Problem 12 - Technical Mathematics with Calculus

Question

Treat the percents given in this exercise as exact numbers, and work to three significant digits. How many liters of a solution containing \(18 \%\) sulfuric acid and how many liters of another solution containing \(25 \%\) sulfuric acid must be mixed together to make 552 liters of solution containing \(23 \%\) sulfuric acid? (All percentages are by volume.)

Step-by-Step Solution

Answer

x liters of 18% solution and y liters of 25% solution where x and y solve the system of equations.

Step 1: Understanding the Problem

We need to find the volume of two solutions (one with 18% sulfuric acid and the other with 25% sulfuric acid) to mix in order to get 552 liters of a 23% sulfuric acid solution.

Step 2: Assign Variables to Unknowns

Let the volume of the 18% solution be x liters and the volume of the 25% solution be y liters.

Step 3: Create the Total Volume Equation

The sum of the volumes of the two solutions must be 552 liters. Therefore, the equation is: \( x + y = 552 \)

Step 4: Create the Total Percentage Equation

The total volume times the desired percent concentration must equal the sum of the amounts of sulfuric acid in both solutions. Thus: \( 0.18x + 0.25y = 0.23 \times 552 \)

Step 5: Solve the System of Equations

We solve the two equations \( x + y = 552 \) and \( 0.18x + 0.25y = 0.23 \times 552 \) simultaneously to find the values of x and y.

Step 6: Simplify the Percentage Equation

Calculate the total amount of sulfuric acid needed: \( 0.23 \times 552 = 127 \) liters. The equations now are \( x + y = 552 \) and \( 0.18x + 0.25y = 127 \).

Step 7: Rearrange the First Equation

From the first equation, express y in terms of x: \( y = 552 - x \).

Step 8: Substitute y into the Second Equation

Substitute \( y = 552 - x \) into the second equation: \( 0.18x + 0.25(552 - x) = 127 \).

Step 9: Solve for x

Simplify and solve the resulting equation to find the value of x.

Step 10: Solve for y

Use the value of x to find the value of y using \( y = 552 - x \).

Key Concepts

Percent Concentration

Percent concentration is a crucial measure in chemistry that dictates how much of a substance is present within a solution. When we talk about percent concentration in algebra, specifically in relation to mixture problems, we're usually referring to the volume or mass percent of a solute in a solvent. For instance, an 18% sulfuric acid solution contains 18 liters of sulfuric acid for every 100 liters of solution.

In the textbook exercise, we're dealing with volume percent, as the percentages are given by volume. This means that when we are preparing to mix solutions with different concentrations, we need to calculate how much of each solution is required to achieve a desired percent concentration in the final mixture. Percent concentration is not only foundational in solving mixture problems but also immensely applicable in fields such as pharmacology, environmental science, and food chemistry.

Solving System of Equations

Solving systems of equations is a classical algebraic tool used when there is more than one unknown variable in a problem, and hence, multiple equations are needed to find those unknowns. In mixture problems, we generally have two unknowns: the amounts of each type of solution we need to mix. The equations represent the relationship between these unknowns and are derived from the conditions given in the problem.

In the provided exercise, we use a system of linear equations to represent the total volume and the percent concentration. The steps to solve these equations typically involve methods such as substitution or elimination. Substitution involves solving one equation for one variable and then substituting this value in the other equation, as shown in steps 7 and 8 of the textbook solution. Once we find a numerical value for one variable, we can easily determine the other.

Chemistry in Mathematics

Chemistry in mathematics often surfaces in algebra through mixture problems where we work with concentrations and volumes to find the right balance of ingredients – a common task in chemical experiments and productions. Integrating chemistry concepts in math enables students to apply mathematical operations to real-world scenarios, making abstract concepts more relatable.

In the exercise under discussion, we apply mathematical techniques to a chemical problem—determining the correct amounts of two sulfuric acid solutions to mix for a desired concentration. This interdisciplinary approach not only enhances problem-solving skills but also prepares students for advanced studies where precise calculations are essential in laboratory and industrial settings.

Volume Calculation

Volume calculation is the process of determining the amount of space occupied by a substance, which is often one of the main variables in mixture problems. Accurately calculating the volume of different solutions to be mixed is essential in ensuring the final solution meets the specifications, such as the desired percent concentration.

Our textbook example required calculating the volume of two sulfuric acid solutions needed to create a new solution with a specific volume and concentration. We represented the total volume by using the equation, as seen in step 3 (\(x + y = 552\)), where x and y represent the volumes of the two original solutions, respectively. This simplicity belies the real-world importance of volume calculations in tasks like dosing medications, creating recipes, and formulating products.