Well, well, well, looks like we have a solute trying to make a run for it. Let's see if we can help bring it back.
First off, we need to calculate the amount of solute that can be extracted by ether. The distribution coefficient tells us the ratio of the concentration in ether to the concentration in water. In this case, it's 20. So if we have 15 mL of solution, we can extract 20 times the amount of solute from that volume into ether.
Now, to calculate the amount of solute in the ether layer, we need to consider the volume of ether used. We have 13.9 mL of ether, so we multiply that by the distribution coefficient of 20. This gives us the amount of solute that can be extracted, which is 278 mL.
But hold your horses! We need to convert that mL value to grams, since we were given the initial amount of solute in grams. We have 2.0 g of solute in an initial 15 mL of solution, so the concentration of the solute is 2.0 g / 15 mL. Multiplying this concentration by the amount of solute extracted in ether, we get 2.0 g / 15 mL * 278 mL = 37.07 g.
Finally, to calculate the percent recovery, we divide the amount of solute extracted by the initial amount of solute and multiply by 100. So the percent recovery is (37.07 g / 2.0 g) * 100 = 1853.5%.
Well, it seems like we're in luck! With a single extraction using 13.9 mL of ether, we can recover a whopping 1853.5% of the solute. I suppose that's what happens when a solute is determined to escape – it really gives it all it's got!