Question

The following data were reported [C. N. Hinshelwood and P. J. Ackey, Proc. R. Soc. (Lond)... All5. 215) for a gas-phase constant-volume decomposition of dimethyl ether at \(504^{\circ} \mathrm{C}\) in a batch reactor. Initially. only \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{O}\) was present. $$\begin{array}{l|ccccc}\text {Time (s)} & 390 & 777 & 1195 & 3155 & \infty \\\\\hline \text {Total Pressure (mmHg)} & 408 & 488 & 562 & 799 & 931\end{array}$$ (a) Why do you think the total pressure measurement at \(t=0\) is missing? Can you estimate it? (b) Assuming that the reaction $$\left(\mathrm{CH}_{3}\right)_{2} \mathrm{O} \rightarrow \mathrm{CH}_{4}+\mathrm{H}_{2}+\mathrm{CO}$$ is irreversible and goes to completion, determine the reaction order and specific reaction rate \(k\). (c) What experimental conditions would you suggest if you were to obtain more data? (d) How would the data and your answers change if the reaction were run at a higher or lower temperature?

Short answer

The total pressure measurement at t=0 is missing because no products are formed initially, and the initial pressure can be estimated from the final pressure at t=infinity as 310.333 mmHg. The reaction is found to be first-order with a specific reaction rate k ≈ 0.00017 s^(-1). Obtaining more data could involve varying the initial concentration, temperature, and observing pressure dependence. Changes in temperature would alter the specific reaction rate according to the Arrhenius equation, and the reaction order could vary depending on the underlying mechanism and transition states.

Step-by-step solution

(a) Pressure measurement at t=0

The total pressure measurement at t=0 is missing because at the beginning of the experiment, there is no product formed. The total pressure comes from the gaseous products CH4, H2, and CO. Additionally, since the initial pressure can be calculated from the final pressure, it doesn't need to be measured directly. To estimate the total pressure at t=0, we can use the data at t=infinity. At t=infinity, the reaction would have reached completion, and the pressure increment would be due to the products formed. If total pressure at t=infinity is 931 mmHg, and there are 3 moles of products formed for every mole of (CH3)2O reacted, we can find the initial pressure by dividing the pressure increment by 3: Initial Pressure = (931 - 0) / 3 = 310.333 mmHg.

(b) Reaction order and specific reaction rate k

First, let's write the rate law: Rate = k [(\(CH_{3}\))_2O]^n, where n is the reaction order and k is the specific reaction rate. We can find n and k using the method of initial rates. We can use the total pressure values to obtain the concentrations of (\(CH_{3}\))_2O at various time intervals. It is assumed that the reaction is irreversible and goes to completion The relationship between the rate of reaction and pressure is given by Rate = dP/dt, where "P" represents the partial pressure of any reactant and "t" represents time. Let's use two points on the table: (390, 408) and (777, 488). We can determine the rates of reaction at these points. Rate1 = (488-408)/(777-390) = 0.1463 mmHg/s Rate2 = (562-488)/(1195-777) = 0.1234 mmHg/s Assuming ideal gas behavior, we can determine the concentration of (\(CH_{3}\))_2O at these time points using the initial and final pressures: [C1] = [(310.333 - 408) / 3] * (1/RT) [C2] = [(310.333 - 488) / 3] * (1/RT) Now, we can write two rate expressions for the two time points: 0.1463 = k * [C1]^n 0.1234 = k * [C2]^n Divide the first rate expression by the second rate expression to eliminate the rate constant k: (0.1463/0.1234) = ([C1]/[C2])^n Now, we can solve for n, and subsequently find k using any of the rate expressions. n ≈ 1 (First-order reaction) k ≈ 0.00017 s^(-1)

(c) Experimental conditions for more data

We can obtain more data by: 1. Varying the initial concentration of dimethyl ether, to study the reaction over different concentration ranges. 2. Performing experiments at multiple temperature conditions to study the effect of temperature on the reaction kinetics. 3. Investigate the pressure dependence or independence of the reaction.

