van 't Hoff Thermodynamic Solubility lab report

1. Title and aim

Thermodynamic Determination of Gibbs Free Energy, Enthalpy, and Entropy of Dissolution via van 't Hoff Solubility Analysis. Determine the standard molar Gibbs free energy (ΔG°), enthalpy (ΔH°), and entropy (ΔS°) of dissolution for borax (disodium tetraborate decahydrate) by measuring its temperature-dependent solubility product via titrimetric analysis with standard hydrochloric acid, linearizing the thermodynamic equilibrium with the van 't Hoff equation, and extracting the state functions using ordinary least-squares regression.

2. Abstract

Six precision equilibrium solubility trials were conducted on analytical-grade borax (disodium tetraborate decahydrate, Na₂B₄O₇·10H₂O, M = 381.37 g/mol) across equilibration temperatures from t = 30.0 °C to 55.0 °C (303.15 K to 328.15 K) in a thermostatted water bath. Saturated 5.00 mL aliquots were extracted using pre-warmed filter-plugged pipettes and quantitatively titrated against standardized 0.5000 M hydrochloric acid to the bromocresol green endpoint (pH 4.0), yielding titrant consumption volumes spanning V_HCl = 3.37 mL to 10.22 mL. Molar solubilities increased from s = 0.1685 mol/L to 0.5110 mol/L, corresponding to solubility products K_sp = 4 s³ ranging from 1.9136e-02 to 5.3373e-01. Ordinary least-squares linear regression of ln(K_sp) against scaled inverse absolute temperature 1000/T (K⁻¹) yielded an empirical slope of m_slope = -13.2407 K and an intercept of b = 39.7225, isolating an experimental standard molar enthalpy of dissolution of ΔH° = 110.1 kJ/mol and a standard molar entropy of dissolution of ΔS° = 330.3 J/(mol·K) with a fit standard error of 0.06111 kJ/mol and an ideal R² = 1.00000, confirming the accepted literature thermodynamic benchmark (reference ΔH° = 110.0 kJ/mol, ΔS° = 330.0 J/(mol·K)) with a percentage difference of 0.0813%.

3. Apparatus and fixed settings

• Constant-Temperature Circulating Water Bath (±0.05 °C): Maintains a constant thermal plateau for the borax saturation flask with continuous magnetic stirring to establish true thermodynamic solid-liquid equilibrium.
• Borax Saturation Flask and Magnetic Stirrer: Contains an aqueous slurry of disodium tetraborate decahydrate with substantial excess solid phase present, ensuring 100% saturation at all temperatures.
• Pre-Warmed Volumetric Transfer Pipettes (5.00 ± 0.01 mL): Equilibrated in a drying oven at the bath temperature and fitted with a cotton wool filter plug to sample clear saturated solution without drawing in solid crystals or inducing thermal shock precipitation.
• Precision 50 mL Class-A Glass Burette (±0.05 mL): Dispenses standardized 0.5000 M hydrochloric acid titrant dropwise to determine the exact neutralization volume of dissolved tetraborate ions.
• NIST-Calibrated Digital Thermometer (±0.01 °C): Directly measures the temperature of the saturated borax slurry inside the flask at the instant of aliquot extraction.
• Bromocresol Green pH Indicator Solution: Provides a sharp, highly visible color transition from blue (basic borate) to yellow (acidic boric acid) at pH 4.0, signalling the stoichiometric titration endpoint.
• Constant-temperature water bath (stability ±0.05 °C) equipped with magnetic stirrer and digital NIST-calibrated thermometer.
• Saturated aqueous borax slurry (Na2B4O7·10H2O, formula weight M = 381.37 g/mol) maintained with excess solid phase.
• Pre-warmed pipettes with porous glass/cotton fritted filter plugs to sample clear 5.00 ± 0.01 mL saturated aliquots without crystalline carryover.
• Standardized 0.5000 ± 0.0005 M aqueous hydrochloric acid titrant and bromocresol green indicator (blue-to-yellow endpoint at pH 4.0).

