**4. Determination of stability constants**

The determination of metal complexes involves several methods including spectroscopic and potentiometric methods. The determination of stability constant is very significant to understand the role and behavior of ligand(s) in stabilizing the metal complexes and found applications in the fields of biology, environmental study, metallurgy, food chemistry, and many other industrial processes. Some of the methods that are used for the determination of stability constants are given as follows.

#### **4.1 Spectroscopic methods**

UV-Vis spectroscopic technique has been used to determine the stability constant and composition of a complex [14]. The formation of metal complex is indicated by the change in absorbance in the UV-Vis spectroscopy. The relationship between absorbance (A) and concentration is given by Beer's law as shown.

$$\mathbf{A} = \varepsilon.\mathbf{c.l.}$$

where ԑ = molar extinction coefficient, l = path length of the absorption cell, c = concentration of the complex.

From the above equation, the concentration (c) of the metal complex can be calculated by measuring absorbance (A) using spectrophotometer and knowing the molar extinction coefficient (ԑ) at that wavelength (λ) and path length (l). For example, consider the formation constant (K*f*) for the following reaction:

$$\mathbf{M}^{\mathbb{n}-} + \perp \xrightarrow{\mathbf{K}\_{\mathsf{f}}} \mathbf{ML^{\mathbb{n}+}}, \quad \mathbf{K}\_{\mathsf{f}} = \frac{[\mathbf{ML^{\mathbb{n}+}}]}{[\mathbf{M^{\mathbb{n}}}] \ [\mathsf{L}]} \tag{7}$$

By knowing the values of [MLn+], [M], and [L], Kf can be calculated. [MLn+], [M], and [L] can be calculated as follows:

$$\mathbf{C\_M} = [\mathbf{M^{n+}}] + [\mathbf{M L^{n+}}] \tag{8}$$

Sr. no. of solution: 1 2 3 4 5 6 7 8 9 10 Volume of metal ion (mL) 0 1 2 3 4 5 6 7 8 9 Volume of ligand (mL) 9 8 7 6 5 4 3 2 1 0

Here the total concentration of the solution is constant, i.e., sum of concentra-

ii. In the next step, optical densities of the solutions prepared in the previous step are to be determined spectrophotometrically at the wavelength of light which is strongly absorbed by metal complex but does not get absorbed by

iii. A plot between mole fraction (*mf*) of the ligand and absorbance or optical

On the extrapolation of the curve, the legs of the curve intersect each other at a

density is to be drawn. The plot obtained is shown in **Figure 4**.

*Determination of the composition of metal complex Job's method of continuous variation.*

CM þ CL ¼ C constant ð Þ (14)

n ¼ CL*=*CM (15)

CL*=*C ¼ *mf* (17)

CM*=*C þ CL*=*C ¼ C*=*C ¼ 1 (16)

tion of the metal, CM, and the ligand, CL, is fixed. Therefore:

metal ion and ligand.

*Stability of Metal Complexes*

*DOI: http://dx.doi.org/10.5772/intechopen.90894*

Eq. (14) can be rewritten as:

From this equation, namely,

**Figure 4.**

**33**

point which is the point of maximum absorbance. Suppose MLn is the formula of the complex, then

$$\mathbf{C}\_{\mathbf{L}} = [\mathbf{L}] + [\mathbf{M}\mathbf{L}^{\mathrm{n+}}] \tag{9}$$

where CL and CM are the total concentrations of the ligand and metal ion, respectively.

From Beer's law

$$\mathbf{A} = \sum\_{\mathbf{ML}^{n+}} \times \mathbf{l} \times [\mathbf{ML}^{n+}] \tag{10}$$

On rearranging

$$\left[\mathbf{ML^{n+}}\right] = \mathbf{A} / \sum\_{\mathbf{ML^{n+}}} \times \mathbf{l} \tag{11}$$

On substituting Eq. (11) in (8), we get

$$\mathbf{C\_{M}} = [\mathbf{M^{n+}}] + \mathbf{A/} \sum\_{\mathbf{ML^{n+}}} \times \mathbf{l}$$

$$\text{or } [\mathbf{M^{n+}}] = \mathbf{C\_{M}} - \mathbf{A/} \sum\_{\mathbf{ML^{n+}}} \times \mathbf{l} \tag{12}$$

Similarly, from Eqs. (9) and (11), we get

$$\mathbf{[L]} = \mathbf{C\_L} - \mathbf{A} / \sum\_{\mathbf{ML^{\*+}}} \times \mathbf{l} \tag{13}$$

By introducing the values of [MLn+], [Mn+], and [L] from Eqs. (11) to (13) in the formation constant Eq. (7), we can determine the value of K*f*.

