Introduction

The stoichiometry and stability constants of metal complexes are important parameters for physical, chemical, biochemical, and environmental studies. Different techniques were used for the determination of these parameters including UV–VIS spectrophotometry (Carmody 1964; Chattopadhyaya and Singh 1974; Dömötör et al. 2018; Gamov et al. 2019; Kluska et al. 2018; Kocyla et al. 2017; Ravichandran et al. 2014), spectrofluorimetry (Chen et al. 2015; Cuprys et al. 2018), potentiometry (Dömötör et al. 2018; Friend and Wall 2019; Jakusch 2018; JANRAO et al. 2014; Kluska et al. 2018; Kocyla et al. 2017; S. Al–Farhan 2018), voltammetry (Geiger et al. 1991; Kamyabi et al. 2016), conductometry (Kazemi et al. 2016; Al–Farhan 2018), gel chromatography (Yoza 1977), ion exchange chromatography (Wacker and Seubert 2014), liquid–liquid extraction (Omoto and Wall 2017), electrophoresis (Ansorge et al. 2018; Holm et al. 2013; TEWARI’ et al. 1993), and isothermal calorimetry (Holm et al. 2013; Kluska et al. 2018). These techniques often require sophisticated and relatively expensive instruments, where the very popular UV–VIS spectrophotometry is most commonly used. Stoichiometry and stability constants can be determined spectrophotometrically by several methods including the Job’s continuous variation method (Carmody 1964), the molar ratio method, and the Bjerrum method. (JANRAO et al. 2014).

Recently, digital imaging devices, e.g., digital cameras (Mohamed and Shalaby 2019; Mohamed et al. 2018), smartphone cameras (Dutta 2019; Mohamed and Shalaby 2019; Mohamed et al. 2018), and scanners (Mohamed and Shalaby 2019; Mohamed et al. 2018; Shokrollahi et al. 2015), served as sensors for colorimetric analysis. Captured images were analyzed to yield the Red, Green, and Blue (RGB) channel intensities (de Morais et al. 2016; Huangfu et al. 2019; Mohamed and Shalaby 2019; Mohamed et al. 2018). Recently, Morais et al. (de Morais et al. 2016) applied the red color channel intensity as a signaling tool for the determination of the Iron-Phenanthroline binding constant. However, the channel intensities of the non-uniform RGB color space were characterized by poor calibration linearity (Mohamed and Shalaby 2019; Mohamed et al. 2018) and therefore were mathematically converted into the Yxyand ΔELUV color space signals that gave excellently linear calibration responses that favorably competed with signals of sophisticated spectrophotometers (Mohamed and Shalaby 2019; Mohamed et al. 2018).

Here we describe a low-cost, simple, sensitive, and accurate method for the determination of the stoichiometry and stability constants of metal–ligand complexes using digital image-based analysis (DIBA). The stoichiometric ratios and stability constants of the Iron complexes with 1,10-Phenanthroline (Phen), 2,4,6-Tris(2-pyridyl)-s-triazine (TPTZ), and Salicylate (SAL) were determined using the molar ratio and the continuous variation methods. A conventional digital camera was used to capture images of cuvettes containing reacting solutions of the iron complexes with various Fe/L ratios or mole fraction. Captured images were analyzed to obtain the non-uniform RGB color space parameters that were converted to the corresponding Yxyand ΔELUV color space analytical signals.(Mohamed and Shalaby 2019; Mohamed et al. 2018) The Yxy color absorbance and ΔELUV color difference values were plotted against the molar ratio or the mole fraction and the stoichiometric ratio and stability constant values were calculated.

Theoretical

The formation of mononuclear complexes can be represented by the following chemical equilibrium M + nL ↔ MLn.

The stability constant βn is given by Eq. (1). In most cases one of the reaction constituents takes part in other equilibrium reaction, e.g., the ligand can be protonated or competed by other metal ions present, so the real concentration of the free ligand may differ widely in solutions with the same total ligand concentration. In this case, the apparent stability constant is called a conditional stability constant “\(\beta^{\prime}_{n}\)” and the overall stability constant can be calculated using Eq. (2) (Chattopadhyaya and Singh 1974; Inczédy 1976). Where \(\alpha_{L\left( H \right)}\) is the side reaction function of ligand protonation and can be calculated using Eq. (3) (Chattopadhyaya and Singh 1974).

