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Chapter 13: Problem 117

In a solution with carbon tetrachloride as the solvent, the compound \(\mathrm{VCl}_{4}\) undergoes dimerization: $$ 2 \mathrm{VCl}_{4} \rightleftharpoons \mathrm{V}_{2} \mathrm{Cl}_{\mathrm{s}} $$ When \(6.6834 \mathrm{~g} \mathrm{VCl}_{4}\) is dissolved in \(100.0 \mathrm{~g}\) carbon tetrachloride, the freezing point is lowered by \(5.97^{\circ} \mathrm{C}\). Calculate the value of the equilibrium constant for the dimerization of \(\mathrm{VCl}_{4}\) at this temperature. (The density of the equilibrium mixture is \(1.696 \mathrm{~g} / \mathrm{cm}^{3}\), and \(K_{f}=29.8^{\circ} \mathrm{C} \mathrm{kg} / \mathrm{mol}\) for \(\mathrm{CCl}_{4} \cdot\) )

Short Answer

Expert verified
The equilibrium constant for the dimerization of \(\mathrm{VCl}_{4}\) at this temperature is \(9.54\).

Step by step solution

01

Calculate the molality of the \(\mathrm{VCl}_{4}\) solution

First, we will determine the molar mass of \(\mathrm{VCl}_{4}\): \(\mathrm{VCl}_{4} = 1 \times \mathrm{V} + 4 \times \mathrm{Cl} = 1 \times 50.94 + 4 \times 35.45 = 193.74 \mathrm{g/mol}\) Now, we will calculate the moles of \(\mathrm{VCl}_{4}\): Moles of \(\mathrm{VCl}_{4}\) = \(\frac{6.6834 \mathrm{~g}}{193.74 \mathrm{g/mol}} = 0.0345 \mathrm{mol}\) Then, we will calculate the molality of the solution: Molality = \(\frac{0.0345 \mathrm{mol}}{100.0 \mathrm{~g} \times \frac{1 \mathrm{kg}}{1000\,\mathrm{g}}} = 0.345 \frac{\mathrm{mol}}{\mathrm{kg}}\)
02

Calculate the molality of dissociated solute (dimerized)

We will use the freezing point depression equation: \(\Delta T_{f} = i K_{f} \cdot m\) , where: - \(\Delta T_{f} = 5.97^{\circ} \mathrm{C}\) (given) - \(i\) = van't Hoff factor - \(K_{f} = 29.8^{\circ} \mathrm{C} \mathrm{kg} / \mathrm{mol}\) (given) - \(m\) = molality of dissociated solute For the dimerization reaction, the van't Hoff factor \(i = 1\) (half the number of original particles). Therefore, we have: \(5.97^{\circ} \mathrm{C} = 1 \times 29.8^{\circ} \mathrm{C} \mathrm{kg} / \mathrm{mol} \cdot m\) Solving for the molality: \(m = \frac{5.97^{\circ} \mathrm{C}}{29.8^{\circ} \mathrm{C} \mathrm{kg} / \mathrm{mol}} = 0.200 \frac{\mathrm{mol}}{\mathrm{kg}}\)
03

Calculate the equilibrium constant of the dimerization reaction

Let \(x\) be the molality of dissociated \(\mathrm{VCl}_{4}\) and \(y\) be the molality of the dimer \(\mathrm{V}_{2}\mathrm{Cl}_{8}\). We have: \(x = 0.345 - 0.200 = 0.145 \frac{\mathrm{mol}}{\mathrm{kg}}\) \(y = 0.200 \frac{\mathrm{mol}}{\mathrm{kg}}\) The equilibrium constant for the dimerization reaction is given by: \(K = \frac{[\mathrm{V}_{2}\mathrm{Cl}_{8}]}{[\mathrm{VCl}_{4}]^{2}} = \frac{y}{x^{2}}\) Calculating the equilibrium constant: \(K = \frac{0.200}{(0.145)^{2}} = 9.54\) Therefore, the equilibrium constant for the dimerization of \(\mathrm{VCl}_{4}\) at this temperature is \(9.54\).

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Most popular questions from this chapter

For the reaction below, \(K_{\mathrm{p}}=1.16\) at \(800 .{ }^{\circ} \mathrm{C}\). $$ \mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g) $$ If a \(20.0-\mathrm{g}\) sample of \(\mathrm{CaCO}_{3}\) is put into a \(10.0\) - \(\mathrm{L}\) container and heated to \(800 .{ }^{\circ} \mathrm{C}\), what percentage by mass of the \(\mathrm{CaCO}_{3}\) will react to reach equilibrium?

A 1.00-L flask was filled with \(2.00\) moles of gaseous \(\mathrm{SO}_{2}\) and \(2.00\) moles of gaseous \(\mathrm{NO}_{2}\) and heated. After equilibrium was reached, it was found that \(1.30\) moles of gaseous NO was present. Assume that the reaction $$ \mathrm{SO}_{2}(g)+\mathrm{NO}_{2}(g) \rightleftharpoons \mathrm{SO}_{3}(g)+\mathrm{NO}(g) $$ occurs under these conditions. Calculate the value of the equilibrium constant, \(K\), for this reaction.

Suppose the reaction system $$ \mathrm{UO}_{2}(s)+4 \mathrm{HF}(g) \rightleftharpoons \mathrm{UF}_{4}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) $$ has already reached equilibrium. Predict the effect that each of the following changes will have on the equilibrium position. Tell whether the equilibrium will shift to the right, will shift to the left, or will not be affected. a. Additional \(\mathrm{UO}_{2}(s)\) is added to the system. b. The reaction is performed in a glass reaction vessel; \(\mathrm{HF}(\mathrm{g})\) attacks and reacts with glass. c. Water vapor is removed.

For the reaction \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g), K_{\mathrm{p}}=0.25\) at a cer- tain temperature. If \(0.040\) atm of \(\mathrm{N}_{2} \mathrm{O}_{4}\) is reacted initially, calculate the equilibrium partial pressures of \(\mathrm{NO}_{2}(g)\) and \(\mathrm{N}_{2} \mathrm{O}_{4}(g)\).

Predict the shift in the equilibrium position that will occur for each of the following reactions when the volume of the reaction container is increased. a. \(\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g)\) b. \(\mathrm{PCl}_{5}(g) \rightleftharpoons \mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g)\) c. \(\mathrm{H}_{2}(g)+\mathrm{F}_{2}(g) \rightleftharpoons 2 \mathrm{HF}(g)\) d. \(\mathrm{COCl}_{2}(g) \rightleftharpoons \mathrm{CO}(g)+\mathrm{Cl}_{2}(g)\) e. \(\mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)\)
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