Can a mole of cans cover the earth?

Answer 1

Assuming these are aluminum soda cans...

For this problem, you want to first know the area one soda can takes up. If the diameter of a typical can is 6.5 cm, then the area of the base is about 33 cm2. Then we find the combined area of one mole of cans. One mole is 6.02 × 1023, so 33 × (6.02 × 1023) = 1.99 × 1025 cm2. This number represents the total area taken up by one mole of cans.*

Now we need to find the surface area of the earth and change either one so the units are compatible. Wikipedia says the surface area of the earth is 510,072,000 km2.

We'll take the area of the earth and change it to cm2 using unit conversion.

510,072,000 km2 × (100,000 cm ÷ 1 km)2 = 5.10072 × 1018 cm2

We can then compare that with the area of cans, and we will realize that the area taken by the cans (1.99 × 1025) is larger than the surface area of the earth (5.10072 × 1018). So we can say conclusively that one mole of cans will be able to cover the earth.

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* Note that if you place cans one after another, a certain amount of space will be taken up in addition to the area of the base of the cylinder because of the circular shape. But because just the area of the base without the extra space was enough to cover the earth, we don't need to do further calculations. If the area of the base only was not enough to cover the earth, then you might want to account for the extra space.