$$\beta^{ }_{n} = \frac{{\left[ {ML_{n} } \right]}}{{\left[ M \right]\left[ L \right]^{n} }}$$
(1)
$$\beta_{n} = \beta^{\prime}_{n} (\alpha_{L\left( H \right)} )^{n}$$
(2)
$$\alpha_{L\left( H \right)} = 1 + \frac{{\left[ {H^{ + } } \right]}}{{K_{1} }} + \frac{{\left[ {H^{ + } } \right]^{2} }}{{K_{1} K_{2} }} + \ldots$$
(3)

The conditional and overall stability constants can be determined from the plots of the molar ratio and continuous variation methods, where at the point of stoichiometry, the ratio between the true observed absorbance (or color absorbance, or color difference) (At) to that extrapolated by the tangents (Aex) equal to the mole fraction of the metal ion or the ligand in the complex, Eq. (4). Thus, the concentration of the complex species can be expressed by Eq. (5), and the concentration of the free metal ion and the free ligand can be expressed by Eqs. (6) and (7), respectively. Thus, by substitution in Eq. (1), the conditional stability constant can be calculated from Eq. (8).(Inczédy 1976)

$$\frac{{A_{t} }}{{A_{ex} }} = \frac{{\left[ {ML_{n} } \right]}}{{C_{M} }} = \frac{{n\left[ {ML_{n} } \right]}}{{C_{L} }}$$
(4)
$$\left[ {ML_{n} } \right] = \frac{{A_{t} }}{{A_{ex} }}C_{M}$$
(5)
$$\left[ M \right] = C_{M} - \frac{{A_{t} }}{{A_{ex} }}C_{M}$$
(6)
$$\left[ L \right] = C_{L} - n\frac{{A_{t} }}{{A_{ex} }}C_{M}$$
(7)
$$\beta '_{n} = \frac{{\left[ {ML_{n} } \right]}}{{\left[ M \right]\left[ L \right]^{n} }} = \frac{{\frac{{A_{t} }}{{A_{ex} }}C_{M} }}{{\left[ {C_{M} - \frac{{A_{t} }}{{A_{ex} }}C_{M} } \right]\left[ {C_{L} - n\frac{{A_{t} }}{{A_{ex} }}C_{M} } \right]^{n} }}$$
(8)

where CM and CL are the analytical concentrations of the metal and ligand, whereas [M], [L], and [MLn] are the equilibrium concentrations of the metal, ligand, and complex, respectively.

Experimental

Apparatus and software

Digital Image-Based Analysis (DIBA) measurements were made using a simple homemade platform. (Mohamed and Shalaby 2019; Mohamed et al. 2018) In brief, the platform consisted of (1) two matched 10 mm glass cells (2) the digital camera and (3) a white cartoon paper as background diffuser. The diffuser, the 3D printed cell-holder and the camera were fixed on a 20 × 20 cm wood plate; each of them was 5-cm distance apart. Digital images were captured, on the bench of our laboratory, using a Canon PowerShot A810 digital camera that is equipped with a 16.0 Mega Pixel CCD sensor, where the conventional fluorescent daylight-lamp fixed to the ceiling served as the light source. (Mohamed and Shalaby 2019; Mohamed et al. 2018) A conventional HP-EliteBook 2540P notebook running under windows 10 was used for treatment of data and analysis. Photoshop CC 2017 and imageJ 1.52c software were used for digital image cropping and RGB channel intensities calculations, respectively. The obtained RGB intensities were converted into the corresponding coordinates of the Yxy and LUV color spaces using the free colormine online converter (Colormine color converter. 2019).

Spectrophotometric measurements, for comparison only, were made on a Shimadzu 1650 UV/VIS spectrophotometer controlled by an UVProbe-2.5 software (Kyoto, Japan).

Eppendorf 10-100 and 100-1000 μL vary-pipettes (Westbury, NY, USA) and a calibrated EDT pH-mV meter (Dover Kent, UK) were used.

Materials and chemicals

All reagents were of ACS grade and were used as received from Sigma-Aldrich (St. Louis, MO, USA), Merck (Darmstadt, Germany) or BDH (Poole, UK). Unless otherwise stated, de-ionized water and aqueous solutions were used throughout. 1,10-phenanthroline, 2,4,6-Tris(2-pyridyl)-s-triazine, sodium salicylate, ferrous ammonium sulfate hexahydrate, ferric ammonium sulfate dodecahydrate, and L-ascorbic acid, sodium acetate, sodium nitrate, and concentrated H2SO4, concentrated HNO3 were also used.

