Consider the hypothetical reaction A(π) + 2 B(π) β 2 C(π), for which πΎπ = 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. (d) The equation from part (c) is a cubic equation (one that has the form ax3 + bx2 + cx + 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.)
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Identify the initial concentrations of the reactants and products. Initially, the concentration of C is 1.00 M (since 1.00 mol in a 1.00 L vessel), and the concentrations of A and B are 0 M.
Set up the expression for the equilibrium constant, Kc, using the reaction stoichiometry: Kc = \frac{[C]^2}{[A][B]^2}. Substitute the known value of Kc, which is 0.25.
Define the change in concentration using the variable x, where x is the increase in concentration of A at equilibrium. Since the reaction shows that 1 mole of A reacts with 2 moles of B to produce 2 moles of C, the concentration of B will increase by 2x and the concentration of C will decrease by 2x.
Write the equilibrium concentration expressions in terms of x: [A] = x, [B] = 2x, and [C] = 1.00 - 2x.
Substitute these expressions into the equilibrium constant expression and solve for x. This will give the equilibrium concentrations of A, B, and C. Check the accuracy of your solution by substituting these values back into the equilibrium expression to see if it equals 0.25.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Equilibrium Constant (Kc)
The equilibrium constant (Kc) is a numerical value that expresses the ratio of the concentrations of products to reactants at equilibrium for a given reaction at a specific temperature. For the reaction A(g) + 2B(g) β 2C(g), Kc = [C]^2 / ([A][B]^2). A Kc value of 0.25 indicates that at equilibrium, the concentration of products is relatively low compared to reactants, suggesting that the reaction favors the reactants.
An ICE table is a tool used to organize the initial concentrations, the changes in concentrations as the reaction proceeds, and the equilibrium concentrations of reactants and products. By defining initial amounts and using a variable (like x) to represent changes, one can systematically calculate the equilibrium concentrations based on stoichiometry and the equilibrium constant.
Stoichiometry refers to the quantitative relationship between reactants and products in a chemical reaction, derived from the balanced chemical equation. In the given reaction, the stoichiometric coefficients indicate that one mole of A reacts with two moles of B to produce two moles of C. Understanding these ratios is essential for calculating the changes in concentrations as the system reaches equilibrium.
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