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Chapter 11: Problem 70

The decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) to \(\mathrm{NO}_{2}\) and \(\mathrm{NO}_{3}\) is a first-order gas-phase reaction. At \(25^{\circ} \mathrm{C}\), the reaction has a half-life of \(2.81\) s. At \(45^{\circ} \mathrm{C}\), the reaction has a half-life of \(0.313 \mathrm{~s}\). What is the activation energy of the reaction?

Short Answer

Expert verified
Answer: The activation energy of the reaction is approximately 60,535 J/mol.

Step by step solution

01

Write down the given information

We are given the following information: 1. The decomposition of N2O5 is a first-order reaction. 2. The half-life at 25°C is 2.81 seconds. 3. The half-life at 45°C is 0.313 seconds. Let's denote the half-life at 25°C as t₁ and the half-life at 45°C as t₂. t₁ = 2.81 s t₂ = 0.313 s T₁ = 25°C + 273.15 = 298.15 K T₂ = 45°C + 273.15 = 318.15 K
02

Find the rate constants (k)

The relationship between the half-life (t) and the rate constant (k) for a first-order reaction is given by: t = ln(2) / k We can use this equation to find the rate constants at both temperatures. k₁ = ln(2) / t₁ = ln(2) / 2.81 = 0.2468 s⁻¹ k₂ = ln(2) / t₂ = ln(2) / 0.313 = 2.2135 s⁻¹
03

Write down the Arrhenius equation

The Arrhenius equation is given by: k = Ae^(-Ea / RT) where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant (8.314 J/mol·K), and T is the temperature in Kelvin.
04

Write down the ratio of the rate constants

Dividing the Arrhenius equations for both temperatures, we get: k₁ / k₂ = e^((Ea / R) * (1 / T₁ - 1 / T₂))
05

Solve for the activation energy (Ea)

Rearrange the equation obtained in Step 4 to solve for Ea: Ea = R * ln(k₁ / k₂) / (1 / T₂ - 1 / T₁) Plug in the values and calculate the activation energy: Ea = 8.314 J/mol·K * ln(0.2468 s⁻¹ / 2.2135 s⁻¹) / (1 / 318.15 K - 1 / 298.15 K) Ea ≈ 60535 J/mol This is the answer: The activation energy of the reaction is approximately 60,535 J/mol.

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Most popular questions from this chapter

Consider the reaction $$\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}(a q)+\mathrm{SCN}^{-}(a q) \longrightarrow \mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5} \mathrm{SCN}^{2+}(a q)+\mathrm{H}_{2} \mathrm{O}$$ The following data were obtained: $$\begin{array}{ccc}\hline\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}{ }^{3+}\right] & {\left[\mathrm{SCN}^{-}\right]} & \text {Initial Rate }(\mathrm{mol} /\mathrm{L} \cdot \mathrm{min}) \\ \hline 0.025 & 0.060 & 6.5 \times 10^{-4} \\\0.025 & 0.077 & 8.4 \times 10^{-4} \\\0.042 & 0.077 & 1.4 \times 10^{-3} \\ 0.042 & 0.100 & 1.8 \times 10^{-3} \\\\\hline\end{array}$$ (a) Write the rate expression for the reaction. (b) Calculate \(k\). (c) What is the rate of the reaction when \(15 \mathrm{mg}\) of \(\mathrm{KSCN}\) is added to \(1.50 \mathrm{~L}\) of a solution \(0.0500 \mathrm{M}\) in \(\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}{ }^{3+}\) ?

The decomposition of sulfuryl chloride, \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\), to sulfur dioxide and chlorine gases is a first-order reaction. $$\mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g)$$ At a certain temperature, the half-life of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is \(7.5 \times 10^{2} \mathrm{~min}\). Consider a sealed flask with \(122.0 \mathrm{~g}\) of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) (a) How long will it take to reduce the amount of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in the sealed flask to \(45.0 \mathrm{~g}\) ? (b) If the decomposition is stopped after \(29.0 \mathrm{~h}\), what volume of \(\mathrm{Cl}_{2}\) at \(27^{\circ} \mathrm{C}\) and \(1.00\) atm is produced?

A reaction has two reactants \(\mathrm{A}\) and \(\mathrm{B}\). What is the order with respect to each reactant and the overall order of the reaction described by each of the following rate expressions? (a) rate \(=k_{1}[\mathrm{~A}]^{3}\) (b) rate \(=k_{2}[\mathrm{~A}] \times[\mathrm{B}]\) (c) rate \(=k_{3}[\mathrm{~A}] \times[\mathrm{B}]^{2}\) (d) rate \(=k_{4}[\mathrm{~B}]\)

The decomposition of nitrogen dioxide is a second-order reaction. At \(550 \mathrm{~K}\), a \(0.250 M\) sample decomposes at the rate of \(1.17 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{min} .\) (a) Write the rate expression. (b) What is the rate constant at \(550 \mathrm{~K}\) ? (c) What is the rate of decomposition when \(\left[\mathrm{NO}_{2}\right]=0.800 \mathrm{M?}\)

Write the rate expression for each of the following elementary steps: (a) \(\mathrm{NO}+\mathrm{O}_{3} \longrightarrow \mathrm{NO}_{2}+\mathrm{O}_{2}\) (b) \(2 \mathrm{NO}_{2} \longrightarrow 2 \mathrm{NO}+\mathrm{O}_{2}\) (c) \(\mathrm{K}+\mathrm{HCl} \longrightarrow \mathrm{KCl}+\mathrm{H}\)
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