7: Thermodynamics of Borax
- Page ID
- 516596
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Before taking your Lab Entry Ticket, ensure you have mastered the following.
1. Prerequisite Math & Theory
- The van 't Hoff Equation: You will graph your data to find enthalpy and entropy. \[ \ln K = -\frac{\Delta H^\circ}{R} \left(\frac{1}{T}\right) + \frac{\Delta S^\circ}{R} \]
- Slope: \(m = -\Delta H^\circ / R\)
- Intercept: \(b = \Delta S^\circ / R\)
- The Trap: \(R\) is in Joules (8.314 J/mol K), but \(\Delta H\) is usually reported in kJ. Don't forget the factor of 1000.
- Stoichiometry: The tetraborate anion (\(\ce{B4O5(OH)4^{2-}}\)) accepts 2 protons from \(\ce{HCl}\). Therefore: \[ \text{moles Borax} = 0.5 \times \text{moles HCl} \]
2. Required Technical Skills (See Appendix 13.1)
- Section A: Measuring Temperature (without hitting "Collect").
- Section D (or C): Titration Data Collection (Manual or Drop Counter).
- Data Skill: Linearizing data in Excel (converting \(T \rightarrow 1/T\) and \(K_{sp} \rightarrow \ln K\)).
3. Critical Safety
- Thermal Hazard: You are heating solutions to 60°C. Glassware will be hot.
- Chemicals: 0.20 M \(\ce{HCl}\) is an irritant.
- 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
\[ \ce{Ag2CrO4} (s) \rightleftharpoons 2 \ce{Ag+} (aq) + \ce{CrO4^{2-}} (aq) \]
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:
\[ K_{sp} = [\ce{Ag+}]^2[\ce{CrO4^{2-}}] \]
and the free energy change for the equilibrium is
\[ \Delta G^\circ = -RT\ln{K_{sp}} = -RT\ln{[\ce{Ag+}]^2[\ce{CrO4^{2-}}]} \]
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 \( \ce{Na2B4O7 . 10H2O} \). However, according to its chemical behavior, a more defining formula for borax is \( \ce{Na2B4O5(OH)4 . 8H2O} \). The anion, tetraborate (\( \ce{B4O7^{2-}} \)), has the structure shown in the figure.
Borax dissolves and dissociates in water according to the equation:
\[ \ce{Na2B4O5(OH)4 . 8H2O} (s) \rightleftharpoons \ce{2Na+} (aq) + \ce{B4O5(OH)^{2-}_{4}} (aq) + \ce{8H2O} (l) \label{boraxdissolution} \]
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:
\[ \ce{B4O5(OH)^{2-}_{4}} (aq) + \ce{2H+} (aq) + \ce{3H2O} (l) \rightleftharpoons \ce{4H3BO3} (aq) \]
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.
In science, straight lines are easier to analyze than curves. The relationship between \(K_{sp}\) and Temperature is exponential (a curve).
- The Trick: By plotting the natural log (\(\ln K\)) versus the inverse temperature (\(1/T\)), we mathematically "flatten" the curve into a straight line.
- The Trap: Because we plot \(1/T\), the x-axis is "reversed." High temperatures (large \(T\)) appear on the left side of the graph (small \(1/T\)). Keep this in mind when interpreting your slope.
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:
\[ \ln{K_{sp}} = -\frac{\Delta H^\circ}{R}\left(\frac{1}{T}\right) + \frac{\Delta S^\circ}{R} \]
- 7.1: Thermodynamics of Borax - Experiment
- This page details safety precautions for handling hydrochloric acid and provides a materials list for its standardization with anhydrous sodium carbonate. It outlines a three-part procedure involving titration, preparation of borax solutions sensitive to temperature, and analysis of the solutions. Additionally, it highlights the importance of identifying outliers in data analysis and the correct disposal methods for chemical waste.
- 7.2: Thermodynamics of Borax - Pre-lab
- This page covers the standardization of hydrochloric acid using sodium carbonate, detailing the required mass for neutralization. It highlights the importance of solid borax in water baths and explains the endothermic dissolution of borax with reference to the Van't Hoff equation. Readers are encouraged to predict graph slopes related to enthalpy changes and reflect on possible calculation errors if slopes show an unexpected sign.
- 7.3: Thermodynamics of Borax - Data and Report
- This page covers experimental data tables for standardizing hydrochloric acid and analyzing borax test solutions. It instructs on calculating molarity and solubility product (Ksp) of borax at different temperatures using Excel for logarithmic computations and plotting. It highlights common pitfalls in data analysis, the significance of the coefficient of determination (R²) for data fitting, and the method to derive standard enthalpy of solution from slope calculations.