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For example, if we have the following Birth-Death process

Birth-Death reaction

Why is the differential equation describing this reaction given by

Equation describing reaction

Where x, y, and a are the number of the reactants (X, Y, and A respectively). The rate is always proportional to the product of the number of reactants, but why not their sum?

M. Z.
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  • What should the rate be if the amount of one reactant is zero? Would it be that if the rate were proportional to the sum? – kaylimekay Jan 18 '21 at 15:50
  • I see your point, because we want the rate to be zero if one of them is zero. It makes a lot of sense. Thank you! – M. Z. Jan 18 '21 at 15:55

1 Answers1

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Where x, y, and a are the number of the reactants (X, Y, and A respectively). The rate is always proportional to the product of the number of reactants, but why not their sum?concentration (or even better chemical acti

'number of the reactants' is an ambiguous term: concentration (or even better chemical activity) is more accurate.

If we take a simple reaction:

$$\text{A}+\text{B}\to \text{C}$$

Then in accordance with kinetic theory the rate of reaction is proportional to:

$$\frac{\text{d}[C]}{\text{d}t}\propto [A]$$ and: $$\frac{\text{d}[C]}{\text{d}t}\propto [B]$$

It follows that: $$\frac{\text{d}[C]}{\text{d}t}\propto [A]\times [B]$$ And: $$\frac{\text{d}[C]}{\text{d}t}=k [A][B]$$

Where the bracketed quantities are concentrations.

Gert
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