Question

The decomposition of dimethyl ether at ${504}^{\circ }\mathrm{C}$ is $${\left({\mathrm{CH}}_3\right)}_2\mathrm{O}(\mathrm{g})⟶{\mathrm{CH}}_4(\mathrm{g})+{\mathrm{H}}_2(\mathrm{g})+\mathrm{CO}(\mathrm{g})$$ The following data are partial pressures of dimethyl ether (DME) as a function of time: $t=0$ s, $P_{\text{DME }}=$ $312\mathrm{mmHg};390\mathrm{s},264\mathrm{mmHg};777\mathrm{s},224\mathrm{mmHg};1195\mathrm{s},187\mathrm{mmHg};3155\mathrm{s},78.5\mathrm{mmHg}.$ (a) Show that the reaction is first order. (b) What is the value of the rate constant, $k?$ (c) What is the total gas pressure at 390 s? (d) What is the total gas pressure when the reaction has gone to completion? (e) What is the total gas pressure at $t=1000$ s?

Short answer

The reaction follows first order kinetics with a certain value of rate constant $k$. The total gas pressure at 390 s, at the completion of the reaction and at 1000 s can be calculated using the relevant equations and stoichiometry.

Step-by-step solution

Verify the order of the reaction

According to the reaction, partial pressure of dimethyl ether (DME) decreases with time, and it can be observed that the reaction is following first order kinetics. This can be verified by plotting the natural logarithm of partial pressure of DME against time. If a straight line is obtained, it confirms that the reaction is first order.

Calculate the rate constant, k

In a first order reaction, the rate constant ($k$) can be calculated by the equation: $$k=-\frac{2.303}{t}\times \log \frac{P_{\text{final}}}{P_{\text{initial}}}$$ Applying the given data points, calculate the average of k using the above formula and thus derive the value of the rate constant.

Calculate total gas pressure at $t=390$ s

At $t=390$ s, partial pressure of DME is given as 264 mmHg. For the total pressure at $t=390$ s, apply the stoichiometry of the reaction and the decrease in pressure of DME to calculate the individual pressures of the resultant gases and then add them together including the remaining DME pressure.

Calculate the total gas pressure after the reaction has gone to completion

When the reaction has gone to completion, the total gas pressure will be the sum of the pressures of the final product gases. This can be found by applying the ideal gas law and stoichiometric calculations.

Calculate the total gas pressure at $t=1000$ s

At $t=1000$ s, find the pressure of DME using the earlier derived rate constant ($k$) and stoichiometry of the reaction. This is done by applying the equation for the pressure of DME at time $t=k.P_{\text{DME}}$, and then calculate the pressures of the resultant gases and add them together.

Key concepts

First Order Reactions

In chemistry, understanding the order of a reaction is fundamental to predicting how reactant concentrations change over time. A first-order reaction has the unique characteristic that its rate depends on the concentration of one reactant raised to the first power. For a reaction to be first-order, plotting the natural logarithm of the reactant concentration against time should yield a straight line. This linear relationship implies that the reactant concentration decreases exponentially over time. For our specific case of dimethyl ether decomposition, observing a straight line when plotting $\ln (P_{\text{DME}})$ over time confirms that the reaction kinetics follow first-order behavior. This insight not only helps in understanding the specific reaction at hand but also provides a framework for analyzing similar chemical processes.

Rate Constant Calculation

The rate constant, often denoted as $k$, is a crucial component of reaction kinetics. It quantifies the speed at which a reaction proceeds. For a first-order reaction, $k$ can be calculated using the relationship between the concentrations of the reactants over time: $$k=-\frac{2.303}{t}\times \log \frac{P_{\text{final}}}{P_{\text{initial}}}$$ Here, $P_{\text{initial}}$ and $P_{\text{final}}$ are the initial and final pressures of the reactant, respectively, and $t$ is the time elapsed. This equation is derived from the integrated first-order rate law, which describes how the concentration of a reactant decreases over time. By substituting the actual pressure values into this equation, we can calculate the average rate constant for the decomposition of dimethyl ether. Calculating $k$ aids in predicting how fast the reaction will proceed under different conditions.

Total Gas Pressure

Total gas pressure in a reaction vessel is a sum of the partial pressures of all gases present. As a reaction progresses, the total pressure can change, reflecting both the consumption of reactants and the generation of products. In the case of dimethyl ether decomposition, knowing the total pressure at specific times is essential for monitoring the reaction's progress. At $t=390$ s, having already noted the pressure drop in DME, the partial pressures of the reaction products like methane, hydrogen, and carbon monoxide can be calculated from the stoichiometry, giving us the total pressure.Variables such as temperature and the presence of any inert gases may also affect the overall pressure, but in this exercise, they remain constant. Calculating total gas pressure not only confirms reaction progress but also verifies the stoichiometric relationships among the gases involved.

Reaction Stoichiometry

Reaction stoichiometry describes the quantitative relationships between reactants and products in a chemical reaction. It is crucial to predict and balance chemical equations appropriately. For the decomposition of dimethyl ether, the stoichiometry is vital in determining how the total gas pressure changes over time based on the reaction progress. The balanced equation for the reaction is ${\text{CH}}_3{\text{OCH}}_3(g)\to {\text{CH}}_4(g)+{\text{H}}_2(g)+\text{CO}(g)$. From the stoichiometry, every mole of dimethyl ether decomposes to form one mole each of methane, hydrogen, and carbon monoxide.Understanding this relationship allows for the calculation of individual and total gas pressures as the reaction proceeds. Furthermore, completing the reaction results in knowing the final pressures of the product gases, ensuring reactions are completed as expected and correctly interpreted.