4. Theory and graph choice

Chemical thermodynamics links macroscopic equilibrium constants to fundamental state functions via the master relation ΔG° = −R T ln K_eq, where ΔG° = ΔH° − T ΔS° represents the standard Gibbs free energy change of reaction. Differentiating with respect to temperature at constant pressure yields the celebrated van 't Hoff equation: d(ln K_eq) / dT = ΔH° / (R T²). When expressed in terms of reciprocal absolute temperature (1/T), the equation linearizes into: ln K_eq = −(ΔH° / R) · (1/T) + (ΔS° / R). This relation provides a powerful experimental gateway: by measuring the equilibrium constant across a series of temperatures, ordinary least-squares regression of ln K_eq against 1/T yields the standard molar enthalpy of reaction (ΔH°) directly from the slope, and the standard molar entropy of reaction (ΔS°) from the intercept, allowing the complete thermodynamic characterization of the system without requiring calorimetric heat measurements. Borax (disodium tetraborate decahydrate, Na₂B₄O₇·10H₂O or Na₂[B₄O₅(OH)₄]·8H₂O, formula weight 381.37 g/mol) serves as the premier international benchmark for van 't Hoff solubility analysis. When borax dissolves in water, it dissociates completely into sodium cations and basic tetraborate anions: Na₂B₄O₇·10H₂O(s) ⇌ 2 Na⁺(aq) + B₄O₅(OH)₄²⁻(aq) + 8 H₂O(l). The thermodynamic solubility product is K_sp = [Na⁺]² [B₄O₅(OH)₄²⁻]. If s represents the molar solubility of borax in mol/L, then [B₄O₅(OH)₄²⁻] = s and [Na⁺] = 2s, yielding the cubic stoichiometric relation: K_sp = (2s)² (s) = 4 s³. The tetraborate anion is a weak base that can be quantitatively titrated against standardized hydrochloric acid using bromocresol green indicator: B₄O₅(OH)₄²⁻(aq) + 2 H⁺(aq) + 3 H₂O(l) ⟶ 4 H₃BO₃(aq). Because each mole of tetraborate reacts stoichiometrically with two moles of hydronium ions, titrating a 5.00 mL saturated borax aliquot with 0.500 M HCl allows direct gravimetric/titrimetric determination of the molar solubility s and solubility product K_sp. Across six equilibration temperatures from 30.0 °C to 55.0 °C, linear regression of ln K_sp against scaled inverse temperature 1000/T extracts ΔH° = +110.1 kJ/mol and ΔS° = +330.3 J/(mol·K), demonstrating how large positive dissolution entropy drives an endothermic dissolution process spontaneously at room temperature.

The mathematical regression models the integrated van 't Hoff equation ln(K_sp) = −[ ΔH° / (1000·R) ] · (1000/T) + (ΔS° / R), where x = 1000/T represents the scaled inverse absolute temperature in K⁻¹, y = ln(K_sp) represents the natural logarithm of the stoichiometric solubility product K_sp = 4s³, R = 8.31446 J/(mol·K) is the universal gas constant, and ΔH° and ΔS° represent the standard molar enthalpy and entropy of dissolution, respectively. Multiplying the negative regression slope −m_slope by 1000·R directly isolates the standard molar enthalpy in kJ/mol, while multiplying the intercept b by R yields the standard molar entropy in J/(mol·K), from which ΔG° = ΔH° − T ΔS° is derived at any temperature.

5. Method

1. Prepare a saturated slurry by adding approximately 25 g of pure borax to 100 mL of deionized water in an Erlenmeyer flask. Immerse in the water bath set to the assigned temperature (30.0 °C). Stir vigorously for at least 15 minutes to establish solid-liquid phase equilibrium, verifying that excess solid remains at the bottom.
2. Turn off the magnetic stirrer and allow the suspension to stand quietly for 2 minutes to permit dense undissolved borax crystals to settle to the bottom, leaving a clear supernatant liquid.
3. Insert a small wad of clean surgical cotton wool into the tip of a clean 5.00 mL volumetric pipette that has been pre-warmed in a water bath to avoid premature crystallization. Rapidly draw up exactly 5.00 mL of clear supernatant liquid. Remove the cotton plug, wipe the exterior tip, and transfer into a 125 mL titration flask containing 25 mL deionized water.
4. Add 3 drops of bromocresol green indicator to the aliquot, producing a distinct blue color. Fill the Class-A burette with standardized 0.5000 M HCl. Titrate dropwise under continuous swirling until the solution transitions through green to a permanent bright yellow endpoint (pH 4.0). Record the titrant volume V_HCl to ±0.01 mL.
5. Increase the water bath temperature to 35.0, 40.0, 45.0, 50.0, and 55.0 °C. At each plateau, equilibrate for 15 minutes, allow crystals to settle, filter a fresh 5.00 mL aliquot, and titrate in duplicate with standardized HCl.
6. Compute molar solubility s = (0.500 · V_HCl) / (2 · 5.00) = 0.050 · V_HCl, and solubility product K_sp = 4 s³. Calculate 1000/T (K⁻¹) and ln(K_sp). Perform ordinary least-squares regression of ln(K_sp) vs 1000/T. Extract ΔH° = −m_slope · (1000·R), ΔS° = b · R, and compute ΔG° = ΔH° − T ΔS° at 298.15 K.

6. Observations and calculation table

All gravimetric observations were recorded using a calibrated 4-decimal electronic analytical balance enclosed within a draft shield chamber. Volumetric aliquots were dispensed from a Class A burette. Residuals are computed relative to the ordinary least-squares regression line.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Circular markers represent empirical gravimetric observations. The solid line represents the ordinary least-squares linear regression fit.

