Three species reversible reaction rate

[babbagee]

The problem statement says to derive the concentration as a function of time for the general three species first order reactions. I have attached an image of the problem, so see the image for the reaction.



I have written the reaction rates for all the reactions but I am having trouble after that.

The reaction rates that I have are,

$\frac{dCA}{dt}$=-k$_{2}$C$_{A}$+k$_{2}$C$_{Q}$-k$_{3}$C$_{A}$+k$_{4}$C$_{S}$

$\frac{dCQ}{dt}$=k$_{1}$C$_{A}$-k$_{2}$C$_{Q}$-k$_{5}$C$_{Q}$+k$_{6}$C$_{S}$

$\frac{dCS}{dt}$=k$_{3}$C$_{A}$-k$_{4}$C$_{S}$+k$_{5}$C$_{Q}$-k$_{6}$C$_{S}$

Where k1 and k2 are the forward and reverse rate coefficient for A$\rightleftharpoons$Q reaction

k4 and k3 are the forward and reverse rate coefficient for A$\rightleftharpoons$S reaction

and
k5 and k6 are the forward and reverse rate coefficient for Q$\rightleftharpoons$S reaction

from stoichiometry, I also know that C$_{A}$+C$_{Q}$+C$_{S}$=C$_{A0}$+C$_{Q0}$+C$_{S0}$=C$_{T0}$ where C$_{T0}$ is the total mols

Initial Conditions:
C$_{A}$(t=0)=C$_{A0}$
C$_{Q}$(t=0)=C$_{Q0}$
C$_{S}$(t=0)=C$_{S0}$

So I have three differential equations which I am having a hard time solving, If anyone could help it would be greatly appreciated.

Thanks In advance.

 

[Navin A S]

The solution to this problem would be to use the method of separation of variables. First, rearrange each differential equation to the form: \frac{dC_{A}}{C_{A}} = -k_{2}dt + k_{2}dt - k_{3}dt + k_{4}dt\frac{dC_{Q}}{C_{Q}} = k_{1}dt - k_{2}dt - k_{5}dt + k_{6}dt\frac{dC_{S}}{C_{S}} = k_{3}dt - k_{4}dt + k_{5}dt - k_{6}dtThen, integrate both sides to get: ln(C_{A}) = -k_{2}t + k_{2}t - k_{3}t + k_{4}t + C_{A0}ln(C_{Q}) = k_{1}t - k_{2}t - k_{5}t + k_{6}t + C_{Q0}ln(C_{S}) = k_{3}t - k_{4}t + k_{5}t - k_{6}t + C_{S0}Finally, take the exponential of both sides to get: C_{A} = C_{A0}e^{-k_{2}t + k_{2}t - k_{3}t + k_{4}t} C_{Q} = C_{Q0}e^{k_{1}t - k_{2}t - k_{5}t + k_{6}t} C_{S} = C_{S0}e^{k_{3}t - k_{4}t + k_{5}t - k_{6}t} This is the concentration as a function of time for the general three species first order reactions.