Question

The rate equation for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) (giving \(\mathrm{NO}_{2}\) and \(\mathrm{O}_{2}\) ) is Rate \(=k\left[\mathrm{N}_{2} \mathrm{O}_{5}\right] .\) The value of \(k\) is \(6.7 \times 10^{-5} \mathrm{s}^{-1}\) for the reaction at a particular temperature. (a) Calculate the half-life of \(\mathrm{N}_{2} \mathrm{O}_{5}\) (b) How long does it take for the \(\mathrm{N}_{2} \mathrm{O}_{5}\) concentration to drop to one tenth of its original value?

Short answer

The half-life is approximately 10,343 seconds. It takes about 34,380 seconds for the concentration to drop to one tenth of its initial value.

Step-by-step solution

Understand the Rate Law

The rate law for the decomposition reaction is given as \( \text{Rate} = k[\mathrm{N}_2\mathrm{O}_5] \), indicating it is a first-order reaction. This means the rate of decomposition depends linearly on the concentration of \( \mathrm{N}_2\mathrm{O}_5 \).

Calculate the Half-Life for a First-Order Reaction

For a first-order reaction, the half-life is given by the formula:\[ t_{1/2} = \frac{0.693}{k}\]Substitute the given value of \( k = 6.7 \times 10^{-5} \text{s}^{-1} \) to find the half-life:\[ t_{1/2} = \frac{0.693}{6.7 \times 10^{-5}}\]Calculating gives:\[ t_{1/2} \approx 10343 \text{s}\]

Calculate Time to Reach One Tenth of Initial Concentration

The time required for a first-order reaction to reach a certain fraction of its initial concentration can be calculated using the formula:\[ t = \frac{1}{k} \ln \left(\frac{[\text{Initial}]}{[\text{Final}]\right)}\]In this case, \([\text{Final}] = \frac{1}{10}[\text{Initial}]\), so:\[ t = \frac{1}{6.7 \times 10^{-5}} \ln(10)\]\[ t \approx 34380 \text{s}\]

Key concepts

Rate Law

In chemistry, rate law is an equation that describes how the rate of a reaction depends on the concentration of the reactants. For first-order reactions like the decomposition of \(\mathrm{N}_2\mathrm{O}_5\), the rate law is formulated as \(\text{Rate} = k [A]\), where \(k\) is the rate constant and \([A]\) is the concentration of the reactant. This equation demonstrates that the rate at which \(\mathrm{N}_2\mathrm{O}_5\) decomposes is proportional to its concentration.
This is significant because it shows that as the concentration decreases, the rate of reaction decreases too. Understanding the rate law for first-order reactions helps predict how long a chemical reaction will take under specific conditions.
Key points about rate law for first-order reactions include:

• The rate constant \(k\) is independent of concentration, but dependent on temperature.
• The reaction order tells us that the concentration influences the rate directly.
• The reaction is exponential in nature, meaning it follows a decay curve over time.

Half-life Calculation

The concept of half-life in chemistry refers to the time required for the concentration of a reactant to decrease by half. For first-order reactions, the half-life \( t_{1/2} \) is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
This formula results in a constant half-life that doesn't depend on the initial concentration.
Let's calculate the half-life for the decomposition of \(\mathrm{N}_2\mathrm{O}_5\) given that the rate constant \(k\) is \(6.7 \times 10^{-5} \mathrm{s}^{-1}\):
Substitute the value of \(k\) into the formula:
\[ t_{1/2} = \frac{0.693}{6.7 \times 10^{-5}} \approx 10343 \text{s}\]
This means that it takes approximately 10343 seconds for the concentration of \(\mathrm{N}_2\mathrm{O}_5\) to reduce to half of its initial amount. This calculation can be applied repeatedly to determine the concentration at any given time point being reduced to half successively.

Decomposition Reaction

A decomposition reaction is a type of chemical reaction where one compound breaks down into two or more simpler substances. In the case of \(\mathrm{N}_2\mathrm{O}_5\) decomposition, it breaks down into \(\mathrm{NO}_2\) and \(\mathrm{O}_2\).
This process can be described by the reaction equation:
\[ \mathrm{N}_2\mathrm{O}_5 \rightarrow 2\mathrm{NO}_2 + \frac{1}{2}\mathrm{O}_2 \]
The decomposition reaction is essential in understanding how time influences the change in concentration of the reactants. For \(\mathrm{N}_2\mathrm{O}_5\), as it decomposes, the rate of formation of \(\mathrm{NO}_2\) and \(\mathrm{O}_2\) can be monitored using rate laws.
Decomposition reactions are:

• Endothermic or exothermic, depending on the necessity of energy to break bonds.
• Influenced by temperature; higher temperatures typically increase the rate of decomposition.
• Commonly observed in unstable compounds or compounds exposed to specific conditions.

Understanding decomposition reactions and their kinetics such as rate laws and half-life helps in predicting how substances behave and change over time, which is valuable in chemical manufacturing and analysis tasks.