Problem 46 - Algebra and Trigonometry

Question

Mixture Problem What quantity of pure acid must be added to 300 \(\mathrm{mL}\) of a 50\(\%\) acid solution to produce a 60\(\%\) acid solution?

Step-by-Step Solution

Answer

Add 75 mL of pure acid.

Step 1: Understanding the Problem

We need to determine how much pure acid (100% acid) should be added to 300 mL of a 50% acid solution to increase its concentration to 60%.

Step 2: Setting Up the Equation

Let \( x \) be the amount of pure acid (100%) to be added. The amount of acid in the initial solution is \( 0.5 \times 300 \) mL, which equals 150 mL. The total volume after adding pure acid is \( 300 + x \) mL. The total amount of acid after adding \( x \) mL is \( 150 + x \). We need the ratio of pure acid in the new solution to be 60%, which gives us the equation: \[ \frac{150 + x}{300 + x} = 0.6 \]

Step 3: Solving the Equation

Multiply both sides by \( 300 + x \) to eliminate the fraction: \( 150 + x = 0.6(300 + x) \). Expand and solve: \( 150 + x = 180 + 0.6x \). Rearrange to combine like terms: \( x - 0.6x = 180 - 150 \). Simplify: \( 0.4x = 30 \). Solve for \( x \) by dividing both sides by 0.4: \( x = \frac{30}{0.4} = 75 \).

Step 4: Conclusion

The solution to the equation shows that 75 mL of pure acid needs to be added to the 300 mL solution to obtain a 60% acid solution.

Key Concepts

Acid Concentration

Understanding acid concentration is key in solving mixture problems. Acid concentration refers to the amount of acid present in a solution, usually expressed as a percentage. For example, a 50% acid solution contains half of its volume as acid, with the remainder typically being water or another solvent. In mixture problems, the goal is often to change the concentration by adding more of one component, like increasing the acid content in a solution. This requires knowledge of initial concentrations and the desired final concentration to set up equations that accurately represent the problem. Grasping the concept of concentration percentages allows you to manipulate and solve these mixture problems effectively.

Solution Ratio

The solution ratio is the comparison between different parts of a solution that define its concentration. In the problem given, you start with a 50% acid solution, meaning half the solution is acid and half is a solvent. Adding pure acid alters this ratio, leading to a more concentrated solution.

To reason out the solution ratio:

• Initial solution: 300 mL at 50% concentration.
• Desired solution: 60% concentration.

You need to calculate the new total volume and ensure the amount of acid divided by this volume matches the 60% target. Keeping track of these ratios helps you understand the impact of adding more of one component, like the pure acid in this case.

Algebraic Equation

Solving mixture problems often involves setting up an algebraic equation, as it's essential in finding the unknown quantity needed to reach a desired concentration. In our case, we formulated an equation to find the amount of pure acid to add.The formula here arises as:\[\frac{150 + x}{300 + x} = 0.6\]Where:

• 150 is the initial amount of acid in mL (since 0.5 * 300 = 150 mL).
• \(x\) is the amount of pure acid added.
• The denominator, \(300 + x\), represents the total volume.

Expanding and simplifying this equation enables us to isolate \(x\) and determine the quantity of pure acid necessary to reach a 60% acid concentration.

Problem Solving Steps

Approaching mixture problems involves a systematic process. It’s about understanding, setting up, solving, and interpreting. 1. **Understand the Problem**: Comprehend the given data and what final result is expected — here, increasing a 300 mL solution from 50% to 60% acid. 2. **Set Up the Equation**: Formulate an equation based on known and unknown elements — using the concentration equation to represent the change. 3. **Solve the Equation**: Use algebraic methods to find the unknown. Multiply to eliminate fractions, rearrange terms, and solve for the unknown. 4. **Conclusion & Verify**: Conclude by checking if the solution makes sense. Ensuring that adding 75 mL of pure acid yields a final concentration of 60% matches the requirements given. Following these steps ensures an organized method to tackle any similar mixture problems efficiently.