Video: Calculating the Volume of a Mass of Gas at Standard Temperature and Pressure Given the Mass of a Given Number of Molecules

The mass of 6.02 × 10²³ molecules of a gas is 16.0 g. What volume does 4.00 g of that gas occupy at standard temperature and pressure? [A] 5.60 litres [B] 8.00 litres [C] 11.2 litres [D] 16.0 litres [E] 22.4 litres

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Video Transcript

The mass of 6.02 times 10 to the 23 molecules of a gas is 16.0 grams. What volume does 4.00 grams of that gas occupy at standard temperature and pressure? A) 5.60 litres, B) 8.00 litres, C) 11.2 litres, D) 16.0 litres, or E) 22.4 litres.

The number 6.02 times 10 to the 23 might seem quite arbitrary. It’s an absolutely huge number of molecules. But the key here is to recognise that 6.02 times 10 to the 23 is Avogadro’s number. One mole is equivalent to an Avogadro’s number of something. And Avogadro’s number, to three significant, figures is 6.02 times 10 to the 23. So, the beginning of our question could be rephrased as the mass of one mole of a gas is 16 grams. From this, we can infer that the molar mass of our gas is 16.0 grams per mole.

Now, let’s have a look at the next bit. We’ve been told the mass of a sample of this gas, 4.00 grams worth. And we need to figure out how much space that gas would naturally occupy at standard temperature and pressure. Standard temperature and pressure means zero degrees Celsius and one bar of pressure, which is very close to one atmosphere. Now, how do we go about figuring out how much space a gas will occupy under these conditions?

For that, we can call on Avogadro’s law, which reminds us that the volume of a gas is proportional to the number of gas particles. In symbolic notation, we can write this as 𝑉. The volume of the gas is directly proportional to 𝑛, the amount of gas particles in moles. When two things are directly proportional, what it means is that you can find the value of one by multiplying the value of the other by a constant. For Avogadro’s law, we assume that we’re dealing with constant pressure and temperature. By rearranging the equation, we get a term which is the volume per mole. This is otherwise known as the molar volume, given the symbol 𝑉 𝑚.

There’s a handy rule of thumb we can use here, which is that at standard temperature and pressure, or STP, the molar volume of any gas is 22.4 litres per mole. Meaning that if we have one mole of gas, its volume will be 22.4 litres. If we have two moles of gas, its volume will be 44.8 litres. This value can be calculated via the ideal gas law. But for this question, we’re just going to use the value 22.4 litres per mole.

So, going back to the question, we have the mass of gas. And where we want to get to is the volume. And in-between, we need to work out the number of particles in that gas. And we can do that using moles. We can convert from mass to amount by dividing by the molar mass. And then, our last step will be converting the amount in moles of our gas to volume by multiplying by the molar volume. So, let’s get started. Our volume is equal to our mass divided by our molar mass. So, we take 4.00 grams and multiply it by one mole per 16 grams. If we evaluated the expression at this point, we’d know that we have 0.25 moles of gas.

Now, we can go from moles to volume by multiplying by the molar volume at standard temperature and pressure, 22.4 litres per mole. This leaves us with 0.25 times 22.4 litres. It may help with the mental arithmetic if you convert to using a fraction. So, 0.25 is equivalent to one-quarter, which is the same as a half multiplied by a half. One-half times 22.4 is 11.2. And if we halve that, we get 5.6. Therefore, the volume our 4.00 grams of gas would occupy at standard temperature and pressure is 5.60 litres.

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