Question

The Rice-Herzfeld mechanism for the thermal decomposition of acetaldehyde \(\left(\mathrm{CH}_{3} \mathrm{CO}(g)\right)\) is \\[ \begin{array}{l} \mathrm{CH}_{3} \mathrm{CHO}(g) \stackrel{k_{1}}{\longrightarrow} \mathrm{CH}_{3} \cdot(g)+\mathrm{CHO} \cdot(g) \\ \mathrm{CH}_{3} \cdot(g)+\mathrm{CH}_{3} \mathrm{CHO}(g) \stackrel{k_{2}}{\longrightarrow} \mathrm{CH}_{4}(g)+\mathrm{CH}_{2} \mathrm{CHO} \cdot(g) \\ \mathrm{CH}_{2} \mathrm{CHO} \cdot(g) \stackrel{k_{3}}{\longrightarrow} \mathrm{CO}(g)+\mathrm{CH}_{3} \cdot(g) \\ \mathrm{CH}_{3} \cdot(g)+\mathrm{CH}_{3} \cdot(g) \stackrel{k_{4}}{\longrightarrow} \mathrm{C}_{2} \mathrm{H}_{6}(g) \end{array} \\] Using the steady-state approximation, determine the rate of methane \(\left(\mathrm{CH}_{4}(g)\right)\) formation.

Short answer

The rate of methane formation in the Rice-Herzfeld mechanism for the thermal decomposition of acetaldehyde can be found using the steady-state approximation and is given by: \(r_{\mathrm{CH}_{4}} = k_{2}\sqrt{\frac{k_{1}}{2k_{4}}[\mathrm{CH}_{3} \mathrm{CHO}]}[\mathrm{CH}_{3} \mathrm{CHO}]\)

Step-by-step solution

List the given reactions and constants

The Rice-Herzfeld mechanism is given as: \[ \begin{array}{l} \mathrm{CH}_{3} \mathrm{CHO}(g) \stackrel{k_{1}}{\longrightarrow} \mathrm{CH}_{3} \cdot(g)+\mathrm{CHO} \cdot(g) \\ \mathrm{CH}_{3} \cdot(g)+\mathrm{CH}_{3} \mathrm{CHO}(g) \stackrel{k_{2}}{\longrightarrow} \mathrm{CH}_{4}(g)+\mathrm{CH}_{2} \mathrm{CHO} \cdot(g) \\ \mathrm{CH}_{2} \mathrm{CHO} \cdot(g) \stackrel{k_{3}}{\longrightarrow} \mathrm{CO}(g)+\mathrm{CH}_{3} \cdot(g) \\ \mathrm{CH}_{3} \cdot(g)+\mathrm{CH}_{3} \cdot(g) \stackrel{k_{4}}{\longrightarrow} \mathrm{C}_{2} \mathrm{H}_{6}(g) \end{array} \]

Apply the steady-state approximation to the reactive intermediates

Using the steady-state approximation, the sum of all rates of formation and consumption of the reactive intermediates (CH₃·(g) and CH₂CHO·(g)) equals zero: For CH₃·(g): \(\frac{d[\mathrm{CH}_{3} \cdot]}{dt} \approx 0 = k_{1}[\mathrm{CH}_{3} \mathrm{CHO}] - k_{2}[\mathrm{CH}_{3} \cdot][\mathrm{CH}_{3} \mathrm{CHO}] + k_{3}[\mathrm{CH}_{2} \mathrm{CHO} \cdot] - 2k_{4}[\mathrm{CH}_{3} \cdot]^{2}\) For CH₂CHO·(g): \(\frac{d[\mathrm{CH}_{2} \mathrm{CHO} \cdot]}{dt} \approx 0 = k_{2}[\mathrm{CH}_{3} \cdot][\mathrm{CH}_{3} \mathrm{CHO}] - k_{3}[\mathrm{CH}_{2} \mathrm{CHO} \cdot]\)

Solve the simultaneous equations for the reactive intermediates

Now solve the two simultaneous equations for [\(\mathrm{CH}_{3} \cdot\)] and [\(\mathrm{CH}_{2} \mathrm{CHO} \cdot\)]: From the second equation: \([\mathrm{CH}_{2} \mathrm{CHO} \cdot] = \frac{k_{2}}{k_{3}}[\mathrm{CH}_{3} \cdot][\mathrm{CH}_{3} \mathrm{CHO}]\) Insert this expression into the first equation and get: \(k_{1}[\mathrm{CH}_{3} \mathrm{CHO}] - k_{2}[\mathrm{CH}_{3} \cdot][\mathrm{CH}_{3} \mathrm{CHO}] + k_{3} \cdot \frac{k_{2}}{k_{3}}[\mathrm{CH}_{3} \cdot][\mathrm{CH}_{3} \mathrm{CHO}] - 2k_{4}[\mathrm{CH}_{3} \cdot]^{2} = 0\) Simplify the equation and get: \(k_{1}[\mathrm{CH}_{3} \mathrm{CHO}] - 2k_{4}[\mathrm{CH}_{3} \cdot]^{2} = 0\) Now solve for [\(\mathrm{CH}_{3} \cdot\)]: \([\mathrm{CH}_{3} \cdot] = \sqrt{\frac{k_{1}}{2k_{4}}[\mathrm{CH}_{3} \mathrm{CHO}]}\)

