Answer
See below
Work Step by Step
a) Chemicals A and B combine in the ratio a:b in producing the chemical C.
Initial amount of A: $A_0$
Initial amount of B: $B_0$
For chemical A: $\frac{a}{a+b}C$
or $(A_0-\frac{a}{a+b}C)$
For chemical B: $\frac{b}{a+b}C$
or $(B_0-\frac{b}{a+b}C)$
Set up the mass action:
$\frac{dC}{dt}=k(A_0-\frac{a}{a+b}C)(B_0-\frac{b}{a+b}C)$
b) Obtain:
$\frac{dC}{dt}=k\frac{a}{a+b}\frac{b}{a+b}(A_0\frac{a+b}{a}-C)(B_0\frac{a+b}{b}-C)\\
\frac{dC}{dt}=k\frac{(a+b)^2}{ab}(A_0\frac{a+b}{a}-C)(B_0\frac{a+b}{b}-C)$
Hence, $\rightarrow r=\frac{(a+b)^2}{ab}k\\
\alpha=\frac{a+b}{a}A_0\\\beta=\frac{a+b}{b}B_0$
Substitute: $\frac{dC}{dt}=r(\alpha-C)(\beta-C)$