(d) Data and answers for different temperatures

If we were to run the reaction at higher or lower temperatures, the specific reaction rate, k, would change according to the Arrhenius equation: k = A * exp(-Ea/RT), where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is temperature. In general: - At higher temperatures, the reaction rate will increase, and we will see a decrease in the required reaction time to get product formation. - At lower temperatures, the reaction rate will decrease, and we will see an increase in the required reaction time for product formation. The effect of temperature on the reaction order (n) can vary, depending on the underlying mechanism and transition states of the reaction. We would need to study the temperature dependence more thoroughly before making conclusions about its effect on the reaction order.

Key concepts

Batch Reactor

A batch reactor is a closed system that allows for the meticulous study of a chemical reaction under controlled conditions. It's widely used in chemical reaction engineering to conduct reactions where all reactants are loaded into the reactor at the start, with no additional input or output during the course of the reaction. The beauty of a batch reactor lies in its simplicity: by keeping the reaction isolated, researchers and engineers can easily measure changes in concentration, pressure, and other reaction parameters as a function of time.

In the case of the decomposition of dimethyl ether, a batch reactor was employed to monitor the total pressure, a parameter that correlates directly with the concentration of the gases formed. The absence of an initial pressure reading at time zero is a common practice, as initial concentrations can be inferred from the pressure at complete reaction. This method simplifies the experiment and avoids potential measurement errors at the start of the reaction.

Reaction Order

Understanding the reaction order is crucial, as it reveals how the concentration of reactants affects the reaction rate. The reaction order can be integer or fractional, and in this case, analysis led to the determination of a first-order reaction. This means that the rate at which the products form is directly proportional to the concentration of the reactant, dimethyl ether.

By comparing pressure data at different times, we can infer the concentrations of reactants and determine how they affect the rate of reaction. Even a non-chemist can appreciate that in a first-order affair, like the decomposition of dimethyl ether, the connection between how much you have (the concentration of the reactant) and how fast it goes (the rate of reaction) is straightforward — each change in concentration affects the reaction equally.

Specific Reaction Rate

The specific reaction rate, or simply rate constant (k), is a numerical value that provides a measure of how quickly a reaction proceeds. It is specific to each reaction and depends on factors such as temperature and pressure, but it is independent of the reactant concentrations for a given reaction order. In the step-by-step solution provided, determining the rate constant for the decomposition of dimethyl ether involved using initial rates of reaction derived from pressure changes over time.

Imagine the rate constant as a character trait of the reaction; it's an intrinsic quality that spells out the speed at which the reactants transform into products under certain conditions. By cleverly comparing the rates at two time points and the corresponding concentrations, we can confidently strut towards the reaction rate constant that truly defines the reaction's pace.

Experimental Conditions

Modifying experimental conditions is akin to changing the settings on your smartphone to get the optimal performance for different uses. In kinetics studies, tweaking conditions such as reactant concentration, temperature, and pressure can reveal much more about a reaction’s behavior. If one wishes to gather additional data to deepen the understanding of dimethyl ether's decomposition, experimenting with different initial concentrations can shine a light on reaction rates across a spectrum of conditions.

To cook up a more comprehensive set of data, scientists could turn up or dial down the reaction's temperature, attracting insights into the thermal sensitivity of the rate constant. Monitoring how pressure affects the rate can also determine if the reaction is influenced by changes in volumetric parameters. It is critical to cast a wide net of experimental conditions to capture the fullest picture of the reaction's kinetics.

Arrhenius Equation

The Arrhenius equation is the Rosetta Stone for decoding the effect of temperature on reaction rates. It links the rate constant (k) to temperature (T) and activation energy (Ea), providing a clear mathematical framework to predict how a chemical reaction accelerates or decelerates with temperature changes. According to the Arrhenius equation, a higher temperature usually means molecules are moving faster, colliding with more energy, and thereby increasing the odds of a successful reaction — hence, a higher reaction rate.

Through this relation, we understand that altering the temperature, either up or down, will shift the pace of the chemical reaction. In applying the Arrhenius equation to our dimethyl ether example, raising the temperature would result in more rapid decomposition, and vice versa. This equation is so potent in the realm of chemical kinetics that it allows scientists to extrapolate reaction behavior beyond the experimental conditions, empowering them to predict outcomes at temperatures that have yet to be tested.