A simple tracer gas technique can determine the ventilation
rate in any room or volume of air. Last
week I discussed natural logarithms with base

*e*. One of the most useful equations involving natural logs is the equation for first-order decay. This sounds complicated but all it really means that the rate of loss or decay of a concentration is directly proportional to the amount remaining. Thus, for example, the rate of decay (or material loss) for a ventilated room volume concentration that is 10 ppm will be twice as high as the loss rate for a room concentration of 5 ppm under the same ventilation conditions. More important, the time it takes to get from 10 ppm to 5 ppm is the half-life. Indeed, it is the same amount of time (i.e., the half-life) that it will take decay or go from 5 ppm to 2.5 ppm, 2.5 ppm to 1.25 ppm, etc. That is really all there is to first-order kinetics. Theoretically you never get to zero but after 7 half-lives the concentration is less than 1% of what you started with and after 10 or so it is vanishingly small.
The decay rate of a gas concentration put into a room typically follows a first order decay process driven by the ventilation rate.

The equation is: -
ln (0.5)/(Q/V) = (0.693)(V/Q) = the
half-life = t

_{1/2}
Thus the ventilation rate (Q) = (0.693)(V/ t

_{1/2})
Consider a room in a home that has a volume (V) of 20 m

^{3}. We put a tracer gas into this room and mix it up for a few minutes with portable fans so that the average concentration is 16 ppm measured with a real time monitor. We get the following data over the next few hours: 12.5 ppm at 0.5 hours, 9.7 at 1 hour, 7.9 at 1.4 hours and 4.1 at 2.8 hours. You would typically have an automatic data-logger to take a reading every 5 minutes or so but the above data points tell the story. The initial concentration goes in half after about 1.4 hours and in half again in the next 1.4 hours; the half-life is about 1.4 hours. Put 20 m^{3}and 1.4 hours in the above equation and Q = 9.9 m^{3}/hr. Another way of expressing this is the ratio Q/V or 0.50 air changes per hour. This is a typical ventilation rate for a home with doors and windows closed in the winter. Crack the windows open and this can go up 10 fold. Industrial rooms are typically higher and in “hot” industrial settings or rooms with a lot of local exhaust ventilation it could be much higher.
What are good tracer gases to use? Freon R-134a used in car air conditioners is
easy to get and it works quite well with a portable flame ionization detector
(FID). If you are

__very__careful not to release too much (say 10 ppm max), carbon monoxide (CO) could work well with a suitably sensitive real-time analyzer that can continuously output the concentration in ppm. In industrial rooms with active gas fired fork lifts there is often enough ambient CO in the room that it represents a “built-in” tracer simply by having the folk lift operators stop for an hour or so (e.g. waiting for lunch or the end of the day) and measuring the CO concentration fall off will give you the data you need.
Next week I will show you more about “

*e”*and natural logarithms; specifically, we will go over how the above equations and similarly useful algorithms are derived. We will also get into how one can use an Excel spreadsheet to get a more precise estimate of half-life from tracer gas studies.
In the meantime, my friend and colleague Dr. Mark Nicas, who
during his career has developed some really useful exposure models and also
teaches at Berkeley, has kindly sent me a document which has 18 pages packed
with quantitative math and science review notes. I would be happy to send it to anyone who
sends me an email request to mjayjock@gmail.com. The stuff in this pdf is pretty
concentrated with useful information and perhaps it would spawn some questions
that I could address in future blogs.

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