7: Thermodynamics of Borax
- Page ID
- 516596
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 (\(8.314\text{ J/mol}\cdot\text{K}\)), but \(\Delta H^\circ\) is usually reported in \(\text{kJ/mol}\). Don't forget the factor of 1000.
- Stoichiometry: The tetraborate anion (\(\mathrm{B_4O_5(OH)_4^{2-}}\)) accepts 2 protons from \(\mathrm{HCl}\). Therefore: \[ \text{moles borax} = 0.5 \times \text{moles HCl} \]
2. Required Technical Skills
- Measuring temperature accurately without starting automated data logging prematurely.
- Titration data collection using a 50-mL buret.
- Data Skill: Linearizing data in Excel or Google Sheets (converting \(T \rightarrow 1/T\) and \(K_{sp} \rightarrow \ln K_{sp}\)).
3. Critical Safety
- Thermal Hazard: You are heating solutions up to \(60\,^\circ\text{C}\). Glassware will be hot. Use beaker tongs or insulated gloves.
- Chemical Hazard: \(0.20\text{ M }\mathrm{HCl}\) is an skin and eye irritant. Wear safety goggles and lab coat.
- To determine the thermodynamic properties (\(\Delta G^\circ\), \(\Delta H^\circ\), and \(\Delta S^\circ\)) of borax dissolution by measuring its molar solubility at various temperatures.
- To calculate the solubility product constant (\(K_{sp}\)) and \(\ln K_{sp}\) of borax at different temperatures using acid-base titration.
- To apply the van 't Hoff 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, chemical equilibrium analysis, and quantitative titration techniques.
INTRODUCTION
The standard free energy change of a chemical process is related to its equilibrium constant according to the equation:
\[ \Delta G^\circ = -RT\ln K \]
where \(R\) is the ideal gas constant (\(8.314\text{ J/mol}\cdot\text{K}\)) and \(T\) is the absolute temperature in kelvins. For a sparingly soluble salt in aqueous solution, the equilibrium constant corresponds to the solubility product constant, \(K_{sp}\).
For example, consider the dissolution equilibrium of silver chromate:
\[ \mathrm{Ag_2CrO_4(s) \rightleftharpoons 2\,Ag^+(aq) + CrO_4^{2-}(aq)} \]
The mass action expression set equal to the solubility product constant is:
\[ K_{sp} = \mathrm{[Ag^+]^2[CrO_4^{2-}]} \]
and the standard free energy change for this equilibrium is:
\[ \Delta G^\circ = -RT\ln K_{sp} = -RT\ln\left(\mathrm{[Ag^+]^2[CrO_4^{2-}]}\right) \]
The free energy change is also defined by the fundamental thermodynamic equation:
\[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \label{deltaGdef} \]
Setting the two free energy expressions equal yields:
\[ -RT\ln K_{sp} = \Delta H^\circ - T \Delta S^\circ \]
Dividing both sides by \(-RT\) and rearranging gives the van 't Hoff equation in standard linear form (\(y = mx + b\)):
\[ \ln K_{sp} = \left(-\frac{\Delta H^\circ}{R}\right)\left(\frac{1}{T}\right) + \frac{\Delta S^\circ}{R} \label{lnKlinear} \]
A plot of \(\ln K_{sp}\) on the y-axis versus reciprocal absolute temperature (\(\frac{1}{T}\) in \(\text{K}^{-1}\)) on the x-axis yields a straight line with a slope \(m = -\frac{\Delta H^\circ}{R}\) and a y-intercept \(b = \frac{\Delta S^\circ}{R}\). From these graphical parameters, \(\Delta H^\circ\) and \(\Delta S^\circ\) for dissolution can be readily determined.
The Borax System
Borax is commonly designated as sodium tetraborate decahydrate, \(\mathrm{Na_2B_4O_7 \cdot 10H_2O}\). However, based on its crystal structure and chemical reactivity, its true chemical formula is \(\mathrm{Na_2B_4O_5(OH)_4 \cdot 8H_2O}\). The structure of the tetraborate anion, \(\mathrm{B_4O_5(OH)_4^{2-}}\), is shown in Figure \(\PageIndex{1}\).
Borax dissolves and dissociates in water according to the reaction equation:
\[ \mathrm{Na_2B_4O_5(OH)_4 \cdot 8H_2O(s) \rightleftharpoons 2\,Na^+(aq) + B_4O_5(OH)_4^{2-}(aq) + 8\,H_2O(l)} \label{boraxdissolution} \]
The solubility product expression for borax is:
\[ K_{sp} = \mathrm{[Na^+]^2[B_4O_5(OH)_4^{2-}]} \]
The tetraborate anion is the conjugate base of the weak acid boric acid and accepts two protons when titrated with a strong acid:
\[ \mathrm{B_4O_5(OH)_4^{2-}(aq) + 2\,H^+(aq) + 3\,H_2O(l) \rightleftharpoons 4\,H_3BO_3(aq)} \]
Because one mole of tetraborate anion is produced for every mole of borax that dissolves, the concentration of \(\mathrm{B_4O_5(OH)_4^{2-}}\) measured by titration represents the molar solubility (\(S\)) of borax. Furthermore, stoichiometric stoichiometry dictates that \(\mathrm{[Na^+]} = 2 \times \mathrm{[B_4O_5(OH)_4^{2-}]} = 2S\). Substituting into the \(K_{sp}\) expression yields:
\[ K_{sp} = (2S)^2(S) = 4S^3 \]
The relationship between \(K_{sp}\) and temperature is exponential. Plotting \(\ln K_{sp}\) versus \(\frac{1}{T}\) linearizes the data into a straight line.
- X-Axis Orientation: Because you plot \(\frac{1}{T}\), higher temperatures (larger \(T\)) correspond to smaller \(\frac{1}{T}\) values and appear on the left side of the graph.
- Units: Convert all temperatures to Kelvin (\(\text{K} = ^\circ\text{C} + 273.15\)) before taking reciprocals.
Gibbs Free Energy:
\[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \]
Free Energy and Equilibrium Constant:
\[ \Delta G^\circ = -RT\ln K \]
Van 't Hoff Equation:
\[ \ln K_{sp} = \left(-\frac{\Delta H^\circ}{R}\right)\left(\frac{1}{T}\right) + \frac{\Delta S^\circ}{R} \]
- 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.