Physics question?: moles and length?

Question

A typical sugar cube has an edge length of 1 cm. If you had a cubical box that contained a mole of sugar cubes, what would its edge length be? (one mole = 6.02x10^23) The answer is 8x10^2 km. Could you show me the steps involved in solving this problem?

Answer 1

So, a mole is just a number -- 6.02x10^23.  A mole of sugar cubes must contain 6.02x10^23 of them.

Let's back up a bit.. what if I said you had a cubical box that contained 8 sugar cubes?  You should be able to work out (just by playing with some cubes), that this would have sides of length 2 (2 by 2 by 2).  Now, what if you had sides of length 3?  Well, the total number of cubes would be 3x3x3.  So, if the sides had length N, then there would be a total of N^3 cubes!

So, you are asked to find a total of 6.023x10^23 cubes -- what are the sides?  N^3 = 6.03x10^23.  So N equals the cube root of 6.02x10^23.

Play with a calculator (or google), and you should find that

(6.02 * (10^23))^(1 / 3) = 84 436 877.3

Really, since you were only given 3 significant digits, you should only retain 3 digits in your answer:  84,400,000.  That is 84.4 million cubes on a side, or 84.4 million cm, or 844 km, or about 524 miles.