Chapter 17, Problem 134CP

Interpretation Introduction

Interpretation : The value of the equilibrium constant for the dimerization of ${\text{VCl}}_{\text{4}}$ should be calculated.

Concept Introduction :

The dimerization is defined as an addition reaction results in producing adduct in which two molecules of same compound reacts with each other.

Answer to Problem 134CP

The value of the equilibrium constant is 33.

Explanation of Solution

Given information :

The compound ${\text{VCl}}_{\text{4}}$ undergoes dimerization. The 6.6834 g of ${\text{VCl}}_{\text{4}}$ dissolved in 100.0 g of carbon tetrachloride with the freezing point is lowered to ${5.97}^{\circ }C$ .The density of the equilibrium is $1.696{\text{g/cm}}^{\text{3}};K_f={29.8}^{\circ }\text{C}\text{kg/mol}$

To calculate the moles of ${\text{VCl}}_{\text{4}}$ as follows,

  $\begin{array}{l}\text{Moles}\text{of}{\text{VCl}}_{\text{4}}\text{=6}\text{.6834}\text{g}{\text{VCl}}_{\text{4}}\text{×}\frac{\text{1}\text{mol}{\text{VCl}}_{\text{4}}}{\text{192}\text{.74}\text{g}{\text{VCl}}_{\text{4}}}\\ \text{Moles}\text{of}{\text{VCl}}_{\text{4}}\text{=3}{\text{.4676×10}}^{\text{-2}}\text{mol}{\text{VCl}}_{\text{4}}\end{array}$

If ${\text{K}}_{\text{f}}$ is the formation constant, $i$ is the number of ions in the solution, $T$ is the temperature and $m$ is the molality of the solute panicles, the total molality of solute particles can be calculated as follows:

  $\begin{array}{l}\text{Total}\text{molality}=\text{im}\\ \text{=}\frac{\text{ΔT}}{{\text{K}}_{\text{f}}}\\ \text{=}\frac{\text{5}\text{.97}\text{C}}{\text{29}\text{.8}\text{C}\text{kg/mol}}\\ \text{Total}\text{molality}=\text{0}\text{.200}\text{mol/kg}\end{array}$

Similarly, the total moles of solute particles can be derived from the molality as follows,

  $\begin{array}{l}\text{Number}\text{of}\text{moles=}\frac{\text{0}\text{.200}\text{mol}}{\text{kg}}\text{×0}\text{.1000}\text{kg}\\ \text{Number}\text{of}\text{moles}=\text{0}\text{.0200}\text{mol}\end{array}$

If $x$ is the change in concentration of ${\text{VCl}}_{\text{4}}$ at equilibrium, the concentration of all the species in the equilibrium reaction as follows,

  $\begin{array}{ccc}& {\text{2VCl}}_{\text{4}}& {\text{V}}_{\text{2}}{\text{Cl}}_{\text{8}}\\ \text{Initial:}& \text{3}{\text{.4676×10}}^{\text{-2}}\text{mol}& \text{0}\\ \text{Equilibrium:}& \text{3}{\text{.4676×10}}^{\text{-2}}\text{-2x}& \text{x}\end{array}$

To calculate the value of x by calculating the total number of moles of solute particles.

  $\begin{array}{l}\text{Total}\text{moles}=\text{Mol}{\text{VCl}}_{\text{4}}\text{+}\text{mol}{\text{V}}_{\text{2}}{\text{Cl}}_{\text{8}}\\ \text{0}\text{.0200}\text{mol}=3.4676\times {10}^{-2}-2x+x\\ x=\text{0}\text{.0147}\text{mol}\end{array}$

To calculate the total volume of solution, calculate the concentration of all the species in solution which is the equilibrium constant.

  $\begin{array}{l}\text{Total}\text{Volume}=106.7\text{g}\times \frac{\text{1}{\text{cm}}^{\text{3}}}{\text{1}\text{.696}\text{g}}\text{×}\frac{\text{1L}}{\text{1}{\text{cm}}^{\text{3}}}\\ \text{Total}\text{Volume}=0.06291\text{L}\end{array}$

Thus, the equilibrium concentrations of solutions are,

  $\begin{array}{l}\left[{\text{VCl}}_4\right]=\frac{3.4676\times {10}^{-2}-2x}{0.06291\text{L}}\\ =\frac{3.4676\times {10}^{-2}-2\left(0.0147\right)}{0.06291L}\\ \left[{\text{VCl}}_4\right]=0.084\text{mol/L}\end{array}$

Similarly,

  $\begin{array}{l}\left[{\text{V}}_{\text{2}}{\text{Cl}}_{\text{8}}\right]=\frac{x}{0.06291L}\\ =\frac{0.0147}{0.06291L}\\ \left[{\text{V}}_{\text{2}}{\text{Cl}}_{\text{8}}\right]=0.234\text{mol/L}\end{array}$

Finally, substituting the values of concentrations to calculate the equilibrium constant (K) as follows:

  $\begin{array}{l}\text{K=}\frac{\left[{\text{V}}_{\text{2}}{\text{Cl}}_{\text{8}}\right]}{{\left[{\text{VCl}}_{\text{2}}\right]}^{\text{2}}}\\ =\frac{0.234}{{\left(0.084\right)}^2}\\ \text{K=}33\end{array}$

Thus, the value of the equilibrium constant is 33.

Conclusion

The value of the equilibrium constant is 33.