Converting Molarity to PPM

Converting molarity to ppm (parts per million) is simple using the factor label method.  Let’s first look at what ppm (or pph, or ppt) stands for.  PPM means parts per million, pph is parts per hundred, and ppt is parts per thousand.  The following expression is another way to express ppm:

\frac{g solute}{g solvent} * 10^6

Lets try an example where we have a solution of 25mM Ca in water.  How would we convert this solution to ppm?  To solve this problem we will set up a factor label problem.  You are probably quite familiar with the factor label method for converting 1 unit (or tag) to another tag (i.e. 300milliliters to liters).  Our calcium problem is a bit more complicated because now we have tags on the top and on the bottom.  On the top we have mmol and on the bottom we have liter.  Both of those tags need to be converted to grams for us to determine ppm.

25mM = \frac{25mmol}{liter H_20}

Step 1:  Convert the top tag (25mmol) to grams.  Since we want to convert a mmol quantity to grams, we will use the element’s molar mass.  For an explanation of molar mass, see this post.  Molar mass has the units \frac{grams}{mol} so we must first convert mmol to mol:

\frac{25mmol Ca}{L H_20} * \frac{1mol Ca}{1000mmol Ca}

Now we can add calcium’s molar mass:

\frac{25 mmol Ca}{L H_20} * \frac{1 mol Ca}{1000 mmol Ca} * \frac{40 g Ca}{1 mol Ca}

We are now done with the top tag; mmol has been converted to grams.  Lets tackle the bottom tag (liter H20).

Step 2: Our next step is to convert  liters of water to grams of water.  The great thing about the factor label method is that we can just continue with our last equation to convert liters to grams.  This step requires the density of water (1g/ml).  To use density, we must first convert liters to milliliters, then we can insert the density of water:

\frac{25mmol Ca}{L H_20} * \frac{1mol Ca}{1000mmol Ca} * \frac{40g Ca}{1mol Ca} * \frac{1L H_20}{1000 mL H_20} * \frac{1mL H_20}{1gH_20}

Notice we flipped the density of water around so that mL is on the top.  This way mL will cancel out and we will be left with grams H20 in the bottom.

Since the tags cross out at each step, our only remaining tags are \frac{g Ca}{gH_20}.  Multiply by 1,000,000 (or 100 if you want pph; 1,000 if you want ppt) and you have converted 25mM Ca into ppm!  Can you convert ppm to molarity?

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