The rate of a particular chemical reaction A+B → C is...

The rate of a particular chemical reaction $A+B \rightarrow C$ is proportional to the concentrations of the reactants $A$ and $B$ : $$ \frac{d C(t)}{d t}=\alpha[A(0)-C(t)][B(0)-C(t)] $$ (a) Find $C(t)$ for $A(0) \neq B(0)$. (b) Find $C(t)$ for $A(0)=B(0)$. The initial condition is that $C(0)=0$.

Transcript

0:00 Here is the question.

00:01 Question is the rate of particular chemical reaction a plus b is to c is proportional to the concentrations of reaction a and we find c t for a 0 is not equal to b 0 and find ct for a 0 mine equal to b 0.

00:18 No, the initial condition of c is 0 equal to 0 and is a constant so here it is we can write in a if a 0 is not equal to b 0 we can differentiate that d t t equal to a no a 0 minus t so we can write here b 0 minus c t so we can write here b 0 minus c t c t d c t divided by we can write here we can write here a zero minus c t we can write here b 0 minus c t equal to d t so we can write here minus 1 b 0 minus a 0 1 c t minus a zero minus one divided by c t minus b 0 equal to d sorry is not equal to d c d c t equal to alpha d t so so we can write here we have to integrate integrating on both sides we can write here 1 divided by a 0 minus b0 log c t minus a 0 minus log c t minus b0 equal to alpha t plus c so we can write here log c t minus a 0 divided by c t minus b 0 equal to a 0 minus b 0 minus b 0 plus c 0 equal to 0 so we can write here log 0 minus a 0 divided by 0 minus b 0 so we can write here a a zero minus b0 into c so we can write here c equal to one device a zero minus b0 log a 0 divided by b0 log log a 0 b0 so we can write here that's why we can write here log c t minus a 0 divided by c t minus b 0 equal to a a c 0 minus 0 minus that is not here c will be not here a 0 b0 multiply multiply alpha t plus 1 by a 0 minus b 0 log a 0 divided by b0 a 0 b 0 so integrate both sides integrate on both sides so we can write here 1 divided by a 0 minus b 0 log c t minus a 0 minus a 0 minus logue c t minus l x minus b t plus c alpha t plus c so we can write here so we can write here a 0 minus b 0 alpha t plus log a 0 b0 so we can write here c t minus a 0 e a c0 minus b c0 alpha t plus log a 0 divided by b 0 so we can write here e a 0 minus b 0 alpha t alpha t into a 0 divided by b 0 so we can write here write here c t 1 minus a 0 divided by b0 e a 0 minus b 0 equal to a 0 minus a 0 e a 0 minus b 0 proportion to t so we can write here c t equal to a 0 1 minus e, a0 minus b 0, alpha t divided by 1 minus a0 divided by b0, ea0 minus b0, alpha t...

Question

The rate of a particular chemical reaction $A+B \rightarrow C$ is proportional to the concentrations of the reactants $A$ and $B$ : $$ \frac{d C(t)}{d t}=\alpha[A(0)-C(t)][B(0)-C(t)] $$ (a) Find $C(t)$ for $A(0) \neq B(0)$. (b) Find $C(t)$ for $A(0)=B(0)$. The initial condition is that $C(0)=0$.

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