Which of the following expressions is equivalent to the number of copper, Cu, atoms present in a pure sample of solid copper with a mass of 50.0 grams, given that the atomic mass of copper is 63.5 u?
What we’re looking for in this question is the expression that, when we evaluate it, will give us the number of copper atoms in 50.0 grams of copper. We don’t need to actually find the final value. We just need to find a way of working it out that looks like one of the expressions. To help us along our way, it’ll help if we identify the source of the numbers in each expression.
50.0 appears in all five options, and it’s equivalent to the mass of our sample without the units of grams. 63.5 appears in four out of the five options, and it’s equivalent to the atomic mass of copper without our units. 29 appears in only one of the options, and if you look at your periodic table, you’ll notice it’s the same as the atomic number of copper. And you should recognize the last number as Avogadro’s number, the number of entities in one mole of entities, given to three significant figures.
Now, let’s have a think about how we would go about working out the number of atoms in a sample. Well, we could start with a mass and then divide it by the mass per atom. This would get us the number of atoms in that sample mass. But we have our mass in grams and our atomic mass in u, unified atomic mass units. One unified atomic mass unit is equivalent to one twelfth of the mass of a carbon-12 atom under specific conditions.
You might see the abbreviation AMU in this context, although atomic mass units are defined slightly differently. One unified atomic mass unit is equivalent to 1.66054 times 10 to the minus 27 kilograms or, if we convert it to grams, 1.66054 times 10 to the minus 24 grams.
So, we’ve got the atomic mass of copper, which is 63.5 unified atomic mass units, which is the same as 1.05 times 10 to the minus 22 grams. So, our expression could look like this, where we have the mass and we divide it by the mass per atom. However, this doesn’t resemble any of the expressions that we can see. So, we need to think up a different route.
Avogadro’s number is related to moles, so maybe we need to convert our mass to moles first. So, going back, we convert mass to moles by taking our mass and dividing by the molar mass, which is the mass per mole. We can then convert from moles to number of atoms by taking our mass divided by our molar mass and multiplying by Avogadro’s constant, which is the number of entities per mole. This is just Avogadro’s number with the units of per mole.
So, let’s start at the beginning. We have our mass of sample, 50.0 grams. The molar mass of copper is simply the atomic mass where we substitute in the units of unified atomic mass units with grams per mole. Now, we can convert from mass to moles of copper atoms by dividing through by the molar mass. And lastly, we can multiply through by Avogadro’s constant in terms of copper atoms per mole of copper. When we divide through, we’re left with our expression.
Be careful here not to do the actual calculation. Just keep the individual numbers. After cancelling units, we get the expression 50.0 multiplied by 6.02 times 10 to the 23 divided by 63.5 copper atoms. This matches with answer E. If we evaluated this expression, we would get the number of copper atoms in 50.0 grams of copper.