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Chapter 13: Problem 100

Consider the following elementary step: $$ \mathrm{X}+2 \mathrm{Y} \longrightarrow \mathrm{XY}_{2} $$ (a) Write a rate law for this reaction. (b) If the initial rate of formation of \(\mathrm{XY}_{2}\) is \(3.8 \times 10^{-3} \mathrm{M} / \mathrm{s}\) and the initial concentrations of \(\mathrm{X}\) and \(\mathrm{Y}\) are \(0.26 \mathrm{M}\) and \(0.88 M\), what is the rate constant of the reaction?

Short Answer

Expert verified
The rate law for the reaction is \(rate = k[X][Y]^{2}\). The rate constant, \(k\), for this reaction is approximately \(2.03 \times 10^{-2} M^{-2}s^{-1}\).

Step by step solution

01

Derive the Rate Law

The general form of a rate law is given by the formula \(rate = k[A]^{m}[B]^{n}\), where \(k\) is the rate constant, \([A]\) and \([B]\) are the molar concentrations of the reactants, and \(m\) and \(n\) are the orders of the reaction with respect to each reactant. The order of the reaction is determined by the coefficients in the balanced chemical equation in case of an elementary reaction. Thus, for the reaction \( X + 2Y \longrightarrow XY_{2}\), the rate law is \(rate = k[X][Y]^{2}\).
02

Apply Given Initial Conditions to Find Rate Constant

Using the given initial rate of formation of \(XY_{2}\) and the initial concentrations of the reactants, we can substitute these values into the rate law to find the rate constant. The given values are: rate = \(3.8 \times 10^{-3} M/s\), [X] = \(0.26 M\), and [Y] = \(0.88 M\). Therefore, \(3.8 \times 10^{-3} = k(0.26)(0.88)^{2}\).
03

Calculate the Rate Constant

Solving the equation from step 2 for \(k\) gives the rate constant for the reaction. \(k = \frac{3.8 \times 10^{-3}}{(0.26)(0.88)^{2}}\).

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Most popular questions from this chapter

The thermal decomposition of phosphine \(\left(\mathrm{PH}_{3}\right)\) into phosphorus and molecular hydrogen is a first-order reaction: $$ 4 \mathrm{PH}_{3}(g) \longrightarrow \mathrm{P}_{4}(g)+6 \mathrm{H}_{2}(g) $$ The half-life of the reaction is \(35.0 \mathrm{~s}\) at \(680^{\circ} \mathrm{C}\). Calculate (a) the first-order rate constant for the reaction and (b) the time required for 95 percent of the phosphine to decompose.

The first-order rate constant for the decomposition of dimethyl ether $$ \left(\mathrm{CH}_{3}\right)_{2} \mathrm{O}(g) \longrightarrow \mathrm{CH}_{4}(g)+\mathrm{H}_{2}(g)+\mathrm{CO}(g) $$ is \(3.2 \times 10^{-4} \mathrm{~s}^{-1}\) at \(450^{\circ} \mathrm{C} .\) The reaction is carried out in a constant-volume flask. Initially only dimethyl ether is present and the pressure is 0.350 atm. What is the pressure of the system after 8.0 min? Assume ideal behavior.

What are the shortest and longest times (in years) that can be estimated by carbon-14 dating?

A factory that specializes in the refinement of transition metals such as titanium was on fire. The firefighters were advised not to douse the fire with water. Why?

List four factors that influence the rate of a reaction.
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