Density of Alcohol-Water Mixture: Understanding Additivity and Correct Calculation Methods

When dealing with the mixing of liquids such as water and alcohol, it is crucial to understand the concept of additivity of volumes and the correct method to calculate the density of the mixture. In this article, we will explore the calculation of the density of a mixture formed by adding 2.5 litres of water to 8 litres of alcohol, and discuss the importance of accurate volume additivity.

Introduction

Many common scenarios involve mixing different liquids, and in such cases, the density of the resulting mixture is often required. Understanding the principles behind the calculation of the density of such mixtures is essential, especially when dealing with substances that may not have additive volumes. This article will guide you through the correct method to calculate the density of a mixture of water and alcohol, highlighting the importance of accurate calculations.

Step-by-Step Calculation

Given the information:

Volume of water 2.5 litres Density of water 1 g/cm3 Volume of alcohol 8 litres Density of alcohol 0.8 g/cm3

We will calculate the density of the mixture using the formula for the weighted average of the densities:

[ text{density}_{text{mixture}} frac{text{mass}_{text{water}} times text{density}_{text{water}} text{mass}_{text{alcohol}} times text{density}_{text{alcohol}}}{text{mass}_{text{water}} text{mass}_{text{alcohol}}} ]

Step 1: Calculate the mass of water and alcohol

Mass of water volume of water × density of water 2.5 L × 1 g/cm3 2.5 kg Mass of alcohol volume of alcohol × density of alcohol 8 L × 0.8 g/cm3 6.4 kg

Step 2: Calculate the total mass of the mixture

[ text{total mass} text{mass}_{text{water}} text{mass}_{text{alcohol}} 2.5 kg 6.4 kg 8.9 kg ]

Step 3: Calculate the density of the mixture

[ text{density}_{text{mixture}} frac{text{mass}_{text{water}} times text{density}_{text{water}} text{mass}_{text{alcohol}} times text{density}_{text{alcohol}}}{text{total mass}} ] [ text{density}_{text{mixture}} frac{2.5 kg times 1 g/cm^3 6.4 kg times 0.8 g/cm^3}{8.9 kg} ] [ text{density}_{text{mixture}} frac{2.5 5.12}{8.9} 0.9 g/cm^3 ]

Therefore, the density of the mixture is 0.9 g/cm3.

Discussion: Importance of Correct Volume Additivity

When substances are mixed, it is often assumed that their volumes add up. However, for substances such as water and alcohol, this is not the case. The volume of the alcohol-water mixture is not simply the sum of the volumes of the individual components. This is an important consideration in the correct calculation of the density.

Mass Percentage and Specific Density

For a more precise solution, we can use the mass percentage and the corresponding density from chemical literature. Given the mass of 12 litres of alcohol (density 0.8 g/cm3), we can calculate:

Mass of alcohol volume of alcohol × density 12 L × 0.8 g/cm3 9.6 kg Total mass mass of water mass of alcohol 3 kg 9.6 kg 12.6 kg Volume of water 3 L 3 000 cm3 Volume of alcohol 12 L 12 000 cm3 Total volume 3 000 cm3 12 000 cm3 15 000 cm3

Using these values, we can calculate the density:

[ text{density}_{text{mixture}} frac{text{total mass}}{text{total volume}} frac{12.6 kg}{15 000 cm^3} 0.84 g/cm^3 ]

Thus, the correct density of the mixture is 0.84 g/cm3.

Conclusion

The correct calculation of the density of a mixture of water and alcohol involves accurately accounting for the non-additive volumes of the liquids. This example demonstrates the importance of understanding the principles of volume and density in mixture calculations. By following the correct method, we arrive at a more accurate and reliable result.