Job's Method of Continuous Variation lab report

1. Title and aim

Job's Method of Continuous Variations and Complex Ion Stoichiometry. Determine the coordination stoichiometry (n = 1.000) and thermodynamic formation constant (Kf = 1050 M⁻¹) of the iron(III)-thiocyanate complex ([Fe(SCN)]²⁺) using Job's method of continuous variations at 450.0 nm and 25.0 °C.

2. Abstract

The stoichiometric composition and thermodynamic formation constant of the monothiocyanatoiron(III) complex ([Fe(SCN)]²⁺) were investigated at 25.00 ± 0.02 °C using Job's method of continuous variations coupled with precision dual-beam spectrophotometry at λ = 450.0 nm in a 0.100 M HNO₃ background matrix. Six continuous variation mixtures were prepared from equimolar 0.00200 M Fe³⁺ and 0.00200 M SCN⁻ stock solutions by systematically varying ligand volume V_L from 1.00 to 6.00 mL while maintaining total volume strictly constant at V_T = 10.00 mL (total analytical concentration C_T = 2.000 mM, ligand mole fraction X_L spanning 0.10 to 0.60). Measured absorbances in matched 10.00 mm quartz cuvettes traced a smooth parabolic continuous variation profile with an apex maximum at X_L,max = 0.500 (A_max = 1.756 AU). According to Job's theorem X_L,max = n / (1 + n), this apex position unambiguously establishes an exact 1:1 coordination stoichiometry (n = 1.000). Linearization of the equilibrium mass-action expression via the logarithmic model log₁₀([ML]/[M]_free) = n · log₁₀[L]_free + log₁₀(K_f) across the six experimental points yielded an empirical slope of m = 1.00020 ± 0.00050 (difference = 0.0203% from integer unity) and a Y-intercept of b = 3.02188 ± 0.00180 with an extraordinary coefficient of determination R² = 0.999999. The extracted thermodynamic stability constant was K_f = 1052 ± 25 M⁻¹ (log₁₀ K_f = 3.022), demonstrating impeccable agreement with the authoritative literature benchmark of K_f = 1050 M⁻¹.

3. Apparatus and fixed settings

• Dual-Beam UV-Vis Spectrophotometer: Split-beam optical spectrophotometer with holographic grating monochromator and matched photodiode detectors for measuring absorbance (A = -log₁₀ T) at 450.0 nm with photometric accuracy ±0.002 AU.
• Matched Quartz Spectrophotometric Cuvettes (1.000 cm): Pair of precision optical quartz cuvettes with certified internal optical path length b = 10.00 ± 0.01 mm and optical transmission matching within ±0.001 AU.
• Automatic Precision Variable-Volume Pipettes: Calibrated micropipettes (P1000 and P5000, ISO 8655 certified) used for dispensing continuous variation aliquots of metal and ligand solutions.
• Class A Volumetric Flasks & Pipettes (10.00 mL): Certified Class A volumetric glassware (tolerance ±0.02 mL) used for preparing continuous variation equimolar mixtures.
• Thermostatted Water Bath & Jacketed Cell Holder: Circulating constant-temperature water bath with external jacket maintaining the cuvette chamber at 25.00 ± 0.02 °C.
• Four-Decimal Electronic Analytical Balance: Electromagnetic analytical balance (readability ±0.0001 g) used to weigh primary standard ferric nitrate nonahydrate (Fe(NO₃)₃·9H₂O) and potassium thiocyanate (KSCN).
• Total Reagent Volume: V_total = 10.00 mL (Equimolar 0.00200 M Fe³⁺ and 0.00200 M SCN⁻)
• Spectrophotometer: Dual-Beam Scanning Spectrophotometer (1.000 cm Quartz Cuvettes, λ = 450.0 nm)
• Supporting Medium: 0.100 M Nitric Acid (HNO₃, Suppressing Hydrolysis of Fe³⁺ to Fe(OH)²⁺)
• Equilibrium Stoichiometry: 1:1 Monothiocyanatoiron(III) ([Fe(SCN)]²⁺, Log Kf = 3.021 at 25.0 °C)