Determination of stoichiometry and stability constants of the iron complexes

For the determination of Fe(II)-Phen stoichiometry using the molar ratio method, the following solutions were used: (1) 100 mL solutions of Fe(II) 6.00 × 10−4 mol L−1, and (2) 100 mL Phen solution 6.00 × 10−4 mol L−1. In a series of dry 10 mm path length glass spectrophotometric cells, 200 µL aliquots of Fe(II) were mixed with 300 µL of 1.0% w/v ascorbic acid [to reduce any Fe(III) traces present in the Fe(II)] and 300 µL of 1.0 mol L−1 acetate buffer (pH = 3.5). Increasing volumes of Phen (0 to 1200 µL) were then added to the reaction cuvettes to give increasing [Phen]/[Fe] ratios from 0.0 to 6.0. The reacting solutions were diluted with water to exactly 3000 µL and then mixed well. All measurements were made at 25 °C with the ionic strength adjusted to 0.1 mol L−1 using sodium nitrate solution.

For the determination of Fe(II)-Phen stoichiometry using the Job’s method, 3.00 × 10−4 mol L−1 solutions of Fe(II) and Phen were used. In another series of dry 10 mm path length glass spectrophotometric cells, various volumes of Fe(II) (100 to 1900 µL) and Phen (1900 to 100 µL) were mixed to give the desired mole fractions. A 300 µL of each of ascorbic acid and acetate buffer (pH = 3.5) was added, and the final volume of each cuvette was then diluted to 3000 µL with water and the reacting solutions were mixed well. Similarly, all measurements were made at 25 °C with the ionic strength adjusted to 0.1 mol L−1 using sodium nitrate solution.

The molar ratio and the continuous variation methods were similarly followed for the Fe(II)-TPTZ and Fe(III)-SAL systems. However, the appropriate experimental conditions of the respective procedures were used. Namely, for the Fe(II)-TPTZ system, 3.80 × 10−4 and 1.00 × 10−4 mol L−1 Fe(II) solutions were used for the molar ratio and the continuous variation methods, respectively. Moreover, acetate buffer of pH 4.5 was used. However, for the Fe(III)-SAL system, 1.80 × 10−3 mol L−1 Fe(III) solution was used for both the molar ratio and the continuous variation methods. Further, 50 µL of 0.10 mol L−1 nitric acid was used to adjust the medium acidity, in the latter system.

Results and discussion

Spectra and Image acquisition

In the molar ratio method, solutions having various [L]/[M] molar ratios were prepared for the Fe(II)-Phen, Fe(II)-TPTZ, and Fe(III)-SAL complexes. A portion from each solution was placed in the spectrophotometric cell and was placed in our homemade platform to capture respective digital images that were automatically outputted in JPEG format with 180 dpi resolution. From each image, a squared area (360 × 360 pixel) was cropped as an area of interest (AOI), Fig. 1 and Figs. S1–S2. However, any other predefined AOI can be selected. Thereafter, the AOI was analyzed, using the freeware ImageJ, to obtain the RGB intensities that were converted to the corresponding Yxy and ΔELUV parameters, based on their excellent responses compared to the original RGB intensities (Mohamed and Shalaby 2019; Mohamed et al. 2018). The color absorbance of x or y parameters and the color difference, ΔELUV, were calculated from Eqs. (9) and (10), respectively.

$$A_{xy} = \, \left| {\log \left( {I_{o} /I} \right)} \right|$$
(9)
$$\Delta E_{\text{LUV}} = \left[ {\left( {\Delta L} \right)^{2} + \left( {\Delta U} \right)^{2} + \left( {\Delta V} \right)^{2} } \right]^{{\frac{1}{2}}}$$
(10)

where Io and I are the x or y value of the blank and sample, respectively, whereas Δ means the difference in the parameter value between the sample and blank, respectively. For validation, the spectrophotometric cell was placed in the spectrophotometer and spectral curves were recorded, Fig. 2a–c. The [L]/[M] ratio was plotted against the color absorbance of the x or y parameters, Fig. 2d–f, or against the color difference (ΔE) calculated from the LUV color space parameters, Fig. 2g–i.

Fig. 1
figure 1

Digital images of Fe(II)-phenanthroline complex at different [phenanthroline]/[Fe(II)] molar ratio (a) and the cropped Area of interest (b). Except for the studied variable, other conditions were those given in th recommended procedure. Captured images were arbitrarily compressed to fit into the page margins; however, for image analysis, the original uncompressed images were used

Full size image
Fig. 2
figure 2

Spectra obtained at different L:M ratio for the studied Fe complexes with Phenanthroline, TPTZ, and Salicylate (a)–(c), Molar ratio graphs plotted using spectrophotometric absorbance at the appropriate wavelength and color absorbance of x or y parameter (d)–(f), and color difference ΔELUV (g)–(i). Other conditions were those of Fig. 1

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For comparison and validation, the [L]/[M] ratio was simultaneously plotted against the spectrophotometric absorbance at the appropriate wavelength for each complex, Fig. 2d–i. The DIBA results showed 1:3, 1:2 and 1:1 (M:L) ratios for the Fe(II)-Phen, Fe(II)-TPTZ and Fe(III)-SAL systems, respectively, in excellent agreement with the spectrophotometric data.

