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Chapter 14: Problem 36

Consider the following reaction, which takes place in a single elementary step: $$2 \mathrm{~A}+\mathrm{B} \underset{k_{-1}}{\frac{k_{1}}{\longrightarrow}} \mathrm{A}_{2} \mathrm{~B}$$ If the equilibrium constant \(K_{\mathrm{c}}\) is 12.6 at a certain temperature and if \(k_{\mathrm{r}}=5.1 \times 10^{-2} \mathrm{~s}^{-1},\) calculate the value of \(k_{\mathrm{f}}\).

Short Answer

Expert verified
The forward rate constant (\(k_{f}\) or \(k_{1}\)) for the given reaction is approximately \(6.31 \times 10^{-2} \mathrm{s}^{-1}\)

Step by step solution

01

Understand Relationship Between Constants

From the equilibrium constant and rate constants, we have the equation \(K_{c}=\frac{k_{\mathrm{1}}}{k_{-1}}\), where \(k_{1}\) is the forward rate constant, \(k_{-1}\) is the reverse rate constant, and \(K_{c}\) is the equilibrium constant.
02

Relate Reaction Rate and Equilibrium Constants

Given that \(k_{\mathrm{r}}=k_{\mathrm{1}}+k_{\mathrm{-1}}=5.1 \times 10^{-2} \mathrm{~s}^{-1}\), we need to use this relationship to solve for the reverse reaction rate constant \(k_{-1}\). This will involve a bit of algebraic manipulation.
03

Calculate Reverse Rate Constant

By rearranging the equation introduced in the previous step, we can find \(k_{-1}\). Substituting the value of \(k_{1}\) from the equation \(K_{c}=\frac{k_{\mathrm{1}}}{k_{-1}}\) into \(k_{r}=k_{1}+k_{-1}\), we get \(k_{-1}=\frac{k_{\mathrm{r}}}{K_{c}+1}\). Now we substitute the given values into this equation.
04

Calculate the Forward Rate Constant

With the reverse rate constant (\(k_{-1}\)) calculated, it can be used to find the forward rate constant (\(k_{1}\)) using the equation \(K_{c}=\frac{k_{\mathrm{1}}}{k_{-1}}\), by rearranging to \(k_{\mathrm{1}}=K_{c} \times k_{-1}\) and substituting the values.

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Most popular questions from this chapter

At \(1024^{\circ} \mathrm{C},\) the pressure of oxygen gas from the decomposition of copper(II) oxide \((\mathrm{CuO})\) is 0.49 atm: $$4 \mathrm{CuO}(s) \rightleftharpoons 2 \mathrm{Cu}_{2} \mathrm{O}(s)+\mathrm{O}_{2}(g)$$ (a) What is \(K_{P}\) for the reaction? (b) Calculate the fraction of \(\mathrm{CuO}\) that will decompose if 0.16 mole of it is placed in a \(2.0-\mathrm{L}\) flask at \(1024^{\circ} \mathrm{C}\). (c) What would the fraction be if a 1.0 mole sample of \(\mathrm{CuO}\) were used? (d) What is the smallest amount of \(\mathrm{CuO}\) (in moles) that would establish the equilibrium?

The equilibrium constant \(K_{\mathrm{c}}\) for the reaction $$\mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g)$$ is 4.2 at \(1650^{\circ} \mathrm{C}\). Initially \(0.80 \mathrm{~mol} \mathrm{H}_{2}\) and \(0.80 \mathrm{~mol}\) \(\mathrm{CO}_{2}\) are injected into a 5.0 - \(\mathrm{L}\) flask. Calculate the concentration of each species at equilibrium.

The equilibrium constant \(K_{\mathrm{c}}\) for the following reaction is 1.2 at \(375^{\circ} \mathrm{C}\). $$\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g)$$ (a) What is the value of \(K_{P}\) for this reaction? (b) What is the value of the equilibrium constant \(K_{\mathrm{c}}\) for \(2 \mathrm{NH}_{3}(g) \rightleftharpoons \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) ?\) (c) What is the value of \(K_{c}\) for \(\frac{1}{2} \mathrm{~N}_{2}(g)+\frac{3}{2} \mathrm{H}_{2}(g)\) \(\rightleftharpoons \mathrm{NH}_{3}(g) ?\) (d) What are the values of \(K_{P}\) for the reactions described in (b) and (c)?

Consider the equilibrium system \(3 \mathrm{~A} \rightleftharpoons \mathrm{B}\). Sketch the changes in the concentrations of \(\mathrm{A}\) and \(\mathrm{B}\) over time for the following situations: (a) Initially only A is present. (b) Initially only B is present. (c) Initially both A and B are present (with A in higher concentration). In each case, assume that the concentration of \(\mathrm{B}\) is higher than that of \(\mathrm{A}\) at equilibrium.

When a gas was heated under atmospheric conditions, its color deepened. Heating above \(150^{\circ} \mathrm{C}\) caused the color to fade, and at \(550^{\circ} \mathrm{C}\) the color was barely detectable. However, at \(550^{\circ} \mathrm{C},\) the color was partially restored by increasing the pressure of the system. Which of the following best fits the above description? Justify your choice. (a) A mixture of hydrogen and bromine, (b) pure bromine, (c) a mixture of nitrogen dioxide and dinitrogen tetroxide. (Hint: Bromine has a reddish color and nitrogen dioxide is a brown gas. The other gases are colorless.)
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