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Chapter 14: Problem 91

The reaction $2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\( is second order in \)\mathrm{NO}$ and first order in \(\mathrm{O}_{2} .\) When \([\mathrm{NO}]=0.040 \mathrm{M}\) and \(\left[\mathrm{O}_{2}\right]=0.035 \mathrm{M},\) the observed rate of disappearance of \(\mathrm{NO}\) is $9.3 \times 10^{-5} \mathrm{M} / \mathrm{s} .(\mathbf{a})\( What is the rate of disappearance of \)\mathrm{O}_{2}$ at this moment? (b) What is the value of the rate constant? (c) What are the units of the rate constant? (d) What would happen to the rate if the concentration of NO were increased by a factor of \(1.8 ?\)

Short Answer

Expert verified
(a) The rate of disappearance of O₂ at this moment is: \[Rate_{\mathrm{O}_{2}} = \frac{1}{2} \cdot (9.3 \times 10^{-5}\, \mathrm{M/s}) = 4.65 \times 10^{-5} \mathrm{M/s}\] (b) The value of the rate constant k is: \[k = \frac{9.3 \times 10^{-5}\, \mathrm{M/s}}{(0.040\, \mathrm{M})^2(0.035\, \mathrm{M})} = 2.07 \times 10^2 \mathrm{M^{-2}s^{-1}}\] (c) The units of the rate constant k are M⁻²s⁻¹. (d) The change in the rate when the concentration of NO is increased by a factor of 1.8 is: \[Rate \, Change = \frac{(1.8)^2}{1} = 3.24\] The rate will increase by a factor of 3.24 when the concentration of NO is increased by a factor of 1.8.

Step by step solution

01

1. Write the Rate Law for the Reaction

: The rate law shows the dependence of the reaction rate on the concentrations of the reactants. Given that the reaction is second order in NO and first order in O₂, we can write the rate law as: \[Rate = k[\mathrm{NO}]^2[\mathrm{O}_{2}]\]
02

2. Determine the Rate of Disappearance of O₂ (a)

: We are given the rate of disappearance of NO, so we will use stoichiometry to find the rate of disappearance of O₂. From the balanced reaction, we have the stoichiometric relationship: \(2\, \mathrm{NO}(g) + \mathrm{O}_{2}(g) \longrightarrow 2\, \mathrm{NO}_{2}(g)\) Using the stoichiometry, the relationship between rates of disappearance is: \[\frac{Rate_{\mathrm{O}_{2}}}{Rate_{\mathrm{NO}}} = \frac{-\Delta[\mathrm{O}_{2}]}{-2\Delta[\mathrm{NO}]}\] Now, plug in the given rate of disappearance of NO and solve for rate of disappearance of O₂: \[Rate_{\mathrm{O}_{2}} = \frac{1}{2} \cdot (9.3 \times 10^{-5}\, \mathrm{M/s})\]
03

3. Calculate the Value of the Rate Constant, k (b)

: Plug in the given reactant concentrations and the calculated rate of disappearance of O₂ into the rate law, and solve for k: \[9.3 \times 10^{-5}\, \mathrm{M/s} = k(0.040\, \mathrm{M})^2(0.035\, \mathrm{M})\] Solve for k, keeping in mind to report to an appropriate number of significant figures.
04

4. Determine the Units of the Rate Constant, k (c)

: We have the rate law in the form: \[Rate = k[\mathrm{NO}]^2[\mathrm{O}_{2}]\] We know that the rate has units of M/s. Thus, the units of k can be found by rearranging the equation: \[k = \frac{Rate}{[\mathrm{NO}]^2[\mathrm{O}_{2}]}\] Determine the units of k using the known units for Rate and concentrations.
05

5. Calculate the Rate Change for an Increased Concentration of NO (d)

: We are asked what would happen to the rate if the concentration of NO were increased by a factor of 1.8. To determine this, we will analyze the rate equation with the new concentration of NO: \[New\ Rate = k[\mathrm{NO}]'^2[\mathrm{O}_{2}]\] Where \([\mathrm{NO}]' = 1.8 \times [\mathrm{NO}]\) The change in the rate can be expressed as: \[Rate \, Change = \frac{New\ Rate}{Old\ Rate}\] Plug in the initial and new concentrations of NO and solve for the Rate Change.

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Most popular questions from this chapter

Many metallic catalysts, particularly the precious-metal ones, are often deposited as very thin films on a substance of high surface area per unit mass, such as alumina \(\left(\mathrm{Al}_{2} \mathrm{O}_{3}\right)\) or silica \(\left(\mathrm{SiO}_{2}\right) .(\mathbf{a})\) Why is this an effective way of utilizing the catalyst material compared to having powdered metals? (b) How does the surface area affect the rate of reaction?

The enzyme urease catalyzes the reaction of urea, $\left(\mathrm{NH}_{2} \mathrm{CONH}_{2}\right)$, with water to produce carbon dioxide and ammonia. In water, without the enzyme, the reaction proceeds with a first-order rate constant of \(4.15 \times 10^{-5} \mathrm{~s}^{-1}\) at $100^{\circ} \mathrm{C}$. In the presence of the enzyme in water, the reaction proceeds with a rate constant of \(3.4 \times 10^{4} \mathrm{~s}^{-1}\) at $21^{\circ} \mathrm{C}$. (a) Write out the balanced equation for the reaction catalyzed by urease. \((\mathbf{b})\) If the rate of the catalyzed reaction were the same at \(100^{\circ} \mathrm{C}\) as it is at \(21^{\circ} \mathrm{C}\), what would be the difference in the activation energy between the catalyzed and uncatalyzed reactions? (c) In actuality, what would you expect for the rate of the catalyzed reaction at \(100^{\circ} \mathrm{C}\) as compared to that at \(21^{\circ} \mathrm{C} ?(\mathbf{d})\) On the basis of parts (c) and (d), what can you conclude about the difference in activation energies for the catalyzed and uncatalyzed reactions?

(a) Develop an equation for the half-life of a zero-order reaction. (b) Does the half-life of a zero-order reaction increase, decrease, or remain the same as the reaction proceeds?

Consider a hypothetical reaction between \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) that is zero order in A, second order in B, and first order in C. (a) Write the rate law for the reaction. (b) How does the rate change when [A] is tripled and the other reactant concentrations are held constant? (c) How does the rate change when [B] is doubled and the other reactant concentrations are held constant? (d) How does the rate change when [C] is tripled and the other reactant concentrations are held constant? (e) By what factor does the rate change when the concentrations of all three reactants are doubled? (f) By what factor does the rate change when the concentrations of all three reactants are cut in half?

The \(\mathrm{NO}_{x}\) waste stream from automobile exhaust includes species such as \(\mathrm{NO}\) and \(\mathrm{NO}_{2}\). Catalysts that convert these species to \(\mathrm{N}_{2}\) are desirable to reduce air pollution. (a) Draw the Lewis dot and VSEPR structures of \(\mathrm{NO}, \mathrm{NO}_{2}\), and \(\mathrm{N}_{2} .(\mathbf{b})\) Using a resource such as Table 8.3 , look up the energies of the bonds in these molecules. In what region of the electromagnetic spectrum are these energies? \((\mathbf{c})\) Design a spectroscopic experiment to monitor the conversion of \(\mathrm{NO}_{x}\) into \(\mathrm{N}_{2}\), describing what wavelengths of light need to be monitored as a function of time.
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