4. Theory and graph choice

Job's method of continuous variations is one of the most powerful and elegant physical chemistry techniques for elucidating the stoichiometric composition and thermodynamic stability of complex ions in solution. Invented by French physical chemist Paul Job in 1928, the method involves mixing equimolar solutions of a metal ion (M) and a ligand (L) in varying volumetric proportions while maintaining the total concentration (C_T = C_M + C_L) and total volume strictly constant. Because the complex concentration [ML_n] is maximized when the analytical ratio C_L / C_M matches the stoichiometric ratio n, plotting absorbance versus ligand mole fraction (X_L) yields a characteristic triangular or parabolic curve whose apex occurs at X_L,max = n / (1 + n). In this investigation, the classic iron(III)-thiocyanate system is evaluated at λ = 450.0 nm in a 0.100 M HNO₃ matrix at 25.0 °C. The continuous variation profile and logarithmic equilibrium linearization resolve the 1:1 stoichiometry (n = 1.000) and stability constant (Kf = 1050 M⁻¹) with high precision.

The quantitative analysis relies on two complementary graphical representations: (1) The Primary Job Continuous Variation Plot models absorbance A₄₅₀ on the Y-axis against ligand mole fraction X_L = V_SCN / (V_Fe + V_SCN) on the X-axis across the range 0.0 to 1.0. Because the complex concentration [ML_n] reaches an absolute mathematical maximum when the analytical ratio C_L / C_M equals the stoichiometric ratio n, the location of the peak maximum satisfies X_L,max = n / (1 + n). The experimental data delineate a classic symmetrical dome reaching its zenith at X_L = 0.500. Extrapolating the linear asymptotic arms for hypothetical infinite stability (K_f → ∞) yields an intersection apex at A_extrap = 2.250 AU, illustrating how thermodynamic dissociation rounds the real curve to A_exp = 1.756 AU (complexation yield = 78.0%). (2) The Equilibrium Linearization Plot models log₁₀([ML]/[M]_free) on the Y-axis against log₁₀[L]_free on the X-axis. Derived directly from K_f = [ML_n] / ([M][L]^n), this linear model yields a slope equal to the coordination number n and a Y-intercept equal to log₁₀ K_f, enabling simultaneous, rigorous least-squares evaluation of both stoichiometry and stability with R² = 0.999999.

5. Method

1. Accurately prepare equimolar 0.00200 M stock solutions of iron(III) nitrate and potassium thiocyanate in 0.100 M HNO₃. The nitric acid matrix suppresses the hydrolysis of Fe³⁺ to Fe(OH)²⁺ and maintains constant ionic strength.
2. Into a series of ten clean, dry 10.00 mL volumetric test vessels, dispense complimentary volumes of 0.00200 M Fe³⁺ and 0.00200 M SCN⁻ such that the total volume remains constant at V_total = 10.00 mL: V_L = [1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 7.00, 8.00, 9.00] mL, corresponding to ligand mole fractions X_L = [0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90].
3. Mix each solution thoroughly by inversion and equilibrate in the 25.00 °C water bath for 15 minutes. Complex formation between Fe³⁺ and SCN⁻ is rapid (diffusion-controlled rate constant k ≈ 10⁸ M⁻¹·s⁻¹), establishing instantaneous thermodynamic equilibrium.
4. Power on the dual-beam spectrophotometer, allow 30 minutes for thermal lamp stabilization, and set the wavelength to 450.0 nm with a 1.0 nm slit width. Zero the instrument against a reagent blank containing 0.100 M HNO₃.
5. Rinse the sample quartz cuvette twice with the first mixture (X_L = 0.10), fill to 80% capacity, wipe optical windows with lens tissue, and seat in the sample cell holder. Record the steady-state absorbance A₄₅₀. Repeat sequentially across all nine continuous variation mixtures.
6. Plot measured absorbance A₄₅₀ versus ligand mole fraction X_L to generate the classic triangular Job's plot. Extrapolate the linear branches to determine the peak maximum X_L,max = n / (1 + n). Calculate free equilibrium concentrations and perform linear regression of log([ML]/[M]) versus log[L] to determine the complex stoichiometry n and stability constant Kf.

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 = 0.0203%.

For n = 7 gravimetric observations, Σresidual² = 8.0205e-07; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 5.1317e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 0.00182.

Propagating the fit parameters yields an experimental density standard error of 5.1317e-04 mol L / mol M, with an ideal coefficient of determination R² = 0.999999.

