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aA + bB → cC + dD

According to chemical kinetics, the reaction rate is given by

[tex]r = -\frac{1}{a} \frac{\textrm{d} C_{\textrm{A}}}{\textrm{d} t} = -\frac{1}{b} \frac{\textrm{d} C_{\textrm{B}}}{\textrm{d} t} = \frac{1}{c} \frac{\textrm{d} C_{\textrm{C}}}{\textrm{d} t} = \frac{1}{d} \frac{\textrm{d} C_{\textrm{D}}}{\textrm{d} t}[/tex]

And the rate equation is (usually) given by

[tex]r = k C_{\textrm{A}}^a C_{\textrm{B}}^b[/tex]

So, for instance, if we want to know the rate at which D is produced we write

[tex]\frac{\textrm{d} C_{\textrm{D}}}{\textrm{d} t} = d k C_{\textrm{A}}^a C_{\textrm{B}}^b[/tex]

and we solve the differential equation.

Now, we assume the same reaction occurs inside an ideal batch reactor, and again we want to know the rate at which D is produced. We perform a mole balance for the species D.

[tex]\frac{\textrm{d} n_{\textrm{D}}}{\textrm{d} t} = rV[/tex]

Where V is the volume of the reactor, and r is the net chemical generation. If we apply n

_{D}= V⋅C

_{D}, we have

[tex]V \frac{\textrm{d} C_{\textrm{D}}}{\textrm{d} t} = rV[/tex]

[tex]\frac{\textrm{d} C_{\textrm{D}}}{\textrm{d} t} = r[/tex]

Finally, we have

[tex]\frac{\textrm{d} C_{\textrm{D}}}{\textrm{d} t} = k C_{\textrm{A}}^a C_{\textrm{B}}^b[/tex]

Although similar, this is not the same result one arrives at when studying chemical kinetics. Maybe it is a ridiculous concern, or I am missing something obvious, but it has been bugging me since I noticed it. Is the r (net chemical generation) used when studying chemical reactor engineering not the same as the r (reaction rate) used when studying chemical kinetics?

Thanks in advance for any input!