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Pharmacokinetics – First Order Rate Equation
Derivation by David McAuley, Pharm D.
concentration time curve By definition, first order process: 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

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