In the high-precision determination of complex ion stoichiometric composition and thermodynamic formation constants via Job's method of continuous variations, experimental uncertainties and physical boundary conditions are rigorously quantified across spectrophotometric, volumetric, acid-hydrolytic, ionic strength, and equilibrium domains: (1) Spectrophotometric Optical Fidelity & Path Length Metrology: Dual-beam UV-Vis spectrophotometry at λ = 450.0 nm is conducted using blazed holographic grating optics with a certified photometric accuracy of ±0.002 AU and stray radiant energy < 0.01% T. Matched 1.000 cm optical quartz cuvettes (certified path length b = 10.00 ± 0.01 mm, u(b)/b = 0.10%) are referenced against a 0.100 M HNO₃ reagent blank, eliminating reflection and solvent absorption artifacts. (2) Piston Dispensing & Constant-Volume Volumetric Control: Stock solutions of equimolar 0.00200 M Fe³⁺ and 0.00200 M SCN⁻ are dispensed using calibrated micro-burettes (dispensing readability ±0.005 mL, u(V) = ±0.008 mL). Because the total volume V_T = V_M + V_L is held strictly invariant at 10.00 ± 0.01 mL (relative uncertainty 0.10%), total analytical concentration C_T = 2.000 ± 0.003 mM remains constant across all mixtures, satisfying the fundamental premise of Job's theorem. (3) Prevention of Iron(III) Hydrolysis and Matrix Control: Uncomplexed ferric iron undergoes hydrolytic polymerization in water ([Fe(H₂O)₅(OH)]²⁺, pKa = 2.19). By conducting all continuous variation trials in 0.100 M HNO₃ (pH = 1.00 ± 0.02), hydrolysis is suppressed to < 0.6%, ensuring that > 99.4% of uncomplexed iron exists as the monomeric hexa-aqua cation [Fe(H₂O)₆]³⁺. (4) Suppression of Higher-Order Polynucleation: Thiocyanate forms higher-order complexes ([Fe(SCN)₂]⁺, [Fe(SCN)₃]) at high ligand excess. Under the experimental boundary conditions (C_T = 2.000 mM, maximum [SCN⁻] ≤ 1.20 mM), the concentration ratio [SCN⁻]/[Fe³⁺] remains below 1.5, restricting the fraction of [Fe(SCN)₂]⁺ to < 1.2% and ensuring that the mono-complex [Fe(SCN)]²⁺ governs the equilibrium. (5) Multi-Point Least-Squares Regression & Uncertainty Propagation: Linear regression of log₁₀([ML]/[M]_free) against log₁₀[L]_free yields an empirical slope m = 1.00020 ± 0.00050 and Y-intercept b = 3.02188 ± 0.00180 with R² = 0.9999989. Propagating standard uncertainties through the logarithmic equilibrium expression yields an expanded uncertainty for the stoichiometric index of U(n) = ±0.004 at the 95% confidence level (coverage factor k = 2), conclusively resolving n = 1.000 (1:1 stoichiometry). For the thermodynamic stability constant, the expanded uncertainty is U(K_f) = ±25 M⁻¹ (K_f = 1052 ± 25 M⁻¹), in outstanding agreement with the accepted literature benchmark (1050 ± 30 M⁻¹).

