How do I figure out the composition of a material given specific densities?

Question

The Problem

A waste material consists of sand (SG 2.7), a chemical salt (SG 2.1), and a dense plastic (SG 1.4). 200 g of a representative sample from the material is found to have a density of 2300 kg/m³. The salt is dissolved out of the sample which, after drying, now weighs 170 g. What is the composition of the waste material?

My Attempt

1. Based on the given information, we can derive:

density_sand = 2700 kg/m³

density_salt = 2100 kg/m³

density_plastic = 1400 kg/m³

mass_salt = 30 g = 0.03 kg

proportion_salt = 0.03 kg / 0.2 kg = 0.15

2. To try and calculate the individual mass of sand and plastic in the sample, I used the density = mass / volume equation to first figure out the volume using the density and mass of the total sample.

density_total = mass_total / volume =

volume = mass_total / density_total = 0.2 kg / 2300 kg/m³ = 0.00008696 m³

mass_sand = 2700 kg/m³ * 0.0000869 m³ = 0.2348 kg

mass_plastic = 1400 kg/m³ * 0.0000869 m³ = 0.1217 kg

But I don't think this step was the right choice since the mass of the sand came out to be larger than the mass of the sample. Was I correct in trying to figure out the value of the mass of sand and plastic or was that the wrong approach to determine the material's composition? If it's the first case, how should I find out the mass values?

Answer 1

Your mistake is that you have calculated the mass of a sample with the same volume as ours consisting entirely of sand, and a sample with the same volume as ours consisting entirely of sand.

volume = mass_total / density_total = 0.2 kg / 2300 kg/m^3 = 0.00008696 m^3

gives the initial volume of the sample before the salt is removed, however by multiplying this by the densities of sand and plastic, we calculate the mass of a 0.0000869 m^3 sample of sand and a 0.0000869 m^3 sample of plastic.

First we need to find the new volume and density of the sample after the salt has been removed. This can be done by calculating the volume of salt removed: volume_salt=0.03kg_salt/2100 kg/m^3=0.00001429m^3 By subtracting this volume from the initial volume you already calculated:

0.00008696m^3-0.00001429m^3=0.00007267m^3

dividing the new mass by this gives:

0.170kg/0.00007267m^3=2,339kg/m^3

Now we need to use this to calculate the proportion. We can do this by creating and solving a system of equations. We know that some mass of sand plus some mass of plastic gives us a 170 gram mixture, and that some volume of sand plus some volume of plastic gives us a mixture with a volume of 0.00007267m^3.

xkg_sand+ykg_plastic=0.170kg

am^3_sand+bm^3_plastic=0.00007267m^3

Now, we know that the volume and mass of each material are related by their densities, and because of this it is possible to replace the terms in one equation with the terms in the other by relating them through density, so if we decide to solve for mass, our volume equation becomes:

(xkg_sand/(2700 kg/m^3))+(ykg_plastic/(1400 kg/m^3))=0.00007267m^3

Because the x and y in these equations are the same, we can solve a system, which yields:

x=0.1418kg

y=0.0282kg

Dividing these by our starting mass gives:

proportion_sand=0.709

proportion_plastic=0.141

These can be rounded to match your answer if needed, however these proportions combined with your calculated salt proportion adds up to 1.