Question

The reaction \(2 \mathrm{NO}_{2} \longrightarrow 2 \mathrm{NO}+\mathrm{O}_{2}\) has the rate constant \(k=0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1}\). Based on the units for \(k\), is the reaction first or second order in \(\mathrm{NO}_{2} ?\) If the initial concentration of \(\mathrm{NO}_{2}\) is \(0.100 \mathrm{M}\), how would you determine how long it would take for the concentration to decrease to \(0.025 \mathrm{M} ?\)

Short answer

The reaction is second-order concerning \(\mathrm{NO}_{2}\). To determine the time it takes for the concentration of \(\mathrm{NO}_{2}\) to decrease from \(0.100 \mathrm{M}\) to \(0.025 \mathrm{M}\), we use the integrated rate law for a second-order reaction: \(\frac{1}{[\mathrm{A}]_t} - \frac{1}{[\mathrm{A}]_0} = kt\). Plugging in the given values, we find that it takes approximately 47.62 seconds for the concentration to decrease.

Step-by-step solution

Determine the reaction order concerning \(\mathrm{NO}_{2}\)

The unit for the rate constant \(k\) is \(\mathrm{M}^{-1}\mathrm{~s}^{-1}\), which indicates that the reaction is second-order concerning \(\mathrm{NO}_{2}\). The second-order reaction rate law is given by: Rate \(=\) \(k[\mathrm{NO}_{2}]^{2}\)

Determine the integrated rate law for a second-order reaction

For a second-order reaction, the integrated rate law is given by: \(\frac{1}{[\mathrm{A}]_t} - \frac{1}{[\mathrm{A}]_0} = kt\) where \([\mathrm{A}]_t\) refers to the concentration of \(\mathrm{NO}_{2}\) at time \(t\), \([\mathrm{A}]_0\) is the initial concentration of \(\mathrm{NO}_{2}\), \(k\) is the rate constant, and \(t\) is the time.

Plug in the given values and solve for time (\(t\))

We are given that the initial concentration of \(\mathrm{NO}_{2}\) is \(0.100 \mathrm{M}\) and it eventually decreases to \(0.025 \mathrm{M}\). We can now plug these values, as well as the rate constant, into the integrated rate law equation: \(\frac{1}{0.025} - \frac{1}{0.100} = (0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1})(t)\) Now, solve for \(t\): \(\frac{1}{0.025 \mathrm{M}} - \frac{1}{0.100 \mathrm{M}} = (0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1})(t)\) \(40 \mathrm{M}^{-1} - 10 \mathrm{M}^{-1} = 0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1} \cdot t\) \(30 \mathrm{M}^{-1} = 0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1} \cdot t\) \(t = \frac{30 \mathrm{M}^{-1}}{0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1}}\) \(t \approx 47.62 \mathrm{~s}\) So, it takes approximately 47.62 seconds for the concentration of \(\mathrm{NO}_{2}\) to decrease from \(0.100 \mathrm{M}\) to \(0.025 \mathrm{M}\).

Key concepts

Rate Laws

Rate laws are mathematical expressions that relate the rate of a chemical reaction to the concentration of its reactants. They provide critical information about how a reaction proceeds and help predict the speed of the reaction under various conditions.
In general, the rate law can be expressed as:

• Rate = k[ _Reactant]
  Superscript represents the order of reaction with respect to that reactant.

For a second-order reaction, like the one in our example, the rate depends on the concentration of NO ₂ squared:

• Rate = k[ NO ₂ ]
  Superscript = 2
• Here, k is the rate constant, and it is crucial for determining the reaction rate.

Understanding rate laws helps in figuring out how the reaction speed changes if the concentrations of reactants are altered.

Integrated Rate Law

The integrated rate law is used to link the concentration of reactants to time, offering a way to calculate the time needed for a certain concentration change under a given set of conditions.
For a second-order reaction, the integrated rate law formula is expressed as:

• \(\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\)

Here:

• \([A]_t\) is the concentration at time \(t\)
• \([A]_0\) is the initial concentration
• \(k\) is the rate constant
• \(t\) represents time

This formula is invaluable for calculating the time required for a reactant concentration to change from one value to another, as seen in our exercise where the concentration of NO₂ is desired to change from 0.100 M to 0.025 M.

Reaction Order

Reaction order is an essential concept that defines how the rate of a chemical reaction depends on the concentration of its reactants. In our example, the reaction is second order concerning NO₂.
The reaction order is determined based on:

• The exponent of the concentration term in the rate law equation.
• For second-order reactions, the concentration is squared.
• The units of the rate constant \(k\), for a second-order reaction, are typically \(M^{-1}s^{-1}\).

Knowing the reaction order is crucial as it helps predict how changes in concentration will affect the reaction rate, and dictates which integrated rate law applies.

Rate Constant

The rate constant, often denoted as \(k\), is a pivotal factor in the rate equation that remains unchanged under identical conditions. It determines the speed of the reaction at a given concentration.
The role and characteristics of the rate constant include:

• Several factors affecting it include temperature and the presence of a catalyst.
• For second-order reactions, \(k\) typically has the unit \(M^{-1}s^{-1}\), which helps identify the reaction order.
• Understanding \(k\) allows chemists to manipulate reaction conditions to achieve desired outcomes.

The consistent value of \(k\) under unchanged conditions allows prediction of how the concentration of reactants will vary over time within the framework of the integrated rate law.