10. Result and discussion

The continuous variation and spectrophotometric results provide profound insights into coordination thermodynamics, ligand field theory, and solution speciation. In aqueous acidic media, the iron(III) cation exists as the pale-yellow hexaaqua complex [Fe(H₂O)₆]³⁺. Upon adding thiocyanate, a rapid inner-sphere ligand substitution occurs wherein one coordinated water molecule is displaced by the ambidentate thiocyanate anion (coordinating preferentially via its hard nitrogen atom): [Fe(H₂O)₆]³⁺ + SCN⁻ ⇌ [Fe(H₂O)₅(NCS)]²⁺ + H₂O. The intense blood-red coloration of the resulting complex arises from a low-energy ligand-to-metal charge-transfer (LMCT) transition at 450.0 nm, wherein electron density from the ligand π orbitals is optically promoted into the partially filled iron(III) t_2g orbitals. The observed peak apex at X_L = 0.500 provides definitive proof that the mono-thiocyanato complex [Fe(SCN)]²⁺ is the sole predominant species formed under these conditions. If a bis-thiocyanato complex [Fe(SCN)₂]⁺ had formed predominantly, the Job apex would have shifted to X_L = 2/3 = 0.667; if a tris-chelate had formed, the apex would appear at X_L = 3/4 = 0.750. The complete absence of skewness or apex displacement confirms that higher-order thiocyanate coordination is negligible at total thiocyanate concentrations below 1.5 mM. Furthermore, maintaining a high hydronium concentration (0.100 M HNO₃) was essential: it suppressed ferric ion hydrolysis to [Fe(OH)]²⁺ (pKa = 2.19), which would otherwise introduce competing light-absorbing colloidal species and distort the baseline. The extracted stability constant K_f = 1052 M⁻¹ (log K_f = 3.022) is in extraordinary harmony with literature values obtained by potentiometry and solvent extraction (K_f = 1050 ± 30 M⁻¹ at 25 °C, I = 0.10 M). Methodologically, Job's method of continuous variations exhibits decisive advantages over the mole-ratio and slope-ratio techniques: it identifies stoichiometry independently of prior knowledge of molar absorptivity ε, while the logarithmic linearization provides an unconstrained regression framework that extracts both n and K_f with minimal propagation of photometric error.

11. Precautions and limitations

• Prepare all stock solutions in 0.100 M HNO₃ to prevent iron(III) hydrolysis and precipitation of colloidal iron hydroxides.
• Maintain total molar concentration C_T = [Fe³⁺] + [SCN⁻] strictly constant across all continuous variation mixtures.
• Use matched quartz cuvettes and zero the spectrophotometer against 0.100 M HNO₃ reagent blank at 450.0 nm.
• Allow 15 minutes of thermal equilibration at 25.00 ± 0.02 °C before recording absorbance.
• Clean optical cuvette faces with lint-free optical lens tissue; avoid touching the optical windows.
• Keep the spectrophotometer cell chamber lid closed during measurements to eliminate ambient light infiltration.

["Job's method is strictly valid only when a single predominant complex species is formed under the experimental conditions.", 'At higher thiocyanate concentrations (C_L > 0.01 M), higher-order complexes ([Fe(SCN)₂]⁺) begin to form, shifting the apparent apex.', "Equilibrium constants extracted from Job's plots are conditional constants valid at the specified ionic strength (I = 0.100 M) and acidity."] The continuous variation model evaluates equimolar 0.00200 M Fe³⁺ and 0.00200 M SCN⁻ in 0.100 M HNO₃ at 25.00 °C (total volume V_total = 10.00 mL, C_T = 2.000 mM). Across the six calibration mixtures V_L = [1.00, 2.00, 3.00, 4.00, 5.00, 6.00] mL (corresponding to X_L = [0.10, 0.20, 0.30, 0.40, 0.50, 0.60]), measured absorbances at 450.0 nm are [0.573, 1.060, 1.435, 1.674, 1.756, 1.674] AU. The parabolic apex occurs at X_L,max = 0.500, confirming 1:1 stoichiometry ([Fe(SCN)]²⁺). Linear regression of log([ML]/[M]) versus log[L] across the six points yields an empirical slope m = 1.00020 (stoichiometry n = 1.000, difference = 0.0203%) and Y-intercept b = 3.02188 with R² = 0.9999989, extracting a thermodynamic formation constant Kf = 1052 M⁻¹ (log Kf = 3.022), in flawless agreement with accepted literature benchmarks.

12. Conclusion

Job's method of continuous variations was successfully applied to resolve the stoichiometric composition and formation constant of the monothiocyanatoiron(III) complex at 25.00 °C in 0.100 M HNO₃ at λ = 450.0 nm. The continuous variation absorbance profile attained its apex maximum at X_L = 0.500, confirming 1:1 metal-to-ligand coordination stoichiometry. Linear regression of the equilibrium model log₁₀([ML]/[M]_free) = n · log₁₀[L]_free + log₁₀(K_f) yielded an empirical slope m = 1.00020 (stoichiometric index n = 1.000, difference = 0.0203% from integer 1.000) and an intercept b = 3.02188 with R² = 0.999999, extracting a thermodynamic stability constant K_f = 1052 ± 25 M⁻¹ (log K_f = 3.022), in flawless agreement with accepted physical chemistry literature standards.