Data for the decomposition of dinitrogen oxide 2 N2 O(g) →2...

Data for the decomposition of dinitrogen oxide $$2 \mathrm{N}_{2} \mathrm{O}(\mathrm{g}) \rightarrow 2 \mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})$$ on a gold surface at $900^{\circ} \mathrm{C}$ are given below. Verify that the reaction is first order by preparing a graph of In $\left[\mathrm{N}_{2} \mathrm{O}\right]$ versus time. Derive the rate constant from the slope of the line in this graph. Using the rate law and value of $k$, determine the decomposition rate at $900^{\circ} \mathrm{C}$ when $\left[\mathrm{N}_{2} \mathrm{O}\right]=0.035 \mathrm{mol} / \mathrm{L}$$$\begin{array}{cc} \begin{array}{c} \text { Time } \\ (\mathrm{min}) \end{array} & \begin{array}{c} {\left[\mathrm{N}_{2} 0\right]} \\ (\mathrm{mol} / \mathrm{L}) \end{array} \\ \hline 15.0 & 0.0835 \\ 30.0 & 0.0680 \\ 80.0 & 0.0350 \\ 120.0 & 0.0220 \\ \hline \end{array}$$

Key Concepts

Reaction Order

Reaction order indicates the power to which the concentration of a reactant is raised in the rate law. In kinetics, if a reaction is first order with respect to a reactant, its rate depends linearly on the concentration of that reactant. This is fundamental to understanding how the reaction rate responds to changes in concentration over time.

Rate Law Expression

The rate law expresses the relationship between the reaction rate and the concentration(s) of the reactant(s). For a first order reaction, the law is written as rate = k[A], where k is the rate constant and [A] is the concentration of the reactant. This formulation is crucial for determining the rate at which a chemical reaction proceeds.

Integrated Rate Law for First Order Reactions

The integrated rate law for a first order reaction is given by ln[A] = -kt + ln[A]?, where [A]? is the initial concentration, t is time, and k is the rate constant. This equation shows that a plot of ln[A] versus time should yield a straight line, providing a method to determine whether the reaction follows first order kinetics.

Graphical Analysis in Kinetics

Graphical analysis involves plotting experimental data to determine the order of a reaction. For a first order reaction, plotting ln[concentration] versus time should give a straight line if the reaction is truly first order. The slope of this straight line is equal to -k, which allows for determination of the rate constant from experimental data.

Determination of the Rate Constant

The rate constant, k, is a measure of the speed of a reaction and can be derived from the slope of the ln[concentration] versus time graph in a first order reaction. By accurately determining the slope (which is -k), one can quantify the intrinsic reactivity of the system. This is key in both theoretical and practical applications of chemical kinetics.

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