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Pharmacokinetics - First Order Rate Equation

Derivation by David McAuley, Pharm D.

concentration time curve
By definition, a first order process can be described as follows:
deltaC/deltaT proportional C

[The rate of change in concentration versus time is directly proportional to the concentration 'C' at any point along the curve]. Note: if you graph the log of the concentration versus time, a linear graph will result.

Using the concept of variation in relation to proportionality you get:

deltaC/deltaT = kC
(The proportionality is replaced by the constant 'k')
'k' is known as the coefficient of variation - "direct proportionality constant". Also, because we are dealing with an elimination process (concentration is decreasing), a negative sign is added:

deltaC/deltaT = - kC or rearranging: deltaC/C = - k deltaT

Next, we will consider a particular time and use the the short hand seen in calculus "with respect to".
d[C]/C = -kdt
Next, using a definite integral and integrating from C -->Co and T --> 0

1st order

Result: lnC - lnCo = -k[ t - 0] or
lnC - lnCo = -kt or
lnC = lnCo -kt

Raising everything to the inverse log (base e) you get: C = Co e-kt

Application

Additional background information can be found here.
Pharmacokinetics