What is the approximate density of C₃H₆ gas at zero degrees Celsius and one atmosphere? A) 1.3 grams per litre, B) 1.9 grams per litre, C) 4.2 grams per litre, D) 18 grams per litre, or E) 42 grams per litre.
It really isn’t that common to think about the density of a gas. Images like this, where we see particles of gas bouncing around inside a box, are quite common. And we’d be seeing side by side a similar picture of a solid in the same box, but much more compact. The truth is actually a lot more exaggerated. For the same material, at one atmosphere, the gas will probably take up about 1000 times as much space as the solid. So while solids typically have densities on the order of one gram per millilitre, gases have densities closer to one gram per litre. So we’d be looking for an answer that’s closer to one gram per litre down to one gram per millilitre.
There aren’t very many organic compounds with the formula C₃H₆. So we can make a good guess that we’re dealing with propene. But what do we know about propene that will allow us to calculate its density? Well, the easiest thing to do is to imagine that propene is an ideal gas. That’s a gas where the particles are infinitely small and they always collide elastically. By “elastically” we mean that when the particles collide with each other or the walls, the total kinetic energy doesn’t change.
We can use this model to our advantage because ideal gases obey Avogadro’s law, which states that the volume of a gas is directly proportional to the number of moles of gas in the sample. This means when pressure and temperature are constant, we can calculate the volume by multiplying the amount of the sample by a constant. The upshot of this is that, for any gas, regardless of the nature of the molecules, the volume divided by the amount is always equal to a constant at specific conditions. We sometimes call this the molar volume, with the symbol 𝑉 m. It just happens to be that the molar volume at zero degrees Celsius and one atmosphere is 22.4 litres per mole to three significant figures.
Now how do we use that to our advantage? For that, we’re going to need to go back to the definition of density. The density is simply the mass of the sample divided by its volume. So to work out the density, what we could do is take a sample of gas, measure its volume, and then measure its mass. But we have a better way of doing this. We know that we can calculate the mass of our gas by multiplying the amount in moles by the molar mass of the gas in grams per mole.
When we combine these two equations, we get the expression density is equal to the amount in moles multiplied by the molar mass divided by the volume. But this bit here looks very similar to the molar volume, only the other way up. If we substituted in, we get the density equal to the molar mass divided by the molar volume.
All that remains is to work out the molar mass of C₃H₆ and plug it into our equation with the molar volume. To work out the molar mass of C₃H₆, we’re going to need the atomic masses of carbon and hydrogen. We’re only after an approximation here. So we can use the rounded values of 12 unified atomic mass units for carbon and one unified atomic mass unit for hydrogen. We then multiply the atomic masses by the number of atoms per molecule of that element and then convert the units to grams per mole. This gives the contribution from each element to the molar mass of C₃H₆. This gives us 42 grams per mole. So one mole of C₃H₆ will have a mass of 42 grams.
So we can calculate the density of propene gas at zero degrees Celsius and one atmosphere by multiplying the molar mass, 42 grams per mole, of C₃H₆ by one mole of C₃H₆ per 22.4 litres. This gives us 42 divided by 22.4 grams per litre, which is about 1.9 grams per litre.
Another way to tackle this question would’ve been to use the ideal gas law. This yields density in terms of the pressure, the molar mass, the ideal gas constant, and the temperature. However, since the question is looking for the approximate density, we can use the rule of thumb that the molar volume for any gas at zero degrees C and one atmosphere is 22.4 litres per mole. Giving us the approximate density of C₃H₆ gas as 1.9 grams per litre.