Video Transcript
A prospector panning for gold in a river collects 15.81 grams of pure gold. Calculate, to two decimal places, the number of Au atoms in this quantity of gold. The molar mass of gold is 197 grams per mole. (A) 6.70 times 10 to the 22nd atoms. (B) 7.50 times 10 to the 24th atoms. (C) 4.83 times 10 to the 22nd atoms. (D) 3.63 times 10 to the 23rd atoms. Or (E) 4.41 times 10 to the 23rd atoms.
To answer this question, we need to determine the number of gold atoms in a given mass of gold. If we knew the average mass of a gold atom, we could simply divide the mass of gold by the average mass of a gold atom and determine the number of gold atoms in this quantity. But the average mass of a gold atom is incredibly small and inconvenient to work with. When performing these types of conversions, instead of using the average mass of individual atoms, we tend to use molar mass, which is the mass of one mole of an entity.
But what is a mole? The mole, frequently abbreviated M-O-L, is the SI unit of amount of substance. As a unit, the mole functions in a similar way to the unit dozen. One dozen donuts is equal to 12 donuts. But one mole of donuts is equal to 6.02214076 times 10 to the 23rd donuts. So the mole is a way to represent a very large number of entities. The value shown here is more precise than is needed for most calculations, so this value is often rounded to 6.022 times 10 to the 23rd.
How does this information help us to solve the question? The question provides us with the molar mass of gold. This molar mass tells us that if we had 197 grams of gold, we would have one mole of gold. And since one mole is equal to 6.022 times 10 to the 23rd entities, this amount of gold would contain 6.022 times 10 to the 23rd atoms of gold. So we now have a relationship between the number of atoms of gold and its mass. We can use this relationship to help us determine the number of gold atoms in 15.81 grams of pure gold.
One way we can solve this problem is to use a proportion. From our discussion of molar mass and the mole, we know that 6.022 times 10 to the 23rd atoms of gold will have a mass of 197 grams. We can set this relationship equal to 𝑥 atoms per 15.81 grams. Solving for 𝑥 will give us the number of gold atoms in the quantity of gold the prospector panned. We’ll need to multiply both sides of the equation by 15.81 grams. 15.81 grams will cancel out entirely on the right-hand side of the equation. In addition, the gram units on the left-hand side of the equation will cancel, leaving us with the unit atoms.
Now we can perform the calculation by multiplying 15.81 by 6.022 times 10 to the 23rd atoms, then divide by 197. This gives us an answer of 4.8328 times 10 to the 22nd atoms. If we round this value to two decimal places, we get 4.83 times 10 to the 22nd atoms, which matches answer choice (C). Therefore, the number of gold atoms in 15.81 grams of pure gold is 4.83 times 10 to the 22nd atoms, or answer choice (C).
