To determine the concentration dependence of the rate of the...

To determine the concentration dependence of the rate of the reaction $$\mathrm{H}_{2} \mathrm{PO}_{3}^{-}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \longrightarrow \mathrm{HPO}_{3}^{2-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell)$$ you might measure $\left[\mathrm{OH}^{-}\right]$ as a function of time using a pH meter. (To do so, you would set up conditions under which $\left[\mathrm{H}_{2} \mathrm{PO}_{3}^{-}\right]$ remains constant, by using a large excess of this reactant.) How would you prove a second-order rate dependence for $\left[\mathrm{OH}^{-}\right] ?$

To determine the concentration dependence of the rate of the reaction
$$\mathrm{H}_{2} \mathrm{PO}_{3}^{-}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \longrightarrow \mathrm{HPO}_{3}^{2-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell)$$
you might measure $\left[\mathrm{OH}^{-}\right]$ as a function of time using a pH meter. (To do so, you would set up conditions under which $\left[\mathrm{H}_{2} \mathrm{PO}_{3}^{-}\right]$ remains constant, by using a large excess of this reactant.) How would you prove a second-order rate dependence for $\left[\mathrm{OH}^{-}\right] ?$

Key Concepts

Integrated Rate Law for Second-Order Reactions

The integrated rate law for a second order reaction involving a single reactant is given by 1/[OH?] = 1/[OH?]? + kt. By measuring the concentration of OH? over time and plotting 1/[OH?] versus time, a straight line indicates a second order kinetic relationship, thus enabling the extraction of the rate constant k from the slope.

Graphical Analysis

A key method to prove second order kinetics is to graph 1/[OH?] against time. A linear plot that passes through the appropriate intercept provides visual evidence for the expected second order behavior and confirms the reaction order with respect to hydroxide ion. This graphical method distinguishes second order kinetics from first order kinetics, which would produce a linear plot of ln[OH?] versus time.

Pseudo-First Order Kinetics

By using a large excess of H?PO??, its concentration is maintained nearly constant during the reaction. This simplifies the rate law to a pseudo-first order reaction in OH?, allowing the observed rate to be effectively described by rate = k'[OH?], where k' = k[H?PO??]. This approach is useful in isolating the effect of OH? concentration on the rate and then deducing the overall second order dependence.

Rate Law

The rate law expresses how the rate of a chemical reaction depends on the concentration of its reactants. In this context, demonstrating that the reaction rate is proportional to [OH?]² shows that the reaction is second order with respect to OH?. This involves formulating the equation rate = k[OH?]² under the conditions when the other reactant concentration remains constant.

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