Understanding the Calculation of Mixtures of Two Solutions with Different Concentrations

Mixing two solutions with different concentrations is a fundamental process in chemistry and engineering. It involves calculating the resulting concentration after combining specific volumes.

This article explores detailed formulas, common values, and real-world applications for accurately determining mixture concentrations. You will find comprehensive tables, step-by-step examples, and expert insights.

• Calculate the final concentration when mixing 200 mL of 10% NaCl with 300 mL of 5% NaCl.
• Determine the volume of 15% acid solution needed to dilute 500 mL of 25% acid to 10% concentration.
• Find the concentration after mixing 100 mL of 8 M HCl with 150 mL of 3 M HCl.
• Calculate the amount of water required to dilute 250 mL of 12% sugar solution to 6% concentration.

Comprehensive Tables of Common Concentration Values and Mixture Parameters

Fundamental Formulas for Calculating Mixtures of Two Solutions

Calculating the concentration of a mixture formed by combining two solutions with different concentrations requires understanding the relationship between volume, concentration, and mass of solute.

1. Basic Concentration Mixture Formula

The most common formula used is based on the conservation of mass of the solute:

• C_f: Final concentration of the mixture (in % or molarity)
• C₁: Concentration of solution 1
• V₁: Volume of solution 1
• C₂: Concentration of solution 2
• V₂: Volume of solution 2

This formula assumes volumes are additive and the solute does not react chemically during mixing.

2. Calculating Volume Required to Achieve a Target Concentration

When the goal is to find the volume of one solution needed to mix with a known volume of another to reach a desired concentration:

• V₂: Volume of solution 2 required
• C_f: Desired final concentration
• C₁, V₁: Concentration and volume of solution 1
• C₂: Concentration of solution 2

This formula is valid only if C₂ ≠ C_f to avoid division by zero.

3. Dilution Formula

When diluting a solution with pure solvent (concentration zero), the formula simplifies to:

• C₁: Initial concentration
• V₁: Initial volume
• C_f: Final concentration
• V_f: Final volume after dilution

From this, the volume of solvent added is V_f – V₁.

4. Mass-Based Calculation

In some cases, it is more accurate to calculate based on the mass of solute and solvent:

• m₁, m₂: Mass of solute in each solution
• C₁, C₂: Concentrations expressed as mass per volume
• V₁, V₂: Volumes of each solution

This approach is essential when dealing with non-ideal solutions or when volume contraction occurs.

Detailed Explanation of Variables and Their Typical Values

• Concentration (C): Usually expressed as percentage weight/volume (% w/v), molarity (mol/L), or molality (mol/kg). Common values range from 0% (pure solvent) to saturation limits depending on solute and solvent.
• Volume (V): Measured in milliliters (mL), liters (L), or other volume units. Volumes must be consistent across calculations.
• Mass (m): Mass of solute, typically in grams (g). Calculated by multiplying concentration by volume when concentration is in % w/v.
• Final Concentration (C_f): The concentration after mixing, which depends on initial concentrations and volumes.

Understanding these variables and their units is critical for accurate calculations and avoiding common errors such as unit mismatches or incorrect assumptions about volume additivity.

Real-World Applications and Case Studies

Case Study 1: Pharmaceutical Solution Preparation

A pharmacist needs to prepare 500 mL of a 12% saline solution by mixing a 20% saline stock solution with distilled water (0% saline). How much of the 20% stock solution and water are required?

Given:

• Desired volume, V_f = 500 mL
• Desired concentration, C_f = 12%
• Stock solution concentration, C₁ = 20%
• Water concentration, C₂ = 0%

Solution:

Using the dilution formula:

Rearranged to find V₁ (volume of stock solution):

Volume of water added:

Interpretation: Mixing 300 mL of 20% saline with 200 mL of water yields 500 mL of 12% saline solution.

Case Study 2: Industrial Chemical Mixing

An industrial process requires mixing 400 mL of a 15 M sulfuric acid solution with 600 mL of a 5 M sulfuric acid solution. What is the final molarity of the mixture?

Given:

• C₁ = 15 M, V₁ = 400 mL
• C₂ = 5 M, V₂ = 600 mL

Solution:

Using the basic concentration mixture formula:

Calculate numerator:

Calculate denominator:

Final concentration:

Interpretation: The resulting solution has a concentration of 9 M sulfuric acid.

Additional Considerations for Accurate Mixture Calculations

• Volume Additivity: In many cases, volumes are assumed additive, but some mixtures exhibit volume contraction or expansion. This is especially true for concentrated solutions or mixtures involving solvents like ethanol and water.
• Temperature Effects: Concentrations and volumes can vary with temperature due to thermal expansion or contraction. Calculations should consider temperature if precision is critical.
• Units Consistency: Always ensure volumes and concentrations are in compatible units before performing calculations to avoid errors.
• Solution Density: For mass-based calculations, knowing the density of solutions can improve accuracy, especially when converting between mass and volume.
• Chemical Reactions: If solutes react upon mixing, simple additive formulas do not apply. Stoichiometric calculations and equilibrium considerations become necessary.

Useful External Resources for Further Study

• Chemguide: Concentrations of Solutions – Detailed explanations of concentration units and calculations.
• Chemistry Explained: Concentration – Overview of concentration concepts and practical examples.
• Engineering Toolbox: Solution Concentration – Engineering-focused resource on solution concentration and mixing.
• NIST: Volume and Density Standards – Authoritative data on volume and density for precise calculations.

Summary of Key Points

• Mixture concentration calculations rely on mass conservation of solute and volume additivity assumptions.
• Formulas vary depending on whether the goal is to find final concentration, required volume, or dilution parameters.
• Real-world applications span pharmaceuticals, industrial chemistry, food science, and environmental engineering.
• Attention to units, temperature, and solution properties ensures accuracy and reliability.
• Extensive tables and examples facilitate quick reference and practical understanding.

Mastering the calculation of mixtures of two solutions with different concentrations is essential for professionals in chemistry, chemical engineering, and related fields. This article provides the technical foundation and practical tools to perform these calculations with confidence and precision.