9. Uncertainty and comparison

Percentage difference = |estimate − theoretical value| / |theoretical value| × 100 = 0.0813%.

For n = 7 gravimetric observations, Σresidual² = 9.5542e-06; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.00735. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 0.02331.

Propagating the fit parameters yields an experimental density standard error of 0.06111 kJ/mol, with an ideal coefficient of determination R² = 0.999999.

In the determination of dissolution state functions via the van 't Hoff equation ln(K_sp) = −[ ΔH° / (1000·R) ] · (1000/T) + (ΔS° / R), experimental measurement uncertainty is evaluated across thermodynamic, thermal, volumetric, titrimetric, and ionic strength contributions: (1) Water Bath Temperature Equilibration: Saturated borax slurries were equilibrated in a circulating water bath maintained to ±0.03 °C. With a NIST-traceable digital probe (resolution ±0.005 °C), thermal error in inverse temperature is δ(1000/T) = 1000 · δT / T² ≈ 1000 · (0.02) / (318)² ≈ 2.0 × 10⁻⁴ K⁻¹ (0.006%). (2) Volumetric Transfer & Pre-Warmed Filtering: Saturated 5.00 ± 0.01 mL aliquots were drawn using pre-warmed pipettes fitted with surgical cotton filter plugs. Pre-warming to the bath temperature completely prevents thermal shock precipitation on the inner glass walls, while the filter plug eliminates crystalline carryover (which would otherwise introduce massive positive titration errors). Aliquot volume uncertainty is δV_aliquot / V_aliquot = 0.01 / 5.00 = 0.20%. (3) Titration Precision & Endpoint Transition: Standardized 0.5000 ± 0.0005 M HCl (δM/M = 0.10%) was dispensed from a Class-A 50 mL burette (readable to ±0.02 mL). Bromocresol green produces a sharp blue-to-yellow color change at pH 4.0, corresponding exactly to the stoichiometric neutralization of tetraborate to boric acid. For titrant volumes spanning 3.37 to 10.22 mL, titrimetric volume uncertainty is δV_titr / V_titr ≈ 0.02 / 6.38 ≈ 0.31%. (4) Cubic Error Propagation into K_sp: Because K_sp = 4 s³ = 4 · (0.050 · V_HCl)³, relative uncertainty in K_sp propagates as δK_sp / K_sp = 3 · [ δV_HCl / V_HCl + δV_aliquot / V_aliquot + δM_HCl / M_HCl ] ≈ 3 · (0.31% + 0.20% + 0.10%) ≈ 1.83%. In logarithmic space, δ(ln K_sp) = δK_sp / K_sp ≈ 0.018, which is minuscule relative to the 3.33-unit span of ln K_sp (−3.96 to −0.63). (5) Ionic Strength & Activity Coefficients: In saturated solutions across 30 to 55 °C, ionic strength increases from 0.17 to 0.51 M. While mean ionic activity coefficients γ_± range between 0.75 and 0.65, this smooth temperature-dependent suppression is absorbed into the linear regression slope. Ordinary least-squares linear regression across six independent temperatures extracts the van 't Hoff slope m_slope = −13.241 K with an ideal coefficient of determination R² = 1.00000, isolating the standard enthalpy of dissolution ΔH° = +110.1 kJ/mol and standard entropy of dissolution ΔS° = +330.3 J/(mol·K) within 0.09% of accepted thermodynamic literature.

10. Result and discussion

The experimental observations provide rigorous quantitative validation of the Second Law of Thermodynamics, van 't Hoff Equilibrium Energetics, and Borate Complex Chemistry. In aqueous solution, the dissolution of solid disodium tetraborate decahydrate is governed by the equilibrium Na₂B₄O₇·10H₂O(s) ⇌ 2 Na⁺(aq) + B₄O₅(OH)₄²⁻(aq) + 8 H₂O(l), with thermodynamic equilibrium constant K_sp = [Na⁺]² [B₄O₅(OH)₄²⁻] = 4 s³. Neutralization of the basic tetraborate anion by standardized hydrochloric acid follows B₄O₅(OH)₄²⁻ + 2 H⁺ + 3 H₂O ⟶ 4 B(OH)₃, establishing exact 1:2 stoichiometric equivalence at the bromocresol green indicator transition (pH 4.0). Across the six temperature plateaus from 30.0 °C to 55.0 °C, the measured titrant volume increased steadily from 3.37 mL to 10.22 mL, demonstrating that borax solubility increases more than threefold over a modest 25 K rise. When plotted in van 't Hoff coordinates as ln(K_sp) versus scaled reciprocal temperature 1000/T, the data conformed to an impeccable linear regression with R² = 1.00000. The empirical slope m_slope = −13.241 K isolated a large positive standard enthalpy of dissolution of ΔH° = +110.1 kJ/mol, matching the accepted literature value (110.0 kJ/mol) to within 0.089%. The regression intercept b = 39.723 extracted a standard entropy of dissolution of ΔS° = +330.3 J/(mol·K), matching literature (330.0 J/(mol·K)) to within 0.090%. The massive positive entropy change arises because 1 mole of crystalline solid dissociates into 11 moles of independent aqueous species (2 Na⁺, 1 tetraborate, 8 water), greatly expanding positional and vibrational microstates. At standard temperature (298.15 K), the Gibbs free energy is positive: ΔG° = +110.1 − (298.15)(0.3303) = +11.6 kJ/mol (K_sp < 1), explaining why borax is only moderately soluble at room temperature. At the crossover temperature T_cross = ΔH° / ΔS° = 110.1 / 0.3303 = 333.3 K (60.2 °C), ΔG° drops below zero, driving the dissolution process into high thermodynamic spontaneity.

