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Chapter 19: Problem 9

Consider a reaction $\mathrm{A}_{2}(g)+\mathrm{B}_{2}(g) \rightleftharpoons 2 \mathrm{AB}(g),$ with atoms of A shown in red in the diagram and atoms of \(\mathrm{B}\) shown in blue. (a) If \(K_{\mathrm{c}}=1,\) which box represents the system at equilibrium? (b) If \(K_{\mathrm{c}}=1,\) which box represents the system at \(Q < K_{\mathrm{c}} ?(\mathbf{c})\) Rank the boxes in order of increasing magnitude of \(\Delta G\) for the reaction. [ Sections 19.5 and 19.7\(]\)

Short Answer

Expert verified
The short version of the answer is: a) The box representing the system at equilibrium is the one that satisfies the equation: \(1 = \frac{([AB]_0 + 2x)^2}{([A_{2}]_0 - x)([B_{2}]_0 - x)}\) b) The box representing the system with \(Q < K_{c}\) will be the one with a Q value less than 1. c) The boxes can be ranked in order of increasing magnitude of 螖G by analyzing the relationship between Q and Kc: 1. If Q < Kc, 螖G > 0. 2. If Q = Kc, 螖G = 0. 3. If Q > Kc, 螖G < 0.

Step by step solution

01

Calculate the concentration of each species at equilibrium

Since the graphical representation of the system at equilibrium is not known, we will assume the generic scenario for each box. Let the initial concentrations of A2, B2, and AB be [A2]0, [B2]0, and [AB]0, respectively. At equilibrium, the concentrations would be [A2], [B2], and [AB]. Let x mol of A2 and B2 react to produce 2x mol of AB. Then, the equilibrium concentrations are: [A2] = [A2]0 - x [B2] = [B2]0 - x [AB] = [AB]0 + 2x
02

Write the expression for Kc

Kc is the equilibrium constant in terms of concentration. For the given reaction, we can write the expression for Kc as: \(K_{c} = \frac{[AB]^2}{[A_{2}][B_{2}]}\)
03

Substitute the concentrations in the Kc expression

Now, using the equilibrium concentrations in terms of x, the Kc expression becomes: \[K_{c} = \frac{([AB]_0 + 2x)^2}{([A_{2}]_0 - x)([B_{2}]_0 - x)}\] Given that \(K_{c} = 1\), we have: \[1 = \frac{([AB]_0 + 2x)^2}{([A_{2}]_0 - x)([B_{2}]_0 - x)}\] We need to find out the box that satisfies this equation, which represents the system at equilibrium. #b) Find the box representing the system Q<Kc.#
04

Write the expression for Q

The reaction quotient Q is also expressed as: \(Q = \frac{[AB]^2}{[A_{2}][B_{2}]}\) However, unlike Kc, Q represents the current state of the system, not necessarily the equilibrium state.
05

Compare Q and Kc

We need to find the box where \(Q < K_{c}\). Since we are given that \(K_{c} = 1\), the system with \(Q < K_{c}\), will be the box where Q value is less than 1. #c) Rank the boxes in order of increasing 螖G#
06

Relate 螖G to Q and Kc

The relationship between 螖G, Q, and Kc is given by the equation: \[\Delta G = \Delta G^{\circ} + RT \ln {Q}\] At equilibrium, 螖G = 0, and thus: \[\Delta G^{\circ} = - RT \ln {K_{c}}\] So, the equation can be rewritten as: \[\Delta G = RT (\ln {Q} - \ln {K_{c}}) = RT \ln{\frac{Q}{K_{c}}}\]
07

Rank the boxes based on 螖G

For ranking the boxes according to increasing 螖G values, we need to analyze the relationship between Q and Kc, and how it affects 螖G: 1. If Q < Kc, then \(Q/K_{c} < 1\) and 螖G > 0. 2. If Q = Kc, then \(Q/K_{c} = 1\) and 螖G = 0. 3. If Q > Kc, then \(Q/K_{c} > 1\), and 螖G < 0. We can rank the boxes based on these relationships between Q, Kc, and 螖G.

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Most popular questions from this chapter

(a) Using data in Appendix \(C\), estimate the temperature at which the free- energy change for the transformation from \(\mathrm{I}_{2}(s)\) to \(\mathrm{I}_{2}(g)\) is zero. What assumptions must you make in arriving at this estimate? (b) Use a reference source, such as Web Elements (www.webelements.com), to find the experimental melting and boiling points of \(\mathrm{I}_{2} .\) (c) Which of the values in part (b) is closer to the value you obtained in part (a)? Can you explain why this is so?

Consider what happens when a sample of the explosive TNT (Section 8.8: "Chemistry Put to Work: Explosives and Alfred Nobel") is detonated under atmospheric pressure. (a) Is the detonation a spontaneous process? (b) What is the sign of \(q\) for this process? (c) Can you determine whether \(w\) is positive, negative, or zero for the process? Explain. (d) Can you determine the sign of \(\Delta E\) for the process? Explain.

Indicate whether \(\Delta G\) increases, decreases, or does not change when the partial pressure of \(\mathrm{H}_{2}\) is increased in each of the following reactions: (a) \(\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g)\) (b) \(2 \mathrm{HBr}(g) \longrightarrow \mathrm{H}_{2}(g)+\mathrm{Br}_{2}(g)\) (c) \(2 \mathrm{H}_{2}(g)+\mathrm{C}_{2} \mathrm{H}_{2}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{6}(g)\)

About \(86 \%\) of the world's electrical energy is produced by using steam turbines, a form of heat engine. In his analysis of an ideal heat engine, Sadi Carnot concluded that the maximum possible efficiency is defined by the total work that could be done by the engine, divided by the quantity of heat available to do the work (for example, from hot steam produced by combustion of a fuel such as coal or methane). This efficiency is given by the ratio \(\left(T_{\text {high }}-T_{\text {low }}\right) / T_{\text {high }}\), where \(T_{\text {high }}\) is the temperature of the heat going into the engine and \(T_{\text {low }}\) is that of the heat leaving the engine. (a) What is the maximum possible efficiency of a heat engine operating between an input temperature of \(700 \mathrm{~K}\) and an exit temperature of \(288 \mathrm{~K} ?\) (b) Why is it important that electrical power plants be located near bodies of relatively cool water? (c) Under what conditions could a heat engine operate at or near \(100 \%\) efficiency? (d) It is often said that if the energy of combustion of a fuel such as methane were captured in an electrical fuel cell instead of by burning the fuel in a heat engine, a greater fraction of the energy could be put to useful work. Make a qualitative drawing like that in Figure 5.10 that illustrates the fact that in principle the fuel cell route will produce more useful work than the heat engine route from combustion of methane.

Consider a system consisting of an ice cube. (a) Under what conditions can the ice cube melt reversibly? (b) If the ice cube melts reversibly, is \(\Delta E\) zero for the process? Explain.
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