#### **4.2 Job's method or method of continuous variations (MCV)**

Job's method of continuous variations (MCV) is used to determine the complex formation as well as stability constants [14, 15]. Job's method is basically used to determine the composition of metal complexes, and this is the modified version of spectroscopic method. This method is applicable in the case of solutions, where the formation of one metal complex takes place.

This method includes the following steps:

i. Make a volume of 10 mL solutions of metal complex containing different proportions of metal ion as well as ligand. The number of solutions should be 10.

*Stability of Metal Complexes DOI: http://dx.doi.org/10.5772/intechopen.90894*

where ԑ = molar extinction coefficient, l = path length of the absorption cell,

From the above equation, the concentration (c) of the metal complex can be calculated by measuring absorbance (A) using spectrophotometer and knowing the molar extinction coefficient (ԑ) at that wavelength (λ) and path length (l). For example, consider the formation constant (K*f*) for the following reaction:

By knowing the values of [MLn+], [M], and [L], Kf can be calculated. [MLn+],

where CL and CM are the total concentrations of the ligand and metal ion,

<sup>A</sup> <sup>¼</sup> <sup>X</sup>

ML<sup>n</sup><sup>þ</sup> ½ �¼ <sup>A</sup>*<sup>=</sup>*

CM <sup>¼</sup> <sup>M</sup><sup>n</sup><sup>þ</sup> ½ �þ <sup>A</sup>*<sup>=</sup>*

or Mn<sup>þ</sup> ½ �¼ CM � <sup>A</sup>*<sup>=</sup>*

½ �¼ L CL � A*=*

formation constant Eq. (7), we can determine the value of K*f*.

**4.2 Job's method or method of continuous variations (MCV)**

X

X

X

By introducing the values of [MLn+], [Mn+], and [L] from Eqs. (11) to (13) in the

Job's method of continuous variations (MCV) is used to determine the complex formation as well as stability constants [14, 15]. Job's method is basically used to determine the composition of metal complexes, and this is the modified version of spectroscopic method. This method is applicable in the case of solutions, where the

i. Make a volume of 10 mL solutions of metal complex containing different proportions of metal ion as well as ligand. The number of solutions should

X

MLn<sup>þ</sup> � l

CM <sup>¼</sup> <sup>M</sup>n<sup>þ</sup> ½ �þ MLn<sup>þ</sup> ½ � (8) CL <sup>¼</sup> ½ �þ <sup>L</sup> ML<sup>n</sup><sup>þ</sup> ½ � (9)

MLn<sup>þ</sup> � <sup>l</sup> � ML<sup>n</sup><sup>þ</sup> ½ � (10)

MLn<sup>þ</sup> � l (11)

MLn<sup>þ</sup> � l (12)

MLn<sup>þ</sup> � l (13)

ð7Þ

c = concentration of the complex.

*Stability and Applications of Coordination Compounds*

[M], and [L] can be calculated as follows:

On substituting Eq. (11) in (8), we get

Similarly, from Eqs. (9) and (11), we get

formation of one metal complex takes place. This method includes the following steps:

be 10.

**32**

respectively.

From Beer's law

On rearranging


Here the total concentration of the solution is constant, i.e., sum of concentration of the metal, CM, and the ligand, CL, is fixed. Therefore:

$$\mathbf{C\_M} + \mathbf{C\_L} = \mathbf{C} \left( \mathbf{constant} \right) \tag{14}$$


On the extrapolation of the curve, the legs of the curve intersect each other at a point which is the point of maximum absorbance.

Suppose MLn is the formula of the complex, then

$$\mathbf{n} = \mathbf{C}\_{\mathbf{L}} / \mathbf{C}\_{\mathbf{M}} \tag{15}$$

Eq. (14) can be rewritten as:

$$\mathbf{C} \mathbf{M} / \mathbf{C} + \mathbf{C} \mathbf{L} / \mathbf{C} = \mathbf{C} / \mathbf{C} = \mathbf{1} \tag{16}$$

From this equation, namely,

$$\mathbf{C}\_{\mathbf{L}}/\mathbf{C} = \mathfrak{m}\mathfrak{f} \tag{17}$$

**Figure 4.** *Determination of the composition of metal complex Job's method of continuous variation.*

On reducing Eq. (16), we get.