Similarly, in the Job’s method of continuous variation, solutions having various mole fractions were prepared for the Fe(II)-Phen, Fe(II)-TPTZ, and Fe(III)-SAL complexes and similarly treated as those of the molar ratio method. Images of these solutions were given in Fig. 3 and Figs. S3–S4, for the Fe(II)-Phen, Fe(II)-TPTZ, and Fe(III)-SAL complexes, respectively. The Job’s plots, Fig. 4a–f, of the studies systems exhibited maxima at 0.25, 0.33 and 0.5, for the Fe(II)-Phen, Fe(II)-TPTZ, and Fe(III)-SAL complexes, respectively, showing that iron reacts with phen, TPTZ, and SAL in molar ratios of 1:3, 1:2 and 1:1, respectively. These data reflect the excellent agreement between the spectrophotometric and DIBA measurements.

Fig. 3
figure 3

Digital images of Fe(II)-phenanthroline complex at different Fe(II) mole fraction (a) and the cropped Area of interest (b). Other conditions were those of Fig. 1

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Fig. 4
figure 4

Continuous variation graphs plotted using the spectrophotometric absorbance and the color absorbance of x or y coordinate of Yxy color (a)–(c) and color difference ΔELUV (d)–(f) for the studied Fe complexes with Phenanthroline, TPTZ, and Salicylate. Conditions and symbols were those of Fig. 1

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It’s worthy to mention that, in addition to the formed trisphenanthroline-Iron(II) complex ion, other monophenanthroline and diphenanthroline complex ions can be formed. However, compared to the very stable trisphenanthroline complex, the mono- and diphenanthroline complexes are relatively unstable and its existence can be neglected at high phanenthroline-to-Iron(II) ratios. For example, the monophenanthroline-Fe(II) complex is formed only at large excess of ferrous (Kolthoff et al. 1950), so that, it can be formed only at iron mole fraction larger than 0.5 and [phenanthroline]/[Fe] molar ratio less than 1.0.

Further, the conditional and overall stability constants of the three studied complexes were determined from the molar ratio plots using Eqs. (8) and (2), respectively. The side reaction functions of each ligand protonation were calculated from Eq (3) using the previously reported pKa values for 1,10-phenanthroline (4.84) (CRC handbook of chemistry and physics 2017), TPTZ (3.53 and 2.73) (Martell and Smith 1989), and Salicylic acid (13.6 and 2.98) (CRC handbook of chemistry and physics 2017). The resulting conditional and overall stability constants of the Fe(II)-Phen, Fe(II)-TPTZ, and Fe(III)-SAL complexes, based on spectrophotometric and DIBA measurements are given in Table 1. The data reflects the excellent agreement not only between the DIBA results and the spectrophotometric measurements but also with the previously reported literature values (Chattopadhyaya and Singh 1974; Martell and Smith 1977, 1989; McKenzie 1955). Thus, our simple camera-based platform can be conveniently applied to the determination of complex stoichiometry and stability constants in a manner competing well with the sophisticated spectrophotometers showing its feasibility for practical application in undergraduate students’ laboratories.

Table 1 Stability constant values of the studied complexes obtained using DIBA and spectrophotometric methods
Full size table

Conclusions

The stoichiometric ratio and binding constant of metal complexes are important physical and analytical chemistry concepts. Conventional methods for the determination of these parameters often require expensive and/or bulky instruments. Herein, we reported a low-cost and simple approach based on digital imaging. The stoichiometric ratio and binding constants of three iron complexes with 1,10-phenanthroline, 2,4,6-Tris(2-pyridyl)-s-triazine, and salicylate were determined using digital imaging and following the molar ratio and continuous variation methods. The color absorbance of non-uniform Yxy color space and the color difference ΔELUV derived from the uniform LUV color space were used as convenient analytical signaling tools. The Yxy color absorbance and the ΔELUV color difference signaling tools can be conveniently applied to investigate a wide range of various colored complexes including not only mononuclear complexes but also polynuclear complexes of various species. Moreover, the current results of composition and conditional and overall stability constants are in excellent agreement with those obtained using sophisticated spectrophotometers and specialized software packages and also in conformity to previously reported literature values. This is very advantageous under resource-limited settings.