How To Convert To and From Parts-Per-Million (ppm)

Parts-per million (ppm) is a common quantity used in many areas of math and science.  It can be somewhat difficult for some to understand because it is not a true unit.  In fact, ppm is a unitless quantity.  It is analogous to percent.  Percent can refer to just about anything.  Percent means part-per-hundred, per cent (cent meaning hundred).  ppm is an exactly analogous quantity, but it is one part per million instead of one part per hundred.  ppm is a ratio between two numbers that have the same units.  Consider the example of a 5% sales tax.  For every hundred dollars I spend, I must pay 5 dollars in sales tax. So:

      5 dollars/100 dollars = 5%


I would much rather pay a 5 ppm sales tax:

     5 dollars/10⁶ dollars = 5 ppm   

We do not usually refer to money in ppm, but we could.  Ppm is more often found as a concentration, for example, ppm by mass or ppm by volume (sometimes referred to as ppmv).  In nuclear magnetic resonance spectroscopy ppm can be used to describe the amount of chemical shift in frequency (Hz/MHz).  This post focuses on the use of ppm as a measure of concentration.

Parts-Per-Million By Volume (ppmv)

Parts-per-million by volume is a common way of expressing a concentration in the gas phase.  Gases are miscible and, in general, once allowed to come to equilibrium a gas is homogeneous, in other words its constituents are equally mixed.  The SI unit of volume is the cubic meter (m³).  Suppose a constituent of a gaseous mixture is at a concentration of 1 ppm.  If one were to separate the mixture into its components, and measure the volume of each component, there would be 1 m³ of the gas of interest for every 10⁶  m³ of the mixture.

      1 m³ constituent/10⁶  m³ of the mixture  = 1 ppm

The Ideal Gas Law


To fully appreciate the value of such a ratio, one needs to consider what happens when the pressure or temperature changes.  The ideal gas law (which is extremely accurate at atmospheric and lower pressures) states:

      PV = nRT

in which p is the pressure, V is the volume, n is the number of moles (proportional to the number of molecules), R is a constant called the universal gas constant, and T is the temperature.  All quantities are in SI units.

Alternatively:

     V = nRT/P

Now consider a trace gas in a mixture.  Let V₁ be the volume of the trace gas and V_T be the total volume of the mixture.  If we measure the concentration in ppmv we are expressing the concentration as a ratio of volumes.

      V₁/V_T = n₁RT₁P_T/n_TRT_TP₁

Consider now that the gas mixture is at the same temperature and pressure.  Notice that temperature and pressure divide out.  If we express a concentration in ppmv, we can change the pressure or the temperature of the mixture and not change the value of the ratio.

     V₁/V_T = n₁/n_T

This feature can be very useful in the atmosphere, where pressure and temperature change.  Sometimes gas phase concentration is measure in milligrams per cubic meter( mg/m³). Although such units can also be useful, the expression is not insensitive to pressure and temperature.  If I have a sample of gas, and I reduce the pressure of the gas, its volume will expand.  The mass of the gas sample, however, is independent of pressure and does not change.  Therefore the same gas mixture at a lower pressure will have a lower amount of a trace gas expressed in mg/m³, whereas its concentration measured in ppm remains unchanged.

Converting Between mg/m³ and ppm

Sometimes it is important to be able to convert between the units mg/m³ and ppmv.  To make such a conversion it is important to know some things about the situation; so it is useful to use a concrete example.  Suppose the concentration of carbon dioxide in air is 380 ppmv, and we want to express this number in mg/m³ at a pressure of 1 atmosphere (101,325 Pa) and a temperature of 298.15 K.   The starting place is the definition of ppmv:
 
     380 ppmv CO₂  =  380 x 10^-6  m³  CO₂/ 1 m³  air.

The volume of air is already in the correct units; so one only needs to convert the volume of CO₂ to mg and the answer presents itself.    If one knows how many moles of CO₂  are present, it is easy to convert moles to mass.  Again, I rearrange the ideal gas law:

      n = PV/RT

Making sure that everything is in SI units:

      n = (101,325)(380 x 10^-6  )/(8.31441)(298.15)

Also consider:

      (Mass of CO₂  in grams) =  n * (molecular mass of CO₂ in grams)

 The molecular mass is simply the sum of the atomic masses (12.01 + 16.00 + 16.00 = 44.01 grams)


     Mass of CO₂  in grams = (44.01)(101,325)(380 x 10^-6  )/(8.31441)(298.15) = 0.684 g

So, at 1 atm and 298.15 K,

     380 ppm CO₂   = 684 mg/m³ CO₂   

*** Exercise for the Reader: Convert 21.2  mg/m³ of ammonia gas (NH₃) at a temperature of 288K and a pressure of 100,001 Pa to ppm ***If you have difficulties, state them in the comments, and I will help.

Parts-Per-Million By Mass


In liquids, it is common to use ppm by mass as a concentration.  In general the calculations are a lot easier than in the gas phase.

     1 ppm solute   =   10^-6  kg solute / 1 kg solution  = 1 mg solute / 1 kg solution


There is a very convenient way to do this calculation in dilute solutions of water.  The density of water is 1 kg/liter (there is a small variation with temperature, but I neglect that here.  Moreover, in dilute aqueous solutions most of the mass is from water:

    1 ppm solute = 1 mg solute / 1 kg solution  ≈  1 mg solute/1 kg water  = 1 mg solute/1 liter water

It is possible to be more accurate by correcting by the known or measured density of the solution.