Solving a Concentration Mixture Problem: The Impossibility of Mixing Solutions

Introduction

In the field of chemistry and pharmacology, understanding the concentration of solutions is critical for a variety of applications, from cosmetics to medical treatments. One common type of problem involves mixing different concentrations of solutions to achieve a desired concentration. In this article, we will explore an example involving the mixing of hyaluronic acid (HA) solutions of different concentrations to achieve a specific concentration. We will also explain why certain solutions cannot be combined to create desired concentrations.

The Problem

Mr. Arsenault has two containers of hyaluronic acid (HA) solutions: a 15% solution and a 3% solution. He needs to create 1500 mL of a 22% HA solution. The challenge is to determine how much of each solution is required. However, there is an important constraint: without concentrating one of the solutions, it is impossible to create a 22% solution from 15% and 3% solutions alone.

Step-by-Step Algebraic Solution

Let's approach this problem algebraically, using linear equations. We'll define two variables: x for the volume of the 15% solution, and y for the volume of the 3% solution.

The total volume must be 1500 mL, giving us the equation:

x y 1500

The amount of HA in each solution must add up to the amount in the final 22% solution. For the 15% solution, the amount is (frac{15}{100}x), and for the 3% solution, the amount is (frac{3}{100}y). The final 22% solution requires (frac{22}{100} times 1500 330) mL of HA.

Setting up the second equation, we have:

(frac{15}{100}x frac{3}{100}y 330)

Multiplying the entire equation by 100 to clear the denominators, we get:

15x 3y 33000

Now, let's solve the system of equations.

Solving the Equations

First, we can solve the first equation for y:

y 1500 - x

Substituting this into the second equation:

15x 3(1500 - x) 33000

15x 4500 - 3x 33000

12x 28500

x 2375

Substituting (x 2375) back into the first equation:

2375 y 1500

y -875

The negative value of (y) indicates that the algebraic solution is not physically possible, as volumes cannot be negative. This confirms that it is impossible to create a 22% HA solution from a 15% and a 3% solution without concentrating one of the solutions through evaporation or another method.

Conclusion

While it is fascinating to explore the algebraic approach to solving such problems, the reality is that certain concentration mixtures are impossible to achieve using only the given solutions. This is because the mixing would require a negative volume, which is not possible. Therefore, if creating a 22% HA solution is needed, other methods such as dilution, concentration, or the use of different starting solutions would be necessary.

This detailed exploration of the problem not only provides insight into the limitations of certain chemical mixtures but also showcases the importance of practical considerations when applying mathematical solutions to real-world problems.