If one puts 1 mg of a salt in 1 liter of water the new volume is going to be very close to 1 L and the new mass is going to be 998203 + 1 = 998204 mg (at 20 °C). The concentration of the salt is thus 1 part salt per 998204 parts solution by weight. That's pretty close to 1 part per million (difference of 0.18%) and so we often use ppm as a approximation to mg/L. Note that a 1 mg/L solution is prepared by putting 1 mg of the salt in a 1 L volumetric flask and adding water up to the mark. The ppm number will vary with temperature and the particular salt as different salts will occupy different volumes requiring more or less water to make 1 L of solution. When you see, for example, 1 ppm sodium it usually means each liter of the solution (at 20 °C) contains 1 mg of the ion even though the actual ppm in that case will not be exactly 1 ppm. To prepare, for example, a (US) gallon of NaCl solution of strength 2 ppm one would observe that a US gallon contains 3.7854 L and add 2*3.7854 mg of NaCl to the gallon of water. The resulting solution is closer to 2 mg/L than 2 ppm but for brewing application the difference is not significant. It is rare to see sodium, chloride, sulfate etc. specified in ppm.
One is more likely to see calcium and magnesium specified in ppm and if one does he must be careful. Calcium and magnesium are mostly specified in terms of mg/L just as other metal ions. But if ppm are specified then it can mean either mg calcium ion per liter (approximately, as with other metals), in which case the report should say 'as the metal' or 'as the ion', or it can mean 'ppm as calcium carbonate' in which case the number should be divided by 50 (half the molecular weight of calcium carbonate, mg/millimol) and multiplied by 20 (mg/milliequivalent), the equivalent weight of calcium or, in the case of magnesium by 12.15, its equivalent weight to get mg/L as the ion. In the case of calcium the ppm number gives the amount of lime which, if dissolved by carbonic acid, would result in the observed or desired calcium concentration.
The other quantity one nearly universally sees listed in ppm as CaCO3 in the American literature is alkalinity. Divide it by 50. This is the number of cc of 0.1 N acid which the analyst added to 100 mL of the water sample to reduce its pH to 4.5.
In summary if an American recipe says to add calcium to the extent of 50 ppm what you really need to worry about is whether he meant the weight of calcium (mg/L as the ion) or 50 ppm as CaCO3 'hardness'. Sometimes it is hard to tell though there are some things to look for. If you see Calcium: 50, Magnesium: 100, Total Hardness 150 then the numbers are as CaCO3. If, OTOH, you see Calcium: 20, Magnesium: 24.3, Total Hardness 150 then the calcium and magnesium numbers are as the ion. Another clue is that if the calcium and magnesium numbers are approximately equal to the alkalinity they are ppm as CaCO3. Sometimes, though, you have to calculate the sum of all the ion charges using both ways. The one that gives you the answer closest to 0 is the correct one.
