Question

The decomposition of ethylene oxide at \(690 \mathrm{K}\) is monitored by measuring the total gas pressure as a function of time. The data obtained are \(t=10 \mathrm{min}, P_{\text {tot }}= 139.14 \mathrm{mmHg} ; 20 \mathrm{min}, 151.67 \mathrm{mmHg} ; 40 \mathrm{min}, 172.65 \mathrm{mmHg} ; 60 \mathrm{min}, 189.15 \mathrm{mmHg} ;\) \(100 \mathrm{min}, 212.34\) \(\mathrm{mmHg} ; 200 \mathrm{min}, 238.66 \mathrm{mmHg} ; \infty, 249.88 \mathrm{mmHg}\) What is the order of the reaction \(\left(\mathrm{CH}_{2}\right)_{2} \mathrm{O}(\mathrm{g}) \longrightarrow \mathrm{CH}_{4}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) ?\)

Short answer

The exact order of the reaction will be determined by checking for linearity in the created plots in step 5. The straight line plot will correspond to the correct order of the reaction. For an accurate determination, a graphical or statistical analysis tool would be needed.

Step-by-step solution

Understand the reaction

The reaction we are analyzing is the decomposition of ethylene oxide (\(C_2H_4O\)) into methane (\(CH_4\)) and carbon monoxide (\(CO\)). No stoichiometric ratios are given, so we assume the reaction happens in a 1:1:1 ratio. That is, one molecule of \(C_2H_4O\) decomposes into one molecule of \(CH_4\) and one molecule of \(CO\).

Find the total pressure after complete reaction

We are given that at \(t = \infty\) the total pressure \(P_{tot}\) is 249.88 mmHg, which is the total pressure when all the \(C_2H_4O\) has decomposed.

Calculate the initial pressure of ethylene oxide

Since the decomposition forms two gas molecules from one, every 1 mmHg decrease in the pressure of \(C_2H_4O\) results in a 2 mmHg increase in the total pressure. So, the initial pressure of \(C_2H_4O\), \(P_{C_2H_4O}\), can be calculated by \(P_{C_2H_4O} = 2 * P_{tot}\(0 min) - P_{tot}(\infty)\)

Calculate the pressure of \(C_2H_4O\)|| at each time interval

We can determine the pressure of \(C_2H_4O\) at each given time interval by rearranging the formula from step 3 to \(P_{C_2H_4O} = 2 * P_{tot}(t) - P_{tot}(\infty)\)

Determine the order of the reaction

By observing how the pressure of \(C_2H_4O\) changes with time, we can determine the order of the reaction. We assume the generic rate law for the decomposition is \(rate = k[P_{C_2H_4O}]^n\), where \(k\) is the rate constant, \(n\) is the order of the reaction. By trying different values for n and applying the rate law, rate versus time plots are made. The plot that gives a straight line will have the correct order. For a zero-order reaction, \(P_{C_2H_4O}\) versustime would give a straight line. For a first-order reaction, \(-ln[P_{C_2H_4O}]\) versus time would give a straight line. For a second-order reaction, \(1/[P_{C_2H_4O}]\) versus time would give a straight line.

Key concepts

Ethylene Oxide Decomposition

Ethylene oxide decomposition is a chemical reaction where ethylene oxide (\( C_2H_4O \)) breaks down to form methane (\( CH_4 \)) and carbon monoxide (\( CO \)). This reaction is quite significant in the field of chemical kinetics, as it allows us to study various fundamental concepts, such as reaction order and rate laws. When ethylene oxide decomposes, each molecule breaks down into one molecule of methane and one molecule of carbon monoxide. Thus, the reaction seems to progress in a 1:1:1 stoichiometric ratio, meaning one molecule of ethylene oxide becomes one molecule of each of the products.

• The overall chemical equation for the decomposition reaction is: \( (CH_2)_2O(g) \rightarrow CH_4(g) + CO(g) \).
• This process can be accompanied by changes in gas pressure, as the total number of gas molecules doubles.
• Understanding how the decomposition affects pressure is essential for determining the order of the reaction.

Studying this decomposition provides us with a real-world application of these theoretical kinetics concepts.

Pressure Measurement

When we talk about pressure measurement in the context of chemical reactions like the decomposition of ethylene oxide, we're often referring to measuring how the pressure changes over time as the reaction proceeds. In many laboratory settings, the pressure is measured because it's a useful indirect way to track the concentration of gases in a reaction mixture. To monitor the reaction, scientists measure the total pressure at different time intervals. As the decomposition progresses, ethylene oxide molecules dissociate into methane and carbon monoxide gases, effectively doubling the number of gas molecules and thereby increasing the total pressure. Each step in the progression of pressure hints at how far the reaction has proceeded.

• Total pressure signifies the cumulative pressure exerted by all gas phase components present.
• Initial pressure can be assumed based on the change in total pressure over time, calculated using known values at equilibrium.
• Pressure data can be used to analyze reaction kinetics, offering insights into the order and rate of the reaction.

This pressure measurement method is particularly helpful when determining reaction order, as it provides tangible data that can be plotted and analyzed.

Rate Law

The rate law of a chemical reaction is an expression that links the reaction rate with the concentration of reactants. In the case of ethylene oxide decomposition, the rate law helps us to understand how quickly ethylene oxide transforms into methane and carbon monoxide and how this speed depends on various factors.For determining the order of the reaction, scientists test different possible reaction orders (\( n \)). The reaction order (\( n \)) tells us how the concentration of the reactant affects the rate of the reaction:

• If the plot of pressure versus time yields a straight line, the reaction follows zero-order kinetics.
• If plotting \(-\ln [P_{C_2H_4O}] \) against time results in a linear graph, the reaction is first order.
• If \( 1/[P_{C_2H_4O}] \) versus time is linear, the reaction follows second-order kinetics.

The process involves checking each plot to see which provides a linear relationship, thereby indicating the correct reaction order. This method is vital because knowing the correct rate law allows scientists to predict the behavior of the reaction under various conditions, facilitating control over industrial and laboratory processes.