Problem 30 - Precalculus Student Solutions Manual 5th

Question

Acid Mixture Marin Caswell needs \(10 \%\) hydrochloric acid for a chemistry experiment. How much \(5 \%\) acid should she mix with \(60 \mathrm{mL}\) of \(20 \%\) acid to get a \(10 \%\) solution?

Step-by-Step Solution

Answer

Mix 120 mL of 5% acid with 60 mL of 20% acid.

Step 1: Define Variables

Let the volume of 5% acid solution needed be denoted by \( x \) mL. We know we have 60 mL of 20% acid solution.

Step 2: Set Up Equations

The amount of pure acid in 5% solution is \(0.05x \) mL. The amount of pure acid in 60 mL of 20% solution is \(0.2 \times 60 = 12 \) mL. The total amount of pure acid in the final mixture should be 10% of the total volume \(x + 60 \) mL.

Step 3: Write the Mixture Equation

To get a 10% solution, the pure acid amount should be equal to 10% of the total mixture volume: \[ 0.05x + 12 = 0.1(x + 60) \]

Step 4: Solve the Equation

Solve the equation for \( x \): \[ 0.05x + 12 = 0.1x + 6 \] Subtract 0.05x from both sides: \[ 12 = 0.05x + 6 \] Subtract 6 from both sides: \[ 6 = 0.05x \] Divide both sides by 0.05: \[ x = \frac{6}{0.05} = 120 \]

Step 5: Verify the Solution

Check the amount of pure acid: \[ 0.05 \times 120 + 12 = 6 + 12 = 18 \] Check the total volume: \[ 120 + 60 = 180 \] Ensure the concentration: \[ \frac{18}{180} = 0.1 = 10\% \]

Key Concepts

Mixture Equations

Mixture equations are used to determine the quantity of different substances needed to create a mixture with specific properties.
In this case, we are mixing two acid solutions with different concentrations to create a final solution with a desired concentration.
By defining variables for the unknown quantities, we can set up an equation that represents the relationship between the substances. This equation helps us find the exact amounts needed.
Here, we denote the volume of 5% acid solution as \( x \) mL. The challenge is to mix it with the given amount of 20% acid solution to achieve the required 10% acid concentration.

Percentage Concentration

Percentage concentration represents the proportion of a substance in a solution, expressed as a percentage.
In our problem, we deal with three different percentage concentrations: 5%, 20%, and 10%.
The 5% means there are 5 parts of acid for every 100 parts of solution. Similarly, for the 20% solution, it's 20 parts of acid for every 100 parts of solution. Our target is a final solution with 10 parts of acid in every 100 parts of solution.
Understanding the percentage concentration helps in identifying how much pure acid is present in each solution, which is crucial for setting up accurate equations.

Solving Algebraic Equations

Solving algebraic equations involves finding the value of the unknown variable that makes the equation true.
In the given exercise, the mixture equation is: \ \[ 0.05x + 12 = 0.1(x + 60) \] \ We need to simplify and solve it to find the value of \( x \), the volume of the 5% solution.
Steps for solving include:

• Expanding and simplifying equations.
• Isolating the variable on one side.
• Performing arithmetic operations to solve for the variable.

We follow these steps until we find \( x = 120 \) mL.

Chemistry Applications

Mixture problems are common in chemistry, especially in preparing solutions with precise concentrations.
By skillfully applying algebraic techniques and understanding of concentrations, chemists can create solutions necessary for various experiments and processes.
This exercise illustrates a typical scenario where precise calculations are crucial. Accurate preparation ensures the reliability and success of experimental results.
Grasping these concepts is essential for anyone studying or working in chemistry.

Solution Preparation

Preparing a chemical solution with a specific concentration involves several careful steps.
For Marin Caswell's experiment, she needs to:

• Determine the quantities of each solution to mix.
• Mix them accurately.
• Verify the final concentration.

Our calculations show that Marin needs 120 mL of 5% acid to mix with 60 mL of 20% acid.
After mixing, the total volume is 180 mL, and the concentration is verified to be 10%.
Accurate solution preparation is crucial for consistency and accuracy in any chemical experiment.