?Consider the hypothetical reaction \(\mathrm{A}(g)+2 \mathrm{~B}(g)

Chapter 15, Problem 15.92

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Consider the hypothetical reaction \(\mathrm{A}(g)+2 \mathrm{~B}(g) \rightleftharpoons\) 2 C(g), for which \(K_{c}=0.25\) at a certain temperature. A 1.00-L reaction vessel is loaded with 1.00 mol of compound C, which is allowed to reach equilibrium. Let the variable x represent the number of mol / L of compound A present at equilibrium.

(a) In terms of x, what are the equilibrium concentrations of compounds B and C?

(b) What limits must be placed on the value of x so that all concentrations are positive?

(c) By putting the equilibrium concentrations (in terms of x ) into the equilibriumconstant expression, derive an equation that can be solved for x.

(d) The equation from part (c) is a cubic equation (one that has the form \(a x^{3}+b x^{2}+c x+d=0\) ). In general, cubic equations cannot be solved in closed form. However, you can estimate the solution by plotting the cubic equation in the allowed range of x that you specified in part (b). The point at which the cubic equation crosses the x-axis is the solution.

(e) From the plot in part (d), estimate the equilibrium concentrations of A, B, and C. (Hint: You can check the accuracy of your answer by substituting these concentrations into the equilibrium expression.)

Text Transcription:

A(g)+2 B(g) \rightleftharpoons

Kc = 0.25

ax3 + bx2 + cx + d = 0