*mf* þ CM*=*C ¼ 1 or CM*=*C ¼ 1 � *mf* (18)

The concentration terms of [ML<sup>+</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.90894*

For electrical neutrality

*Stability of Metal Complexes*

], [M<sup>+</sup>

In the potentiometric or pH-metric determination of stability constant, a ligand and an acid such as nitric acid are titrated against standard NaOH during which period the pH of the solution has to be measured after each addition. Throughout

trically by using a pH meter and by substituting the values of [ML+

in the formation constant (K*f*); Eq. (20) can determine the value of K*f*.

the experimental studies, several conditions must be maintained such as

] are negligible as compared to [Na<sup>+</sup>

The above equation is used to calculate the value of pKa of ligand.

pH meter and knowing the concentrations of Mn+, H+

When a ligand is titrated with a solution containing Mn+ ion and an equivalent

HL, L�, Mn+, M(n�1)+L, M(n�2)+L2, etc. By measuring the pH values with the help of

the stepwise stability constants K1, K2, K3, etc. During the evaluation of equilibrium

, the resulting solution will have an equilibrium mixture of H<sup>+</sup>

The total concentration of the ligand is given as follows:

The protonation of the ligand can be represented as

The equilibrium constant Ka is given as

By combining Eqs. (21)–(23), we get

[OH�] and [H<sup>+</sup>

amount of H+

**35**

Hence Eq. (24) becomes

], and [L] can be calculated potentiome-

H<sup>þ</sup> ½ �þ Na<sup>þ</sup> ½ �¼ OH� ½ �þ L� ½ � (21)

LTotal ¼ ½ �þ HL L� ½ � (22)

], [M<sup>+</sup>

], and [L]

ð23Þ

, OH�,

] at near neutral solution.

, HL, etc., one can calculate

By dividing (17) by (18), we get.

$$\mathbf{C\_L/C X C/C\_M} = m f/\mathbf{1} - m f$$
 
$$\mathbf{C\_L/C\_M} = m f/\mathbf{1} - m f$$
 
$$\mathbf{n} = m f/\mathbf{1} - m f \tag{19}$$

The composition of the complex, MLn, can be determined by the value of n as obtained from Eq. (19). There are some drawbacks of this method. One of the drawbacks is that this method is applicable where, under experimental conditions, only one complex is formed. Also, the total volume of the solutions which contain metal ion and ligand should not be changed. The method of continuous variations has also found tremendous application in the field of organometallic chemistry [16].

#### **4.3 Bjerrum's method**

The type and extent of interaction existing between the metal ion and ligand can be investigated by various experimental methods [17–21], and each method requires different experimental conditions and resulted in differences in the interpretation of reaction mechanism and stability constants. Bjerrum's method to determine the stability constant is also known as potentiometric method. This method is based on the competition between hydrogen ion and metal ion for ligand which is a weak base. Consider a metal ion and an acid such as nitric acid are added to a ligand in aqueous solution; the following equations are obtained:

When acid reacts with ligand

$$\text{H}^+ + \text{H}^+ \xrightarrow{\text{H}\_{\text{g}}} \text{HCl}^+; \text{ K}^\text{-} + \frac{[\text{H}^+][\text{H}^+]}{[\text{H}^+]} $$

where Ka is acid association constant of the ligand When metal ion reacts with ligand

$$\mathbf{L}\_{\star} + \mathbf{M}^{\dagger} \xrightarrow{\mathbf{K}\_{\mathbf{f}}} \mathbf{ML} \; ; \quad \text{where } \mathbf{K}\_{\mathbf{f}} - \frac{\mathbf{DM}^{+} \mathbf{f}}{|\mathbf{L}| |\mathbf{M}^{+}|} \tag{20}$$

where K*<sup>f</sup>* is the formation constant.

Let us consider CH, CM, and CL are the total concentrations of acid, metal ion, and ligand, respectively. Then

$$\mathbf{C}\_{\text{H}} = [\text{H}^{+}] + [\text{HL}^{+}]$$

$$\mathbf{C}\_{\text{L}} = [\text{L}] + [\text{ML}^{+}] + [\text{HL}^{+}]$$

$$\mathbf{C}\_{\text{M}} = [\text{M}^{+}] + [\text{ML}^{+}]$$

By solving above three equations and using the acid association constant Ka, we get

*Stability of Metal Complexes DOI: http://dx.doi.org/10.5772/intechopen.90894*

On reducing Eq. (16), we get.

*Stability and Applications of Coordination Compounds*

By dividing (17) by (18), we get.

**4.3 Bjerrum's method**

or CL*=*CM ¼ *mf =*1 � *mf*

aqueous solution; the following equations are obtained:

where Ka is acid association constant of the ligand

When acid reacts with ligand

When metal ion reacts with ligand

where K*<sup>f</sup>* is the formation constant.

and ligand, respectively. Then

we get

**34**

*mf* þ CM*=*C ¼ 1

or CM*=*C ¼ 1 � *mf* (18)

CL*=*CXC*=*CM ¼ *mf =*1 � *mf*

or n ¼ *mf =*1 � *mf* (19)

The composition of the complex, MLn, can be determined by the value of n as obtained from Eq. (19). There are some drawbacks of this method. One of the drawbacks is that this method is applicable where, under experimental conditions, only one complex is formed. Also, the total volume of the solutions which contain metal ion and ligand should not be changed. The method of continuous variations has also found tremendous application in the field of organometallic chemistry [16].

