Question

The following gas-phase reaction was studied at \(290^{\circ} \mathrm{C}\) by observing the change in pressure as a function of time in a constant-volume vessel: $$ \mathrm{ClCO}_{2} \mathrm{CCl}_{3}(g) \longrightarrow 2 \mathrm{COCl}_{2}(g) $$ Determine the order of the reaction and the rate constant based on the following data: $$ \begin{array}{rc} \text { Time }(\mathrm{s}) & \mathrm{P}(\mathrm{mmHg}) \\ \hline 0 & 15.76 \\ 181 & 18.88 \\ 513 & 22.79 \\ 1164 & 27.08 \end{array} $$ where \(P\) is the total pressure.

Short answer

The order of reaction and rate constant could not be determined without performing the actual calculations using the provided data set. Perform the described steps to determine these parameters.

Step-by-step solution

Calculate Average Rates

One possible way to approach this is to first calculate the average rates between successive time points. The rate for a reaction is given by the change in pressure over the change in time. Compute this for every adjacent pair of time points from the given data. Use the formula \(\frac{\Delta P}{\Delta t}\) i.e., subtract the pressure of the current time point from the next and divide by the difference in time points.

Determine The Order Of Reaction

With the average rates from the previous step, you now determine the order of the reaction. Look at how the average rates change with respect to the total pressure at each time point. If the rate increases linearly with increasing pressure, then the reaction is first-order. If the rate increases exponentially with increasing pressure, it's a second order reaction. If it's not changing at all, reaction is zero order.

Use The Order To Determine The Rate Constant

After we've determined the order of reaction, we can find the rate constant using following formulas based on the order of reaction: For a zeroth-order reaction: \(k = \frac{Rate}{[A]^{0}}\), For a first-order reaction: \(k = \frac{Rate}{[A]^{1}}\), For a second-order reaction: \(k = \frac{Rate}{[A]^{2}}\) where A is the initial pressure at time t=0.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with understanding the speed or rate at which chemical reactions occur. It is a pivotal subject for students who wish to comprehend how different conditions affect the rate of a reaction. When examining a gas-phase reaction, such as the decomposition of ClCO₂CCl₃(g) into COCl₂(g), it's crucial to gather data that can describe the rate at which reactants are transformed into products. This is generally measured by observing changes in properties, such as pressure or concentration over time.

By calculating average rates, students can determine how quickly the reaction is happening. This introductory step is essential before moving on to determine the order of the reaction, which describes how the rate is dependent on the concentration of reactants. It's important for students to understand that this is not merely about finding the rate but also about understanding the pattern by which the rate changes with varying reactant concentrations or pressures in the case of gas-phase reactions.

Rate Constant Calculation

The rate constant is a numerical value that is inherently tied to the specific reaction you are studying. It is crucial in the field of chemical kinetics because it provides a direct quantitative measure of how fast a reaction proceeds under given conditions. To calculate the rate constant, initially, the order of the reaction must be determined. This order can show whether the reaction rate is proportional to the concentration of the reactants to the power of zero, one, or two corresponding to zeroth, first, or second-order reactions, respectively.

For first-order reactions, the formula to find the rate constant is simply the rate divided by the initial concentration. For a zeroth-order reaction, the concentration term's exponent is zero. This may be counterintuitive, but mathematically, any number to the power of zero is one; thus, the rate itself is equal to the rate constant. For second-order reactions, it becomes a bit more complex as you divide the rate by the concentration squared. Learning to manipulate these equations is a critical skill for students who are looking to excel in chemistry and related fields.

Gas-phase Reaction

Gas-phase reactions are chemical reactions that occur between substances in the gaseous state. One unique feature of these reactions is that they often involve changes in pressure, which can be relatively straightforward to measure with the appropriate equipment. In educational settings, understanding gas-phase reactions is instrumental for students to grasp how different states of matter react under various conditions.

When dealing with gas-phase reactions, it's crucial to consider the total pressure exerted by the mixture of gases involved in the reaction. As the reaction progresses, the pressure changes can be indicative of the reaction rate. Gas laws, such as the ideal gas law, are often utilized to relate the volume, temperature, and number of moles of gas to the pressure change. These relationships are vital when analyzing reaction kinetics for gases, as they can give students insights into how variations in each parameter affect the reaction rate. With the problem-solving steps provided, students enhance their understanding of how to interpret and use pressure data, apply gas laws, and eventually determine the order of a gas-phase reaction.