7: Thermodynamics of Borax
- Page ID
- 516596
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)PURPOSE
- To determine the thermodynamic properties ( \( \Delta G^\circ \), \( \Delta H^\circ \), and \( \Delta S^\circ \) ) of borax dissolution by measuring its solubility at various temperatures.
- To calculate the solubility product ( \( \ln{K_{sp}} \) ) of borax at different temperatures using acid-base titration.
- To apply the relationship between \( \ln{K_{sp}} \) and \( \frac{1}{T} \) to extract enthalpy and entropy changes for a sparingly soluble salt.
- To gain hands-on experience in solution preparation, equilibrium analysis, and titration techniques as tools for thermodynamic investigation.
INTRODUCTION
The free energy change of a chemical process is proportional to the natural logarithm of its equilibrium constant according to the equation:
\[ \Delta G^\circ = -RT\ln{K} \]
where \( R \) is the ideal gas constant, 8.314 J/mol⸱K, and \( T \) is the absolute temperature in kelvins. The equilibrium constant, \( K \), is expressed for the equilibrium system when the reactants and products are in their standard states. For a sparingly soluble salt in an aqueous solution, the precipitate and the ions in the solution correspond to the standard states of the reactants and products, respectively.
For instance, the standard state equilibrium silver chromate is
\[ \ce{Ag2CrO4} (s) \rightleftharpoons 2 \ce{Ag+} (aq) + \ce{CrO^{2-}_{4}} (aq) \]
The solubility product, \( \ln{K} \), is set equal to the product of the molar concentrations of the ions, all raised to the power of their respective coefficients in the balanced equation. This is the mass action expression for the system:
\[ K_{sp} = [\ce{Ag+}]^2[\ce{CrO^{2-}_{4}}] \]
and the free energy change for the equilibrium is
\[ \Delta G^\circ = -RT\ln{K_{sp}} = -RT\ln{[\ce{Ag+}]^2[\ce{CrO^{2-}_{4}}]} \]
The free energy change of a chemical process is also a function of the enthalpy change and the entropy change of the process:
\[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \label{deltaGdef} \]
When the two free energy expressions are set equal, then
\[ -RT\ln{K_{sp}} = \Delta H^\circ - T \Delta S^\circ \]
Solving for \( \ln{K} \) and rearranging yields
\[ \begin{align} \begin{split} \textcolor{green}{\ln{K_{sp}}} &= \textcolor{red}{-\frac{\Delta H^\circ}{R}} \textcolor{blue}{\left(\frac{1}{T}\right)} + \textcolor{purple}{\frac{\Delta S^\circ}{R}} \\ \LARGE \textcolor{green}{y} &= \quad \LARGE \textcolor{red}{m} \; \textcolor{blue}{x} \; + \; \textcolor{purple}{b} \end{split} \label{lnKlinear} \end{align} \]
This equation can prove valuable in determining the thermodynamic properties of a chemical system, such as that of a sparingly soluble salt. A linear relationship exists when the values of \( \textcolor{green}{\ln{K_{sp}}} \) obtained at various temperatures are plotted as a function of the reciprocal temperature, \( \textcolor{blue}{\frac{1}{T}} \). The slope of the line equals \( \textcolor{red}{-\frac{\Delta H^\circ}{R}} \), and the y-intercept equals \( \textcolor{purple}{\frac{\Delta S^\circ}{R}} \). Since \(R\) is a constant, the \( \Delta H^\circ \) and the \( \Delta S^\circ \) for the equilibrium system can easily be calculated.
The Borax System
Borax is often given the name sodium tetraborate decahydrate and the formula \( \ce{Na2B4O7} \cdot \ce{10H2O} \). However, according to its chemical behavior, a more defining formula for borax is \( \ce{Na2B4O5(OH)4} \cdot \ce{8H2O} \). The anion, tetraborate (\( \ce{B4O^{2-}_7} \)), has the structure shown in the figure.
Borax dissolves and dissociates in water according to the equation:
\[ \ce{Na2B4O5(OH)4} \cdot \ce{8 H2O} (s) \rightleftharpoons \ce{2 Na+} (aq) + \ce{B4O5(OH)^{2-}_{4}} (aq) + \ce{8 H2O} (l) \label{boraxdissolution} \]
The mass action expression, set equal to the solubility product at equilibrium, for the solubility of borax is:
\[ K_{sp} = [\ce{Na+}]^2[\ce{B4O5(OH)^{2-}_{4}}] \]
The tetraborate anion is the conjugate base of the weak acid boric acid and is capable of accepting two protons from a strong acid in an aqueous solution:
\[ \ce{B4O5(OH)^{2-}_{4}} (aq) + \ce{2 H+} (aq) + \ce{3 H2O} (l) \rightleftharpoons \ce{4 H3BO3} (aq) \]
Therefore, the molar concentration of the tetraborate anion in a saturated borax solution can be measured with titration with a standardized hydrochloric acid solution.
This analysis is also a measure of the molar solubility of borax in water at a given temperature; according to stoichiometry, one mole of the tetraborate anion forms for every mole of borax that dissolves. Also, according to the stoichiometry of the dissolution of the borax (see equation \( \ref{boraxdissolution} \)) , the molar concentration of the sodium ion in the saturated solution is twice that of the experimentally determined tetraborate anion concentration:
\[ [\ce{Na+}] = 2 \times [\ce{B4O5(OH)^{2-}_{4}}] \]
The solubility product for borax at a measured temperature is, therefore,
\[ \begin{align} \begin{split} K_{sp} &= [\ce{Na+}]^2[\ce{B4O5(OH)^{2-}_{4}}] \\ K_{sp} &= [2 \times \ce{B4O5(OH)^{2-}_{4}}]^2[\ce{B4O5(OH)^{2-}_{4}}] \\ K_{sp} &= 4 \times [\ce{B4O5(OH)^{2-}_{4}}]^3 \\ K_{sp} &= 4 \times (\text{the molar solubility of borax})^3 \\ K_{sp} &= 4S^3 \end{split} \end{align} \]
To obtain the thermodynamic properties for the dissolution of borax, values for the molar solubility and the solubility product for borax are determined over a range of temperatures. The natural log of \( K_{sp} \) is plotted versus \( \frac{1}{T} \) to determine the standard enthalpy change, \( \Delta H^\circ \), the standard entropy change, \( \Delta S^\circ \), and ultimately to calculate the standard free energy change, \( \Delta G^\circ \), for the dissolution of borax. See equations \( \ref{lnKlinear} \) and \( \ref{deltaGdef} \), respectively.
Gibbs Free Energy:
\[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \]
Gibbs Free Energy and the Equilibrium Constant:
\[ \Delta G^\circ = -RT\ln{K} \]
Van't Hoff Equation:
\[ {\ln{K_{sp}}} = {-\frac{\Delta H^\circ}{R}} {\left(\frac{1}{T}\right)} + {\frac{\Delta S^\circ}{R}} \]