Extraordinary Differential Equations #10

Calculus Level 3

Consider two reactants A and B in the following reaction in a 1:1 stoichiometric ratio to give product C: \[A+B \rightarrow C\] The reaction is first order with respect to both A and B. You are given that the rate of formation of C is thus directly proportional to the concentrations of both A and B, i.e. we have the following differential equation: \[Rate=\frac{d[C]}{dt}=k[A][B]\] where \([C]\) is the concentration of C (etc.), and \(k\) is a proportionality constant called the rate constant. For this particular reaction \(k=2.50\times10^{-4} \text{ dm}^{3} \text{ mol}^{-1} \text{ s}^{-1} \). If the initial concentrations of A and B are 3 \(\text{mol dm}^{-3}\) and 1 \(\text{mol dm}^{-3}\) respectively, and that there is no C in the beginning, calculate the concentration of C in \(\text{mol dm}^{-3}\) after 1 hour.

(This problem is part of the set Extraordinary Differential Equations.)

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