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Chapter 14: Problem 28

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?

Short Answer

Expert verified
Solution: (a) The rate law is given by Rate = k * [B]^2 * [C]. (b) The rate remains unchanged when [A] is tripled. (c) The new rate is 4 times the initial rate when [B] is doubled. (d) The new rate is 3 times the initial rate when [C] is tripled. (e) The rate changes by a factor of 8 when all three reactants' concentrations are doubled. (f) The rate changes by a factor of \(\frac{1}{8}\) when all three reactants' concentrations are cut in half.

Step by step solution

01

(a) Write the rate law for the reaction

The rate law for a reaction can be expressed as: Rate = k * [A]^m * [B]^n * [C]^p where m, n, and p are the orders of the reactants A, B, and C respectively, and k is the rate constant. In this case, the reaction is zero order in A, second order in B, and first order in C. Therefore, the rate law is: Rate = k * [A]^0 * [B]^2 * [C]^1 This simplifies to: Rate = k * [B]^2 * [C]
02

(b) How does the rate change when [A] is tripled and the other reactant concentrations are held constant?

Since the reaction is zero order in A, tripling the concentration of A will not have any impact on the rate of the reaction. Thus, the rate remains unchanged.
03

(c) How does the rate change when [B] is doubled and the other reactant concentrations are held constant?

The reaction is second order in B, so if the concentration of B doubles, we can represent this change in the rate law: New Rate = k * [2B]^2 * [C] New Rate = k * (2^2) * [B]^2 * [C] New Rate = 4 * (k * [B]^2 * [C]) The new rate is 4 times the initial rate.
04

(d) How does the rate change when [C] is tripled and the other reactant concentrations are held constant?

The reaction is first order in C, so if the concentration of C triples, we can represent this change in the rate law: New Rate = k * [B]^2 * [3C] New Rate = 3 * (k * [B]^2 * [C]) The new rate is 3 times the initial rate.
05

(e) By what factor does the rate change when the concentrations of all three reactants are doubled?

We will have the concentrations of A, B, and C doubled: New Rate = k * [2A]^0 * [2B]^2 * [2C]^1 New Rate = k * [B]^2 * [C] * (2^2) * 2 The factor by which the rate changes is (2^2) * 2 = 8.
06

(f) By what factor does the rate change when the concentrations of all three reactants are cut in half?

We will have the concentrations of A, B, and C halved: New Rate = k * [\(\frac{1}{2}\)A]^0 * [\(\frac{1}{2}\)B]^2 * [\(\frac{1}{2}\)C]^1 New Rate = k * [B]^2 * [C] * (\(\frac{1}{2}\))^2 * \(\frac{1}{2}\) The factor by which the rate changes is (\(\frac{1}{2}\))^2 * \(\frac{1}{2}\) = \(\frac{1}{8}\).

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Most popular questions from this chapter

(a) A certain first-order reaction has a rate constant of $2.75 \times 10^{-2} \mathrm{~s}^{-1}\( at \)20^{\circ} \mathrm{C}\(. What is the value of \)k$ at \(60^{\circ} \mathrm{C}\) if $E_{a}=75.5 \mathrm{~kJ} / \mathrm{mol} ?(\mathbf{b})\( Another first-order reaction also has a rate constant of \)2.75 \times 10^{-2} \mathrm{~s}^{-1}\( at \)20^{\circ} \mathrm{C}$. What is the value of \(k\) at \(60^{\circ} \mathrm{C}\) if $E_{a}=125 \mathrm{~kJ} / \mathrm{mol} ?(\mathbf{c})$ What assumptions do you need to make in order to calculate answers for parts (a) and (b)?

(a) Consider the combustion of ethylene, \(\mathrm{C}_{2} \mathrm{H}_{4}(g)+\) $3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) .\( If the concentration of \)\mathrm{C}_{2} \mathrm{H}_{4}$ is decreasing at the rate of \(0.025 \mathrm{M} / \mathrm{s}\), what are the rates of change in the concentrations of \(\mathrm{CO}_{2}\) and $\mathrm{H}_{2} \mathrm{O}\( ? (b) The rate of decrease in \)\mathrm{N}_{2} \mathrm{H}_{4}$ partial pressure in a closed reaction vessel from the reaction $\mathrm{N}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g)$ is \(10 \mathrm{kPa}\) per hour. What are the rates of change of \(\mathrm{NH}_{3}\) partial pressure and total pressure in the vessel?

For each of the following gas-phase reactions, write the rate expression in terms of the appearance of each product and disappearance of each reactant: (a) $\mathrm{O}_{3}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{O}_{2}(g)+\mathrm{H}_{2}(g)$ (b) $4 \mathrm{NH}_{3}(g)+5 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{NO}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)$ (c) $2 \mathrm{C}_{2} \mathrm{H}_{2}(g)+5 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)$ (d) $\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{NH}_{2}(g) \longrightarrow \mathrm{C}_{3} \mathrm{H}_{6}(g)+\mathrm{NH}_{3}(g)$

You perform a series of experiments for the reaction $\mathrm{A} \rightarrow 2 \mathrm{~B}$ and find that the rate law has the form, rate \(=k[\mathrm{~A}]^{x} .\) Determine the value of \(x\) in each of the following cases: (a) The rate increases by a factor of \(6.25,\) when \([\mathrm{A}]_{0}\) is increased by a factor of \(2.5 .(\mathbf{b})\) There is no rate change when \([\mathrm{A}]_{0}\) is increased by a factor of \(4 .(\mathbf{c})\) The rate decreases by a factor of \(1 / 2,\) when \([\mathrm{A}]_{0}\) is cut in half.

The rate of a first-order reaction is followed by spectroscopy, monitoring the absorbance of a colored reactant at \(520 \mathrm{nm}\). The reaction occurs in a 1.00-cm sample cell, and the only colored species in the reaction has an extinction coefficient of $5.60 \times 10^{3} \mathrm{M}^{-1} \mathrm{~cm}^{-1}\( at \)520 \mathrm{nm} .$ (a) Calculate the initial concentration of the colored reactant if the absorbance is 0.605 at the beginning of the reaction. (b) The absorbance falls to 0.250 at $30.0 \mathrm{~min} .\( Calculate the rate constant in units of \)\mathrm{s}^{-1}$. (c) Calculate the half-life of the reaction. (d) How long does it take for the absorbance to fall to \(0.100 ?\)
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