Find the rate of methane formation

Since the rate of methane formation is determined by the second reaction, we can write: \(r_{\mathrm{CH}_{4}} = k_{2}[\mathrm{CH}_{3} \cdot][\mathrm{CH}_{3} \mathrm{CHO}]\) Substitute the expression for [\(\mathrm{CH}_{3} \cdot\)]: \(r_{\mathrm{CH}_{4}} = k_{2}\sqrt{\frac{k_{1}}{2k_{4}}[\mathrm{CH}_{3} \mathrm{CHO}]}[\mathrm{CH}_{3} \mathrm{CHO}]\) Hence, the rate of methane formation is given by: \(r_{\mathrm{CH}_{4}} = k_{2}\sqrt{\frac{k_{1}}{2k_{4}}[\mathrm{CH}_{3} \mathrm{CHO}]}[\mathrm{CH}_{3} \mathrm{CHO}]\)

Key concepts

Rice-Herzfeld Mechanism

The Rice-Herzfeld mechanism is a series of chemical reactions that describe the thermal decomposition of certain organic compounds. It's a fundamental concept for students learning about reaction mechanisms in chemical kinetics. This mechanism, particularly for the decomposition of acetaldehyde into methane and other products, is essential when understanding how complex reactions can be broken down into simpler steps.
In the realm of organic chemistry, this mechanism serves as a classic example of radical reactions where transient species, such as free radicals, play a pivotal role. Studying such mechanisms helps in understanding the dynamics of chemical processes and allows chemists to manipulate these reactions for various applications, such as synthesis of new compounds or materials.
Key Steps in the Rice-Herzfeld Mechanism:

• Initiation of the radical process by homolytic bond dissociation.
• Propagation where radicals react with stable molecules to form new radicals.
• Termination when two radicals combine to form a stable product.

For students, grasping the Rice-Herzfeld mechanism's steps and recognizing how radical species influence reaction rates is crucial for mastering topics in chemical kinetics and organic chemistry.

Thermal Decomposition of Acetaldehyde

The thermal decomposition of acetaldehyde (\( \text{CH}_3\text{CHO} \text{(g)} \) is a chemical reaction where acetaldehyde is broken down into simpler molecules upon heating. This process is fundamental to both industrial chemistry and laboratory studies, where understanding the breakdown of compounds at high temperatures is necessary for various applications such as combustion, pyrolysis, and material synthesis.
Understanding the kinetics and mechanisms, such as the Rice-Herzfeld mechanism, behind the thermal decomposition of acetaldehyde helps us control the reaction conditions to favor the desired products. For instance, predicting the formation of methane (\( \text{CH}_4 \text{(g)} \) or ethane (\( \text{C}_2\text{H}_6 \text{(g)} \) during the decomposition process has significant implications in energy production and environmental impact assessment.
The breakdown of acetaldehyde involves radical intermediates, and the reaction is typically carried out in a controlled environment where temperature can affect the rate and products of decomposition. Students looking to specialize in physical chemistry or chemical engineering would benefit from a deep understanding of such reactions.

Rate of Methane Formation

The rate of methane formation from the decomposition of acetaldehyde is an essential concept for students exploring reaction kinetics within the context of energy-related processes. Methane, being a fundamental hydrocarbon, is widely used as a fuel and as a starting material for various chemical syntheses.
In the provided exercise, the steady-state approximation assists in simplifying the complex set of reactions to make it possible to calculate the rate of methane (\( \text{CH}_4 \text{(g)} \) formation mathematically. This approximation assumes that the concentration of intermediate radicals stays constant throughout the reaction. By applying this principle, it's possible to derive a mathematical expression that links the rate of methane formation to the concentrations of reactants and the rate constants of the intermediate steps.
This understanding is not only fundamental in academic settings but also critical in industries that rely on the production of methane and other hydrocarbons, where maximizing yield and efficiency are of utmost importance. Deep diving into this topic, students can draw connections between theoretical calculations and practical applications in real-world scenarios.

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes and the factors affecting these rates. It's a cornerstone of both academic research and practical applications in chemical and pharmaceutical industries, environmental science, and materials engineering.
The Rice-Herzfeld mechanism and the study of methane formation from acetaldehyde decomposition serve as a model for teaching the principles of kinetics. Through this example, students encounter core concepts such as reaction mechanisms, rate laws, and the steady-state approximation. These principles enable the prediction and control of the speed at which reactions occur, which is vital for designing chemical reactors and optimizing production processes.
By understanding chemical kinetics, students gain a valuable toolset to analyze reactions critically, derive rate equations, and apply these concepts to novel situations. Notably, the ability to manipulate rates of reaction is instrumental in creating more efficient chemical processes, reducing waste, and improving safety in industrial settings.