its volume and its concentration. So, we can express the amounts of acid as follows:

  • Amount of acid in 0.5 gallons of 10% solution: 0. 5 * 0.10
  • Amount of acid in x gallons of 35% solution: x * 0.35
  • Amount of acid in the final (0.5 + x) gallons of 15% solution: (0.5 + x) * 0.15

Using the principle that the total amount of acid in the final solution is the sum of the amounts in the initial solutions, we can set up the equation:

0.5 * 0.10 + x * 0.35 = (0.5 + x) * 0.15

Expanding the Equation

Let's break down what each part of the equation represents:

  • 0.5 * 0.10: This calculates the amount of pure acid in the 0.5 gallons of the 10% solution. Multiplying the volume (0.5 gallons) by the concentration (0.10) gives us the volume of pure acid.
  • x * 0.35: This calculates the amount of pure acid in the x gallons of the 35% solution. Again, multiplying the volume (x gallons) by the concentration (0.35) gives us the volume of pure acid.
  • (0.5 + x) * 0.15: This calculates the amount of pure acid in the final mixture. The total volume of the mixture is the sum of the volumes of the two initial solutions (0.5 + x gallons), and multiplying this by the desired concentration (0.15) gives us the volume of pure acid in the final mixture.

The Full Equation

Putting it all together, the equation we need to solve is:

0.05 + 0.35x = 0.075 + 0.15x

This equation accurately represents the relationship between the volumes and concentrations of the acid solutions in the mixture. The next step involves solving this equation for x to find the number of gallons of the 35% acid solution needed to achieve the desired 15% concentration. This equation sets the stage for the algebraic manipulation that will lead us to the solution.

Solving for x

Now that we have the equation, the next step is to solve for x, which represents the number of gallons of the 35% acid solution Eli needs to add. Let's walk through the steps to isolate x.

Rearranging the Equation

Our equation is:

0.05 + 0.35x = 0.075 + 0.15x

First, we want to group the terms with x on one side and the constants on the other. We can do this by subtracting 0.15x from both sides and subtracting 0.05 from both sides. This gives us:

0.35x - 0.15x = 0.075 - 0.05

Simplifying the Equation

Now, let's simplify the equation by combining like terms:

0. 20x = 0.025

We've reduced the equation to a much simpler form where we have a single term with x on one side and a constant on the other. This makes it easier to isolate x.

Isolating x

To isolate x, we need to divide both sides of the equation by 0.20:

x = 0.025 / 0.20

Calculating x

Now, let's perform the division:

x = 0.125

The Solution

So, x = 0.125 gallons. This means Eli needs to add 0.125 gallons of the 35% acid solution to the 0.5 gallons of the 10% acid solution to get a 15% acid solution. The answer makes sense in the context of the problem – a relatively small amount of the more concentrated solution is needed to bring the overall concentration up to 15%.

Final Answer and Verification

We've solved for x, and we found that Eli needs to add 0.125 gallons of the 35% acid solution. But before we call it a day, let's verify our answer to make sure it's correct.

Restating the Answer

Eli should add 0.125 gallons of the 35% acid solution to the 0.5 gallons of the 10% acid solution to obtain a 15% acid solution.

Verifying the Solution

To verify, we'll plug x = 0.125 back into our original equation and see if it holds true:

0.5 * 0.10 + 0.125 * 0.35 = (0.5 + 0.125) * 0.15

Let's break this down:

  • Left side:
    • 0.5 * 0.10 = 0.05
    • 0.125 * 0.35 = 0.04375
    • 0.05 + 0.04375 = 0.09375
  • Right side:
    • 0.5 + 0.125 = 0.625
    • 0.625 * 0.15 = 0.09375

Both sides of the equation equal 0.09375, which confirms that our solution is correct. This verification step is crucial in problem-solving, especially in chemistry and mathematics, to ensure that the answer not only makes sense but is also accurate.

Why Verification Matters

Verifying your solution is a great habit to get into because it helps you catch any mistakes you might have made along the way. Whether it's a simple arithmetic error or a misunderstanding of the problem setup, verification can save you from incorrect answers. It also builds confidence in your problem-solving abilities, knowing that you've taken the time to ensure your solution is solid.

Practical Implications and Other Applications

Mixing solutions to achieve a desired concentration isn't just a math problem; it has real-world applications in various fields. Understanding how to set up and solve these equations is valuable in chemistry, biology, medicine, and even everyday tasks like cleaning and cooking.

Chemistry and Biology

In chemistry labs, solutions are mixed all the time to create reagents for experiments. The correct concentration is crucial for accurate results. Similarly, in biology, preparing solutions with specific concentrations is essential for cell cultures, drug dilutions, and other experiments.

Medicine

In the medical field, pharmacists and healthcare professionals often need to dilute medications to achieve the correct dosage for patients. This requires precise calculations and an understanding of solution concentrations.

Everyday Applications

Even in everyday life, we encounter situations where mixing solutions is important. For example, when you mix cleaning solutions, you need to ensure you're using the right proportions to avoid damaging surfaces or creating harmful fumes. In cooking, mixing ingredients to achieve the right flavor often involves understanding ratios and proportions, which is similar to working with concentrations.

Adapting the Equation for Different Scenarios

The equation we used to solve Eli's problem can be adapted to different scenarios involving mixing solutions. The key is to identify the components, write down the knowns and unknowns, and set up the equation based on the principle that the total amount of solute in the final solution is the sum of the amounts in the initial solutions. For instance, if you were mixing two solutions of the same solute but different concentrations to achieve a specific volume and concentration, the same principles would apply.

Conclusion

So, there you have it! We've broken down how to formulate an equation to solve a mixture problem, specifically focusing on acid solutions. Remember, the key is to understand the relationship between volume, concentration, and the amount of solute. By setting up the equation correctly and solving for the unknown variable, you can tackle similar problems with confidence. And don't forget to verify your answer – it's always a good idea to double-check your work! Whether you're in a chemistry lab or just tackling a homework problem, these skills will definitely come in handy. Keep practicing, and you'll become a pro at mixing solutions in no time!