How to Determine the Density of a Mixed Sample Using Mass Fractions

Understanding how to determine the density of a mixed sample is crucial in many scientific and industrial applications. This article explains both the traditional method using mass and volume and the more practical method involving mass fractions. For accurate measurement and analysis, this guide provides clear steps and examples to ensure you achieve the correct density.

Traditional Method: Using Mass and Volume

The traditional method for determining the density of a mixed sample involves measuring the mass and volume of each component and then calculating the overall density. The formula used is:

Density (ρ) Mass (m) / Volume (V)

For a mixed sample composed of multiple components, the overall density can be calculated using the following steps:

Determine the Mass of Each Component: Measure or calculate the mass of each component in the mixed sample. Determine the Volume of Each Component: If the volume of each component is known, you can use it directly. If not, you can calculate the volume using the density of each component using the formula: Total Mass:

mtotal m1 m2 m3 … mn

Total Volume:

Vtotal V1 V2 V3 … Vn

Calculate the Overall Density:

ρmixed mtotal/Vtotal

Practical Method: Using Mass Fractions

Often, the mass fractions of components are known, but not their exact masses. In such cases, the density of the mixed sample can be calculated using mass fractions with the following formula:

ρt 1 ÷ (Xa/ρa × Xb/ρb)

Where:

ρt is the total or net density. a and b are the components. Xa is the mass fraction of component a. ρa is the density of component a.

Proving the above formula:

ρt 1 ÷ (Xa/ρa × Xb/ρb)

Proof:

ρt Mt ÷ (Ma/ρa × Mb/ρb)

Where:

Mt is the total mass.

Then,

1/ρt (Ma/ρa × Mb/ρb)Mt

1/ρt MaMbMt(1/ρa)(1/ρb)

1/ρt (Ma/Mt(1/ρa) × (Mb/Mt(1/ρb)

1/ρt Xa(1/ρa) × Xb(1/ρb)

ρt 1 / (Xa(1/ρa) × Xb(1/ρb))

Example:

Suppose you have a mixed sample consisting of two substances:

Substance A: 200 g with a density of 2 g/cm3 Substance B: 300 g with a density of 1 g/cm3

Calculate the volume of each substance:

VA 200 g / 2 g/cm3 100 cm3 VB 300 g / 1 g/cm3 300 cm3

Calculate total mass and volume:

Total Mass: mtotal 200 g 300 g 500 g Total Volume: Vtotal 100 cm3 300 cm3 400 cm3

Calculate overall density:

ρmixed 500 g / 400 cm3 1.25 g/cm3

This method allows you to find the density of a mixed sample accurately, especially when working with mass fractions.

Additional Tips:

Always ensure the mixture is on a mass scale to read its mass accurately. If it is a liquid mixture, subtract the mass of the container holding the liquid. Divide the mass by the volume to determine the density.

Understanding the formula and the practical application of these methods can greatly enhance your ability to determine the density of mixed samples efficiently and accurately.