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Komplexe Reaktionen

Parallelreaktion: Integration des Differentialgleichungssystems

Beziehungen

\frac{d a}{d t} = - k_{1} a - k_{2} a

\frac{d b}{d t} = k_{1} \cdot a

\frac{d c}{d t} = k_{2} \cdot a

Konzentration von A

Integration von Gl.(1):

\frac{d a}{a} = - (k_{1} + k_{2}) d t

\int_{a_{0}}^{a} \frac{1}{a} d a = - \int_{t_{0} =0}^{t} (k_{1} + k_{2}) d t

ln \frac{a}{a_{0}} = - (k_{1} + k_{2}) t

a = a_{0} \cdot e^{- (k_{1} + k_{2}) t}

Konzentration von B

Gl.(7) in Gl.(2) einsetzen:

d b = k_{1} \cdot a_{0} \cdot e^{- (k_{1} + k_{2}) t} d t

Integration

\int_{b_{0} =0}^{b} 1d b = k_{1} \cdot a_{0} \int_{t_{0} =0}^{t} e^{- (k_{1} + k_{2}) t} d t

{[b]}_{0}^{b} = k_{1} \cdot a_{0} {[- \frac{1}{k_{1} + k_{2}} e^{- (k_{1} + k_{2}) t} + \frac{1}{k_{1} + k_{2}}]}_{0}^{t}

b = \frac{k_{1}}{k_{1} + k_{2}} a_{0} (1 - e^{- (k_{1} + k_{2}) t})

Konzentration von C

Gl.(7) in Gl.(3) einsetzen:

d c = k_{2} \cdot a_{0} \cdot e^{- (k_{1} + k_{2}) t} d t

Integration

\int_{c_{0} =0}^{c} 1d c = k_{2} \cdot a_{0} \int_{t_{0} =0}^{t} e^{- (k_{1} + k_{2}) t} d t

{[c]}_{0}^{c} = k_{2} \cdot a_{0} {[- \frac{1}{k_{1} + k_{2}} e^{- (k_{1} + k_{2}) t} + \frac{1}{k_{1} + k_{2}}]}_{0}^{t}

c = \frac{k_{2}}{k_{1} + k_{2}} a_{0} (1 - e^{- (k_{1} + k_{2}) t})

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