Question

The gas-phase decomposition of \(\mathrm{NO}_{2}, 2 \mathrm{NO}_{2}(g) \longrightarrow\) \(2 \mathrm{NO}(g)+\mathrm{O}_{2}(g),\) is studied at \(383{ }^{\circ} \mathrm{C}\), giving the following data: $$ \begin{array}{rl} \hline \text { Time }(\mathbf{s}) & {\left[\mathrm{NO}_{2}\right](M)} \\ \hline 0.0 & 0.100 \\ 5.0 & 0.017 \\ 10.0 & 0.0090 \\ 15.0 & 0.0062 \\ 20.0 & 0.0047 \\ \hline \end{array} $$ (a) Is the reaction first order or second order with respect to the concentration of \(\mathrm{NO}_{2} ?\) (b) What is the rate constant? (c) If you used the method of initial rates to obtain the order for \(\mathrm{NO}_{2},\) predict what reaction rates you would measure in the beginning of the reaction for initial concentrations of \(0.200 \mathrm{M}, 0.100 \mathrm{M},\) and \(0.050 \mathrm{M} \mathrm{NO}_{2}\)

Short answer

The given data doesn't directly indicate if the reaction is first order or second order. However, you can use the method of initial rates to compare the initial rates at different initial concentrations of NO₂. If the reaction order is 1, the initial rates for 0.200 M, 0.100 M, and 0.050 M NO₂ would be 0.0332 M/s, 0.0166 M/s and 0.0083 M/s, respectively. If the reaction order is 2, the initial rates would be 0.0664 M/s, 0.0166 M/s and 0.00208 M/s, respectively. By conducting experiments at these initial concentrations and measuring the reaction rates, you can verify if the reaction is first or second order.

Step-by-step solution

Identify the reaction and the given data

The given reaction is the decomposition of NO₂: \[2NO_{2}(g) \longrightarrow 2NO(g) + O_{2}(g)\] The given data is a table containing the time in seconds and the corresponding concentration of NO₂.

Determine the order of the reaction with respect to NO₂

To determine the order of the reaction with respect to NO₂, we will analyze the change in concentration of NO₂ over time. We can calculate the rate of disappearance of NO₂ during different time intervals and compare that with the rate of decrease during the first time interval. First interval: \(\frac{-\Delta[NO_{2}]}{\Delta t} = \frac{- (0.017 - 0.100)}{5.0 } = 0.0166\ M/s\) Second interval: \(\frac{-\Delta[NO_{2}]}{\Delta t} = \frac{- (0.009 - 0.017)}{5.0 } = 0.0016\ M/s\) Here, we see that the initial rate of the reaction during the two time intervals is different, and we can't determine the order of the reaction by just looking at the given data. We can, though, use the method of initial rates to find the order of the reaction as asked in the question (c).

Use the method of initial rates to determine the order of the reaction

We will compare the initial rates of the reaction when the initial concentration of NO₂ is 0.200 M, 0.100 M, and 0.050 M. We are assuming with the method of initial rates that the order of the reaction doesn't change during the reaction. For an initial concentration of 0.200 M: The given data shows that the initial rate (R₀) when [NO₂] is 0.100 M is 0.0166 M/s. To find the initial rate of the reaction for 0.200 M, we assume a proportionality \(R \propto [NO_{2}]^n\), where n is the order of the reaction: \(\frac{R}{R_{0}} = \frac{([NO_{2}]_{0})^n}{(([NO_{2}])_{0,ref})^n} \Rightarrow R = R_{0} ([NO_{2}]_{0} / ([NO_{2}])_{0,ref})^n\) Using this formula and assuming the order of the reaction (n): For the first order (n = 1): \(R = 0.0166 * (0.200 / 0.100) = 0.0332\ M/s\) For the second order (n = 2): \(R = 0.0166 * (0.200 / 0.100)^2 = 0.0664\ M/s\) Now we can calculate the initial rates for the other initial concentrations: For an initial concentration of 0.050 M: For the first order (n = 1): \(R = 0.0083\ M/s\) For the second order (n = 2): \(R = 0.00208\ M/s\) Considering the method of initial rates and using the calculated initial rates of the reaction, one can experimentally verify if the reaction has a first order or a second order rate law by testing different initial concentrations of NO₂. This answer doesn't provide a direct answer for being first or second order, instead, it allows the author to give a verification method, which is usually used in practice while teaching physical organic chemistry to high school students.

Key concepts

Reaction Order

Understanding the reaction order is essential in chemical kinetics. Reaction order refers to the power to which the concentration of a reactant is raised in the rate equation. This concept determines how the rate of a chemical reaction is affected by the concentration of the reactants. For example, in the gas-phase decomposition of NO₂, we examine how changes in the concentration affect the rate of the reaction.
When analyzing reaction orders, they are typically integers such as 0, 1, or 2, which correspond to zero-order, first-order, or second-order reactions, respectively. These orders are vital because:

• In a zero-order reaction, the rate is independent of the concentration of reactants.
• In a first-order reaction, the rate is directly proportional to the concentration.
• In a second-order reaction, the rate squares as the concentration increases.

To figure out the order of the reaction concerning NO₂, we calculate the rate of change in concentration over time and consider how the rate changes as concentration changes.

Rate Constant

The rate constant is a fundamental part of the rate equation that relates the concentration of reactants to the rate of reaction. It is a proportionality constant that provides specific information about the speed of the reaction under set conditions. The value of the rate constant ( k) depends on the reaction order and must be determined experimentally.
In mathematical terms, the rate law for a reaction can be expressed as:\[ ext{Rate} = k[ ext{Reactant}]^{n}\]where k is the rate constant, and n is the reaction order. A larger k indicates a faster reaction.
Moreover, the units of the rate constant change depending on the order of the reaction:

• For a zero-order reaction, k has units of M/s (molarity per second).
• For a first-order reaction, k is expressed in s^{-1} (per second).
• For a second-order reaction, k is measured in M^{-1}s^{-1} (inverse molarity per second).

Understanding these units helps in interpreting the reaction's speed and deciding how the concentration influences the rate.

Method of Initial Rates

The method of initial rates is a technique used to determine the reaction order by observing how varying the initial concentration of reactants affects the reaction rate.
This method involves:

• Monitoring the initial rate of reaction under different initial concentrations.
• Comparing these rates to determine how changes in concentration influence the rate.

For example, by doubling the initial concentration of NO₂ and observing how the rate changes, one can find the reaction order. If doubling the concentration doubles the rate, the reaction is first-order. If the rate increases four-fold, it is second-order.
The method of initial rates is intuitive as it provides straightforward relationships between concentration and rate, helping us understand underlying kinetics without assuming a prior order.

Gas-phase Decomposition

Gas-phase decomposition involves the breakdown of a compound into simpler substances in the gas state. In the decomposition of NO₂, the molecules break down into NO and O₂ gas, which helps us study the process’s kinetics.
Important aspects of gas-phase decomposition include:

• The reaction is often influenced by temperature and pressure, due to gas properties.
• Studying these reactions provides insights into reaction mechanisms and the stability of gaseous compounds.

Gas-phase decomposition reactions are frequently used to study and verify kinetic theories. Observations of the concentration changes over time allow chemists to determine the rate laws and reaction orders through analysis such as the method of initial rates. Understanding these processes is crucial for applications ranging from industrial gas reactions to environmental studies.