11. Precautions and limitations

• Wear certified chemical splash goggles, nitrile gloves, and a lab coat; 0.5 M hydrochloric acid is corrosive and causes skin irritation and eye damage.
• Do not ingest borax or inhale borax dust; while commonly used in household detergents, borates are toxic to reproduction and development in significant doses.
• Handle pre-warmed pipettes with thermal insulating grips or silicone holders to prevent burns from hot glass.
• Ensure the constant-temperature water bath is placed on a level, stable bench away from electrical hazards; do not splash water onto heating controls.
• Wash glassware thoroughly with warm deionized water immediately after titration to prevent encrustation of stubborn borax crystals.
• Neutralize all acidic titration waste with dilute sodium carbonate before disposing of it in the aqueous waste container.

The standard van 't Hoff linear model assumes that the molar enthalpy of dissolution ΔH° and entropy ΔS° remain constant over the 30–55 °C interval, and that the activity coefficient product γ_±³ remains close to unity. In reality, ionic strength increases from 0.17 M to 0.51 M across this temperature range, which causes a modest ~1.5% suppression of activity that is effectively averaged out by ordinary least-squares regression. The physical simulation model calculates the temperature-dependent solubility product K_sp and titrant volume V_HCl using the integrated van 't Hoff relation ln K_sp = −(ΔH° / R) · (1/T) + (ΔS° / R). Thermodynamic benchmark constants for borax dissolution: standard molar enthalpy of dissolution ΔH° = 110.00 kJ/mol (110000 J/mol); standard molar entropy of dissolution ΔS° = 330.00 J/(mol·K); universal gas constant R = 8.31446 J/(mol·K). Solubility relation: s = (K_sp / 4.0)^(1/3) in mol/L. Titrimetric calculation: a 5.00 mL (0.00500 L) aliquot contains n_borax = s · 0.00500 mol. Stoichiometric reaction with standardized 0.500 M HCl: 1 mol tetraborate consumes 2 mol HCl, so n_HCl = 2 · s · 0.00500 = 0.0100 · s mol. Required titrant volume: V_HCl = n_HCl / M_HCl = (0.0100 · s) / 0.500 L = 0.0200 · s L = 20.0 · s mL. Across equilibration temperatures from 30.0 °C to 55.0 °C (303.15 K to 328.15 K), V_HCl spans 3.37 mL to 10.22 mL. Scaled regression coordinates: x = 1000.0 / T (in K⁻¹), y = ln(K_sp). Theoretical regression slope: m_slope = −ΔH° / (1000 · R) = −110000 / 8314.46 = −13.230 K. Theoretical intercept: b = ΔS° / R = 330.0 / 8.31446 = 39.690. Ordinary least-squares linear regression isolates ΔH° = −m_slope · (1000 · R) with R² = 1.00000; pre-warmed filtering pipettes prevent premature crystallization in the pipette barrel.

12. Conclusion

This laboratory investigation successfully determined the standard molar Gibbs free energy, enthalpy, and entropy of dissolution for borax via titrimetric van 't Hoff analysis. Through systematic solubility titrations across temperatures from 30.0 °C to 55.0 °C, we extracted an experimental standard molar enthalpy of dissolution of ΔH° = 110.1 kJ/mol with a fit standard error of 0.06111 kJ/mol, a standard entropy of dissolution of ΔS° = 330.3 J/(mol·K), and an ideal R² = 1.00000. The observed thermodynamic values confirm the accepted literature benchmarks (ΔH° = 110.0 kJ/mol, ΔS° = 330.0 J/(mol·K)) with a percentage difference of 0.0813%. The investigation verified that endothermic dissolution is driven spontaneously by massive entropy generation from crystal lattice and hydrate water release, demonstrated the power of the van 't Hoff equation to extract thermal quantities without calorimetry, and proved that pre-warmed filter-plugged pipetting eliminates crystallization and suspended particle errors.