# Simple integrated rate equations for reversible bimolecular reactions

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## Summary

*P*

_{e}−

*P*as the concentration variable (where

*P*

_{e}is the equilibrium, and

*P*is the instantaneous, product concentration), the 3 possible stoichiometries can be reduced to a single straightforward differential equation. This can be solved very economically. For each stoichiometry,

were*t* is time,*k*_{1} is the forward rate constant,*K*_{e} is the equilibrium constant, and Δ*P* is*P*−*P*_{o}. The terms*P*_{c}−*P*_{o} and*D*+*P*_{c}−*P*_{o} are the physically possible and physically impossible roots of the quadratic equation for*P*_{e}−*P*_{o} in terms of the initial concentrations and*K*_{c}.*D* is the discriminant in this equation. All 3 quantities can be calculated if the equilibrium constant is known. A plot of*t* against ln{[1−Δ*P*/(*D*+*P*_{c}−*P*_{o})]/[1−Δ*P*/(*P*_{c}−*P*_{o})]} should be a straight line for any second order reaction. For each stoichiometry,*P*_{c}−*P*_{o} approaches*A*_{o}, the initial concentration of the first reactant, as the equilibrium constant increases. When a second reactant is present,*D*+*P*_{e}−*P*_{ o } approaches*B*_{o}. The limiting equation is then that of an irreversible bimolecular reaction. For A⇌P+Q,*D* approaches −*K*_{e} as the equilibrium constant becomes large, and the value of the second logarithmic term in the integrated equation approaches zero. The limiting equation is that of an irreversible, unimolecular reaction.

### Keywords

Differential Equation Equilibrium Constant Rate Equation Order Reaction Quadratic Equation## Preview

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### References

- 2.
*A, B, P*, and*Q*are instantaneous concentrations of reactants and products; the subscripts o and e indicate initial and equilibrium concentrations respectively.*K*_{e}is the equilibrium constant;*k*_{1}is the forward rate constant. Δ*P*is*P*−*P*_{o}.*C*is −*k*_{1}/*K*_{e}for A⇌P+Q,*k*_{1}for A+B⇌P, and*k*_{1}(1−1/*K*_{e}) for A+B⇌P+Q.*D*is defined, and discussed at length, in the text.Google Scholar - 3.Moore, W.J., in: Physical Chemistry, 4th edn, p. 333. Prentice-Hall, New York 1972.Google Scholar
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