Mixing Solutions to Achieve a Desired Concentration: A Practical Guide

Solution mixing is a fundamental process in various fields, from chemistry and engineering to pharmaceuticals. Understanding how to achieve a specific concentration by mixing different solutions can help in various applications, such as creating a 60% solution from a 30% and a 70% solution. This article will guide you through the process of solving such problems using algebraic methods, ensuring a clear and practical approach.

Problem Statement

Let's consider the problem: How many liters of 70% solution (70% concentration of solute) do we need to add to 5 liters of 30% solution to achieve a final concentration of 60%?

Solution Methodology

Step 1: Determine the Amount of Solute in Each Solution

In the 30% solution:

Volume 5 liters Concentration 30% Amount of solute 5 liters × 0.30 1.5 liters

In the 70% solution:

Volume X liters (to be determined) Concentration 70% Amount of solute X liters × 0.70 0.7X liters

Step 2: Total Volume and Total Solute After Mixing

The total volume after mixing will be 5 X liters.

The total amount of solute after mixing will be 1.5 0.7X liters.

Step 3: Set Up the Equation for the Final Concentration

We want the final concentration to be 60%. Therefore, we can set up the equation:

1.5 0.7X5 X 0.60

Step 4: Solve the Equation

Multiply both sides by (5 X):

1.5 0.7X 0.60(5 X)

Expand the right side:

1.5 0.7X 3 0.6X

Rearrange the equation:

1.5 0.7X - 0.6X 3

1.5 0.1X 3

0.1X 3 - 1.5

0.1X 1.5

Divide both sides by 0.10:

X 15

Conclusion: You need to add 15 liters of the 70% solution to the 5 liters of the 30% solution to obtain a 60% solution.

Alternative Solutions

Let's consider a few alternative methods to solve this problem and verify the previous solution:

Method 1: Direct Proportion Using NV Constant

Using the NV constant method:

5 liters × 30% X liters × 70% × V

150 60V

V 15 liters

Method 2: Combined Volume and Solute Equation

Let V be the volume of the 70% solution required to be mixed:

70V 150 60(5 V)

70V 150 300 60V

10V 150

V 15 liters

Method 3: Simple Solute Equation

Let V be the volume of the 70% solution to be added:

0.30 × 5 0.70V 0.60(5 V)

1.5 0.70V 3 0.60V

0.10V 1.5

V 15 liters

Conclusion

By using various methods, we have confirmed that 15 liters of the 70% solution must be added to 5 liters of the 30% solution to produce a 60% solution. This detailed breakdown of the problem helps in understanding the underlying principles of solution mixing and ensures the correct application of algebraic methods.