Question

The rate law for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is rate \(=\) \(k\left[\mathrm{~N}_{2} \mathrm{O}_{5}\right] .\) If \(k=1.0 \times 10^{-5} \mathrm{~s}^{-1},\) what is the reaction rate when the \(\mathrm{N}_{2} \mathrm{O}_{5}\) concentration is \(0.0010 \mathrm{~mol} \mathrm{~L}^{-1}\) ?

Short answer

The reaction rate is \(1.0 \times 10^{-8} \mathrm{~mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{s}^{-1}\.}

Step-by-step solution

Understand the Given Rate Law

Recognize that the given rate law for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is first-order, because the rate of reaction is directly proportional to the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) raised to the first power.

Identify the Rate Constant and the Concentration

The rate constant \(k\) is given as \(1.0 \times 10^{-5} \mathrm{~s}^{-1}\) and the concentration of \(\mathrm{N}_{2O}_{5}\) is given as \(0.0010 \mathrm{~mol} \mathrm{~L}^{-1}\). These values will be used to calculate the rate of reaction.

Key concepts

First-Order Reaction

Understanding the nature of a first-order reaction is fundamental when delving into chemical kinetics. In a first-order reaction, the rate at which the reaction proceeds is directly proportional to the concentration of one of the reactants. In mathematical terms, we express this as the rate of reaction being equal to the rate constant, k, multiplied by the concentration of the reactant to the first power. This is reflected in rate laws that take the form:

\[ \text{Rate} = k [\text{Reactant}] \]
Here, the brackets signify the concentration of the reactant. The key characteristic of a first-order reaction is that the time it takes for a given fraction of the reactant to decompose does not depend on the initial concentration. This means that the half-life of the reaction, which is the time taken for half of the reactant to decompose, is constant. This can be invaluable in predicting the behavior of reactions in diverse scenarios, from industrial processes to pharmacokinetics.

Rate Constant

The rate constant, typically represented by the symbol 'k', is an element of the reaction rate equation and is a measure of how quickly a reaction proceeds. It is particular to each chemical reaction and can vary significantly depending on temperature, the presence of catalysts, and other environmental conditions.

The specific value of the rate constant provides insight into the speed of a reaction; a larger value of 'k' implies a faster reaction while a smaller value indicates a slower process. This makes the rate constant an essential parameter for chemists and engineers designing and controlling chemical processes. It's important to note that the units of the rate constant will vary depending on the order of the reaction; for a first-order reaction, the units are per time, often s^-1. In the given exercise, the value of the rate constant is \(1.0 \times 10^{-5} \mathrm{s}^{-1}\), which enables us to predict how rapidly the decomposition of \(\mathrm{N_2O_5}\) will occur at any given concentration.

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes and the factors that affect these rates. It examines the collision frequency of molecules, the energy and orientation of collisions, and the transition state, which is the state corresponding to the highest potential energy along the reaction path.

Kinetics can shed light on the reaction mechanism, the step-by-step pathway from reactants to products, revealing which molecular events are the slowest and therefore rate-determining. This knowledge is not only academically intriguing but also incredibly powerful in practical applications. For instance, by understanding chemical kinetics, pharmaceutical companies can better design drugs that are both effective and have optimal metabolism rates.