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Chapter 14: Problem 43

As described in Exercise 14.41 , the decomposition of sulfuryl chloride \(\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right)\) is a first-order process. The rate constant for the decomposition at \(660 \mathrm{~K}\) is $4.5 \times 10^{-2} \mathrm{~s}^{-1}\(. (a) If we begin with an initial \)\mathrm{SO}_{2} \mathrm{Cl}_{2}\( pressure of \)60 \mathrm{kPa}$, what is the partial pressure of this substance after 60 s? (b) At what time will the partial pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) decline to one-tenth its initial value?

Short Answer

Expert verified
(a) The partial pressure of the sulfuryl chloride after 60 seconds is approximately 16.16 kPa. (b) The time it takes for the partial pressure to decrease to one-tenth of its initial value is approximately 50.67 seconds.

Step by step solution

01

(a) Calculate the partial pressure of SO₂Cl₂ after 60 seconds

Using the first-order decay equation: \(P(t) = P_0 e^{-kt}\) Plug in the given values: \(P(60) = 60~kPa \times e^{- (4.5 \times 10^{-2}~s^{-1})(60~s)}\) Calculate the result: \(P(60) \approx 16.16 ~kPa\) The partial pressure of the sulfuryl chloride after 60 seconds is approximately 16.16 kPa.
02

(b) Calculate the time for the partial pressure to decrease to one-tenth of its initial value

Since we want to find the time when the partial pressure of SO₂Cl₂ is one-tenth of its initial value (60 kPa), we can set up the equation as follows: \(\frac{1}{10}(60~kPa) = 60~kPa \times e^{-(4.5 \times 10^{-2}~s^{-1})t}\) Divide both sides by the initial pressure (60 kPa): \(\frac{1}{10} = e^{-(4.5 \times 10^{-2}~s^{-1})t}\) Take the natural logarithm of both sides: \(ln(\frac{1}{10}) = -(4.5 \times 10^{-2}~s^{-1})t\) Divide by the rate constant: \(\frac{ln(\frac{1}{10})}{-(4.5 \times 10^{-2}~s^{-1})} = t\) Calculate the result: \(t \approx 50.67~s\) The time it takes for the partial pressure to decrease to one-tenth of its initial value is approximately 50.67 seconds.

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Most popular questions from this chapter

(a) The gas-phase decomposition of sulfuryl chloride $\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right), \mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g)\( is first order in \)\mathrm{SO}_{2} \mathrm{Cl}_{2}\(. At \)300^{\circ} \mathrm{C}$ the half-life for this process is two and a half days. What is the rate constant at this temperature? (b) At \(400^{\circ} \mathrm{C}\) the rate constant is \(0.19 \mathrm{~min}^{-1}\). What is the half-life at this temperature?

The following mechanism has been proposed for the reaction of \(\mathrm{NO}\) with \(\mathrm{H}_{2}\) to form \(\mathrm{N}_{2} \mathrm{O}\) and $\mathrm{H}_{2} \mathrm{O}$ : $$ \begin{aligned} \mathrm{NO}(g)+\mathrm{NO}(g) & \longrightarrow \mathrm{N}_{2} \mathrm{O}_{2}(g) \\ \mathrm{N}_{2} \mathrm{O}_{2}(g)+\mathrm{H}_{2}(g) & \longrightarrow \mathrm{N}_{2} \mathrm{O}(g)+\mathrm{H}_{2} \mathrm{O}(g) \end{aligned} $$ (a) Show that the elementary reactions of the proposed mechanism add to provide a balanced equation for the reaction. (b) Write a rate law for each elementary reaction in the mechanism.(c) Identify anyintermediatesin the mechanism. (d) The observed rate law is rate \(=k[\mathrm{NO}]^{2}\left[\mathrm{H}_{2}\right]\). If the proposed mechanism is correct, what can we conclude about the relative speeds of the first and second reactions?

The reaction between ethyl iodide and hydroxide ion in ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) solution, $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{I}(a l c)+\mathrm{OH}^{-}(a l c) \longrightarrow\( \)\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)+\mathrm{I}^{-}(\( alc \))\( has an activation energy of \)86.8 \mathrm{~kJ} / \mathrm{mol}\( and a frequency factor of \)2.1 \times 10^{11} \mathrm{M}^{-1} \mathrm{~s}^{-1}$ (a) Predict the rate constant for the reaction at \(30^{\circ} \mathrm{C}\). (b) A solution of KOH in ethanol is made up by dissolving \(0.500 \mathrm{~g} \mathrm{KOH}\) in ethanol to form $500 \mathrm{~mL}\( of solution. Similarly, \)1.500 \mathrm{~g}\( of \)\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{I}\( is dissolved in ethanol to form \)500 \mathrm{~mL}$ of solution. Equal volumes of the two solutions are mixed. Assuming the reaction is first order in each reactant, what is the initial rate at \(30^{\circ} \mathrm{C} ?(\mathbf{c})\) Which reagent in the reaction is limiting, assuming the reaction proceeds to completion? ((d) Assuming the frequency factor and activation energy do not change as a function of temperature, calculate the rate constant for the reaction at $40^{\circ} \mathrm{C} .$

Based on their activation energies and energy changes and assuming that all collision factors are the same, rank the following reactions from slowest to fastest. (a) $E_{a}=75 \mathrm{~kJ} / \mathrm{mol} ; \Delta E=-20 \mathrm{~kJ} / \mathrm{mol}$ (b) $E_{a}=100 \mathrm{~kJ} / \mathrm{mol} ; \Delta E=+30 \mathrm{~kJ} / \mathrm{mol}$ (c) $E_{a}=85 \mathrm{~kJ} / \mathrm{mol} ; \Delta E=-50 \mathrm{~kJ} / \mathrm{mol}$

(a) What is a catalyst? (b) What is the difference between a homogeneous and a heterogeneous catalyst? (c) Do catalysts affect the overall enthalpy change for a reaction, the activation energy, or both?
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