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.
V = nRT/P
Now consider a trace gas in a mixture. Let V1 be the volume of the trace gas and VT 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/VT = n1RT1PT/nTRTTP1
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/VT = n1/nT
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)
(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.