Problem 20 - A Graphical Approach to College Algebra

Question

$$\begin{aligned} &\text {Solve each problem analytically, and support your solution}\\\ &\text {graphically.} \end{aligned}$$ Acid Mixture Bill Charlson needs \(10 \%\) hydrochloric acid for a chemistry experiment. How much \(5 \%\) acid should he mix with 60 milliliters of \(20 \%\) acid to obtain a \(10 \%\) solution?

Step-by-Step Solution

Answer

Mix 120 ml of the 5% acid with 60 ml of the 20% acid.

Step 1: Understand the Problem

Bill needs a solution that has 10% concentration of acid. He will mix a 5% acid solution with 60 ml of a 20% acid solution. We need to find out how much of the 5% acid solution is required.

Step 2: Define Variables

Let \( x \) be the amount of 5% acid solution needed in milliliters. Bill already has 60 ml of a 20% acid solution.

Step 3: Set Up the Equation

The total volume of the final solution will be \( x + 60 \) ml. The amount of pure acid in the 5% solution is \( 0.05x \), and the amount of pure acid in the 20% solution is \( 0.20 \times 60 = 12 \) ml. We want the final mixture to be 10% acid, so the equation is:\[ 0.05x + 12 = 0.10(x + 60) \]

Step 4: Solve the Equation

Start by expanding and simplifying the equation:\[ 0.05x + 12 = 0.10x + 6 \]Subtract \( 0.05x \) from both sides:\[ 12 = 0.05x + 6 \]Subtract 6 from both sides:\[ 6 = 0.05x \]Now, divide by 0.05:\[ x = \frac{6}{0.05} = 120 \]

Step 5: Verify the Solution

To check, calculate the total acid in the mixture: Using 120 ml of 5% acid gives \( 0.05 \times 120 = 6 \) ml of acid. With the 20% acid, you're getting 12 ml from 60 ml. So, \( 6 + 12 = 18 \) ml in 180 ml total solution. Check that the percentage is 10%:\[ \frac{18}{180} \times 100 = 10 \% \]This confirms our solution is correct.

Step 6: Graphical Representation

Graphically, you can represent this by plotting two scenarios on a graph: one line representing the concentration of the 5% solution and another line representing the decrease in concentration of the 20% solution as more 5% solution is added, aiming for intersecting at 10% concentration. The intersection point on the x-axis should be at 120 ml.

Key Concepts

Concentration Solutions

Concentration solutions are a fundamental concept in chemistry, particularly when creating mixture solutions for experiments. A solution's concentration is the amount of solute present in a given quantity of solvent or solution. In the context of acid solutions, the concentration is usually expressed as a percentage, indicating the fraction of the solution that is pure acid.

In the given problem, the concentration percentage helps us understand how strong the acid is within the solution. A 5% hydrochloric acid solution means that 5% of its total volume is pure acid, while a 20% solution contains a higher amount—20% pure acid. It's crucial to balance these concentrations to achieve a desired effect, such as a 10% acid concentration.

The task is to mix a lower percentage acid (5%) with a higher percentage one (20%) to dilute the overall concentration down to a specific target (10%). This process will require thoughtful calculations and an understanding of how different concentrations combine to reach a new, desired concentration level.

Algebraic Equations

Algebraic equations are powerful tools in solving mixture problems involving concentrations. They allow us to systematically find the unknown quantity by setting up relationships between different components of the problem. In this problem involving acid mixtures, the unknown variable is the volume of the 5% acid, labelled by the variable \( x \). We aim to determine how much of this acid is needed to achieve the desired final concentration of 10% when mixed with a known volume of 20% acid solution.

To set up the equation, we calculate the amount of pure acid in each part of the solution:

• Pure acid from 5% solution: \( 0.05x \)
• Pure acid from 20% solution: \( 0.20 \times 60 \)

We then create an equation accounting for both sources of acid equaling the total pure acid needed for the final 10% mixture:\[ 0.05x + 12 = 0.10(x + 60) \]

Simplifying the equation gives us the amount of 5% acid required. This algebraic approach not only gives a direct numerical answer but also enhances the logical understanding of how concentrations mix.

Chemistry Experiments

Chemistry experiments frequently require precise solutions, emphasizing the importance of correct formulae and calculations. When working with acid mixtures, as in Bill Charlson's experiment, it’s critical to achieve the exact concentration needed for the experiment's purpose. A slight deviation in concentration can lead to drastically different results.

In practical experimental scenarios, once the desired concentration has been determined through calculation, the next step is to ensure accuracy through careful measurement and mixing of solutions.

The process of mixing involves:

• Accurately measuring liquids with appropriate laboratory tools and techniques
• Using graduated cylinders or pipettes to ensure precision
• Stirring the solution thoroughly for uniform distribution of acid molecules

It is essential to verify the solution's concentration with suitable methods, often employing titration or pH testing, to confirm the desired specifications have been achieved before application in any further experimental procedures or observations. This verification step is crucial to ensure experimental validity and repeatability.