A Simple Derivation of the First Order Kinetic Model for Ventilation

Last week I promised to really break down for you the details of the derivation of the equation for first order decay.   You will remember that this means that the amount or rate of decay is proportional to the amount remaining.   The basic equation describing this situation is below:

C is the concentration at any time (t),  C0 is the initial concentration and k is the rate of decay.  The units are mg/m³ for C and C0, hour for t and 1/hours for k.

Let’s start with the curve y = e^-kt .  Where y (the vertical axis below) equals the proportion of the starting (or time = zero) concentration that remains at any time (t) which is the horizontal axis.   Multiply the numbers on the y or verticle axis by 100 and you have the percentage remaining.  You will remember from a previous blog that e is the base number of natural logarithms.   Let’s look at this curve for the value of k = 1/hr which means one mixing volume loss change per hour.    The value k is also equal to ventilation rate in a room expressed as Q/V where Q is the ventilation rate in m³/hr and V is the room volume in m³.  

Proportion Remaining versus time (hours)

If we rearrange the above equation a little we get  C/C0  = e^-kt.   When C/C0 = 0.5 we have the half-life equal to t in the expression  e^-kt .    If you take the natural log of each side of the equation you get:  ln(0.5) =   -0.693 = -kt_1/2 thus  t_1/2   = 0.693/k  = 0.693/(Q/V).   This should look familiar relative to last week’s blog.  Indeed, it is the same equation slightly rearranged to solve for Q.   What we have here is that the half-life at one air change per hour is 0.693 hours or every 41.6 minutes the concentration will go in half.   You can see that after 5 half-lives or about 3.5 hours the concentration has decayed over 95%.   As some say “the solution to pollution is dilution”.

This is the same math used for simple radio-active decay.  A prime example is Carbon14 (C14) dating.  C14 is a component of all living plants and animals and the percentage of  C14 isotope in the total Carbon in these living things is related to how much C14 is in the atmosphere when they were alive.  As it turn out the amount of C14 in the atmosphere is constant enough (or can be determined from historical records of tree rings) so the ratio or proportion of C14 in tissue is known when the plant or animal was alive.  C14 starts to decay with a half-life of 5730 years after the plant or animal dies.  Thus we can date the carbonaceous material from dead things using these same first order kinetics.   

I promised to get into showing you how an Excel spreadsheet technique that can be used to do a precise calculation of ventilation rate using tracer gas data but  I think this was enough for one post and I am going to let that go until next time.