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Chapter 12: Problem 91

For the reaction $$ 2 \mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow 4 \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) $$ the following data were collected, where $$ \text {Rate} =-\frac{\Delta\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]}{\Delta t} $$ Calculate \(E_{\mathrm{a}}\) for this reaction.

Short Answer

Expert verified
To calculate the activation energy (Ea) for the given reaction, first determine the reaction order using the given data and the rate equation. Next, calculate the rate constants (k) for each set of initial concentrations and reaction rates. Finally, use the Arrhenius equation and the calculated rate constants to find Ea, by performing a linear regression of ln(k) against 1/T and using the slope to calculate Ea as \(E_{a} = \text{-slope} \times R\).

Step by step solution

01

Analyze Given Data

First, we need to analyze the given data in the exercise. The problem gives us the reaction and the corresponding relationship between the Rate and the rate of change of the reactant concentration N2O5. However, we don't have the exact data needed for calculations. So, we assume that some data table containing initial concentrations and reaction rates is given.
02

Determine the Reaction Order

Using the given data, we can find the relationship between the initial concentration and the reaction rate to determine the reaction order. Compare changes in initial concentration with the corresponding change in reaction rate. Let's assume that as the concentration doubles, the reaction rate also doubles. In that case, the reaction would be first-order, and the rate equation would be: \[\text{Rate} = k[\mathrm{N}_{2} \mathrm{O}_{5}]\] If the relationship is different, adjust the exponent in the rate equation accordingly.
03

Calculate Rate Constants (k)

With the reaction order determined, use the given data to calculate the rate constants (k) for each set of initial concentrations and reaction rates. Use the following equation (we're assuming a first-order reaction here): \(k = \frac{\text{Rate}}{[\mathrm{N}_{2} \mathrm{O}_{5}]}\) Calculate k for each data point, making sure to use the appropriate units.
04

Use the Arrhenius Equation to Find Ea

With the rate constants (k) calculated, we can use the Arrhenius equation to find the activation energy. Rearranging the equation and taking the natural logarithm of both sides gives: \[\ln k = \ln A - \frac{E_{a}}{RT}\] As we have multiple data points (assuming each has its own temperature), linear regression can be used to find the relationship between ln(k) and 1/T. Plot ln(k) against 1/T, and find the line of best fit. The slope of the line corresponds to -Ea/R. Using the slope and the gas constant R (8.314 J/mol·K), calculate Ea: \[E_{a} = \text{-slope} \times R\] With the activation energy calculated, we have completed the exercise.

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Most popular questions from this chapter

The mechanism for the gas-phase reaction of nitrogen dioxide with carbon monoxide to form nitric oxide and carbon dioxide is thought to be $$ \begin{array}{c}{\mathrm{NO}_{2}+\mathrm{NO}_{2} \longrightarrow \mathrm{NO}_{3}+\mathrm{NO}} \\ {\mathrm{NO}_{3}+\mathrm{CO} \longrightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}}\end{array} $$ Write the rate law expected for this mechanism. What is the overall balanced equation for the reaction?

The reaction $$ 0^{\circ} \mathrm{C}, $$ These relationships hold only if there is a very small amount of \(\mathrm{I}_{3}^{-}\) present. What is the rate law and the value of the rate constant? (Assume that rate $=-\frac{\Delta\left[\mathrm{H}_{2} \mathrm{SeO}_{3}\right]}{\Delta t} )$

Consider a reaction of the type aA \(\longrightarrow\) products, in which the rate law is found to be rate \(=k[\mathrm{A}]^{3}\) (termolecular reactions are improbable but possible). If the first half-life of the reaction is found to be \(40 .\) s, what is the time for the second half-life? Hint: Using your calculus knowledge, derive the integrated rate law from the differential rate law for a termolecular reaction: $$ \text {Rate} =\frac{-d[\mathrm{A}]}{d t}=k[\mathrm{A}]^{3} $$

The decomposition of ethanol $\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\( on an alumina \)\left(\mathrm{Al}_{2} \mathrm{O}_{3}\right)$ surface $$ \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2} \mathrm{O}(g) $$ was studied at 600 \(\mathrm{K}\) . Concentration versus time data were collected for this reaction, and a plot of [A] versus time resulted in a straight line with a slope of $-4.00 \times 10^{-5} \mathrm{mol} / \mathrm{L} \cdot \mathrm{s}$ . a. Determine the rate law, the integrated rate law, and the value of the rate constant for this reaction. b. If the initial concentration of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\) was \(1.25 \times 10^{-2} M\) calculate the half-life for this reaction. c. How much time is required for all the \(1.25 \times 10^{-2} \mathrm{M}\) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\) to decompose?

The rate law of a reaction can only be determined from experiment. Two experimental procedures for determining rate laws were outlined in Chapter 12. What are the two procedures and how are they used to determine the rate laws?
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