Stoichiometry Tutorials: Composition of Mixtures

Composition of Mixtures Movie Text

On another page, we talked about using elemental analysis as a qualitative analysis technique. In that case, we had a pure mineral and we wanted to determine what type of mineral it was. By performing an elemental analysis, we could determine the empirical formula. This can tell us, for instance, whether the mineral is As₂S₃ or As₄S₄, or something else. That approach is very useful if you think you have a pure mineral. But let's say we have a sample that we know is a mixture of two substance, such as As₂S₃ and As₄S₄. What experiment can be done to determine the relative proportion of each of these substances in our sample?

I can take a chunk of the sample and weight it. Let's say I weigh a piece of sample and find it has a mass of 13.86 grams. I then know the total mass of the mixture, but I don't know how much of it is As₂S₃ versus As₄S₄.

This is another example where chemical reactions provide a useful basis for an analytical chemistry technique. If I burn the sample, the substances react with oxygen to produce SO₂ gas, leaving As metal behind. The experiment I'll perform is to burn my 13.86 grams of sample and measure the amount of SO₂ gas that comes off. To see why this may be useful, consider the way that each of these minerals reacts with oxygen. The balanced reactions are shown here:

As₂S₃ + 3 O₂ → 2 As + 3 SO₂

As₄S₄ + 4 O₂ → 4 As + 4 SO₂

Since the two different minerals have different amounts of S in them, they produce different amounts of SO₂. This is why measuring the amount of SO₂ given off may be useful: the amount given off depends on the relative amount of the two substances in the sample. By measuring the amount of SO₂ produced, I should be able to determine the amount of As₂S₃ versus As₄S₄ in the sample.

Let's say I do this experiment and my 13.86 grams of sample produces 9.23 grams of SO₂. What do I do next? I don't know how many grams I have of each of these substances, I only know the total mass of the mixture and the amount of SO₂ produced. This is a pretty complicated problem, and one where algebra turns out to be very useful. What algebra allows me to do is introduce a variable (let me call it x) that precisely captures what I don't know. I then write all the information I do know in terms of this unknown x. If all goes well, meaning if I have enough information from my experiment, I'll end up with an algebraic expression that I can solve for the unknown x.

The first choice I need to make here is to precisely define my unknown. Let me choose to say that x is the mass of As₂S₃.

x = mass of As₂S₃ in grams

If I knew this value, I would know everything about the sample. Before doing the algebraic solution, let's see what would happen if the sample contained 1.00g of As₂S₃. In other words, let me assume that the value of the unknown x is 1.00.

Since my total mass is 13.86 g, and I have 1.00 grams of As₂S₃, there must be 12.86 g of As₄S₄. I can then use reaction stoichiometry to determine the amount of SO₂ produced by burning this mixture:

Let's begin with the As₂S₃. I first use the molecular weight to convert from the number of grams of this substance to the number of molecules expressed in moles. I then look at my reaction and see that there are 3 moles of SO₂ produced for every mole of As₂S₃ consumed. This gives me the number of SO₂ molecules that are produced. To get the mass of SO₂, I again use the molecular weight. This gives me the mass of SO₂ produced from the 1.00 g of As₂S₃. A similar procedure gives me the mass of SO₂ produced from the As₄S₄. I again use the molecular weight to convert to the moles of As₄S₄. From the chemical reaction, I know that 4 moles of SO₂ are produced for every mole of As₄S₄ consumed, giving me the moles of SO₂ produced. I then use the molecular weight of SO₂ to convert from moles to grams. This gives me the amount of SO₂ produced by burning the As₄S₄ portion of the sample. Experimentally, I am burning the mixture all at once, so I collect the sum of these two or 8.48 g of SO₂.

This means that if my mixture contained 1.00 g of As₂S₃, I would expect to collect 8.48 g of SO₂. In reality I collected 9.23 g of SO₂. If I change the 1.00 g of As₂S₃ to 2.00 g, so that there is now 11.86 g of As₄S₄, and redo all the math, I get 8.66 g of SO₂ produced.

This is still not equal to observed 9.23 g of SO₂ collected experimentally. I could keep guessing at the mass of As₂S₃ present in the sample until I get the observed 9.23 g of SO₂, but it is much easier to use algebra. As mentioned above, algebra allows me define an unknown x, write everything I know in terms of this unknown, and then solve for x.

In this case, our unknown x is the mass of As₂S₃ in the mixture. Since the total mass is 13.86 g, the mass of As₄S₄ must be 13.86-x grams. Reaction stoichiometry allows me to write the amount of SO₂ produced from the As₂S₃ as 0.781 x grams. Similarly, I get the amount of SO₂ produced from the As₄S₄ in terms of x, as 0.599 (13.86-x) grams. I can then sum these to get the total amount of SO₂ expressed in terms of my unknown x.

I now now the total amount of SO₂ produced by burning the mixture, in terms of the unknown x. Experimentally, I collected 9.23 g of SO₂, so I can set this algebraic expression equal to 9.23

0.781 x + 0.599 (13.86-x) g SO₂ total = 9.23 g SO₂ total

Solving this for x gives x = 5.10. x referred to the mass of As₂S₃ present in the sample, so I now know that my mixture contained 5.10 grams of As₂S₃. The remainder of the 13.86 total mass, or 8.76 g, must then be As₄S₄. I now know everything I wanted to know about the sample. I can check my result by calculating the amount of SO₂ produced by this mixture, and I see that I predict the observed 9.23 g of SO₂.

It may also be interesting to quote this result in terms of the relative proportions of the two minerals present in the mixture. As₂S₃ comprises 5.10 g of the 13.86 g mixture, corresponding to 36.8 %. As₄S₄ comprises 8.76 g of the 13.86 g mixture, or 63.2 %.

I've successfully used a reaction to distinguish how much of a sample is As₂S₃ versus As₄S₄. This works because the reaction stoichiometry is different for burning As₂S₃ versus As₄S₄. Burning 1.00 gram of As₂S₃ produces 0.781 g of SO₂, whereas burning 1.00 gram of As₄S₄ produces 0.599g of SO₂.

Since the mass of SO₂ produced on burning 1.00 g of each of these minerals is different, measuring the amount of SO₂ produced on burning a mixture gives us the information we need to determine the relative amount of As₂S₃ and As₄S₄ present in the sample. This technique is a nice example of the role that reaction stoichiometry can play in quantitative analysis.