The type and extent of interaction existing between the metal ion and ligand can be investigated by various experimental methods [17–21], and each method requires different experimental conditions and resulted in differences in the interpretation of reaction mechanism and stability constants. Bjerrum's method to determine the stability constant is also known as potentiometric method. This method is based on the competition between hydrogen ion and metal ion for ligand which is a weak base. Consider a metal ion and an acid such as nitric acid are added to a ligand in

Let us consider CH, CM, and CL are the total concentrations of acid, metal ion,

CH ¼ H<sup>þ</sup> ½ �þ HL<sup>þ</sup> ½ � CL ¼ ½ �þ L ML<sup>þ</sup> ½ �þ HL<sup>þ</sup> ½ � CM ¼ M<sup>þ</sup> ½ �þ ML<sup>þ</sup> ½ �

By solving above three equations and using the acid association constant Ka,

ð20Þ

$$\begin{aligned} \text{[ML^{-}]} \quad & \mathbf{C\_{U}} \cdot \mathbf{C\_{H}} + [\mathbf{H}^{-}] \cdot \frac{\mathbf{K\_{a}} \mathbf{H\_{-}} \mathbf{J\_{-}}}{\mathbf{K\_{a}} \mathbf{H\_{-}} \mathbf{J\_{-}}} \\\\ [\mathbf{M^{-}}] = \mathbf{C\_{M}} \cdot [\mathbf{M^{-}}] \end{aligned} $$
 
$$\begin{aligned} [\mathbf{L^{-}}] = \frac{\mathbf{C\_{H}} \cdot [\mathbf{H^{-}}]}{\mathbf{K\_{a}} [\mathbf{H^{-}}]} \end{aligned} $$

The concentration terms of [ML<sup>+</sup> ], [M<sup>+</sup> ], and [L] can be calculated potentiometrically by using a pH meter and by substituting the values of [ML+ ], [M<sup>+</sup> ], and [L] in the formation constant (K*f*); Eq. (20) can determine the value of K*f*.

In the potentiometric or pH-metric determination of stability constant, a ligand and an acid such as nitric acid are titrated against standard NaOH during which period the pH of the solution has to be measured after each addition. Throughout the experimental studies, several conditions must be maintained such as

For electrical neutrality

$$\left[\mathbf{H}^+\right] + \left[\mathbf{Na}^+\right] = \left[\mathbf{OH}^-\right] + \left[\mathbf{L}^-\right] \tag{21}$$

The total concentration of the ligand is given as follows:

$$\mathbf{L}\_{\text{Total}} = [\mathbf{HL}] + [\mathbf{L}^-] \tag{22}$$

The protonation of the ligand can be represented as

$$\mathbb{H} \rightleftharpoons \mathbb{H}^+ + \mathbb{M}$$

The equilibrium constant Ka is given as

$$\mathbf{K\_{a}} = \frac{[\mathbf{H^{+}}][\mathbf{L^{+}}]}{[\mathbf{HL}]} \tag{23}$$

By combining Eqs. (21)–(23), we get

$$\log \mathsf{K}\_{\mathsf{a}} = -\log \mathsf{[\mathsf{H}^{+}]} \quad + \quad \log \left( \frac{\mathsf{L}\_{\mathsf{Total}} \cdot \left( \left[ \mathsf{N}\_{\mathsf{a}} \right] + \left[ \mathsf{H}^{+} \right] \cdot \left[ \mathsf{OH}^{-} \right] \right)}{\left[ \mathsf{N}\_{\mathsf{a}} \right] + \left[ \mathsf{H}^{+} \right] \cdot \left[ \mathsf{OH}^{-} \right]} \right)$$

[OH�] and [H<sup>+</sup> ] are negligible as compared to [Na<sup>+</sup> ] at near neutral solution. Hence Eq. (24) becomes

$$\log\_{\mathsf{H}}\mathsf{K}\_{\mathsf{H}} = -\log\mathsf{I}\mathsf{H}\_{\mathsf{H}}^{-1}\mathsf{I}\_{\mathsf{H}} + \log\left(\frac{\mathsf{L}\_{\mathsf{H}^{\mathsf{H}}}\mathsf{L}\_{\mathsf{H}}^{-1}\mathsf{L}\_{\mathsf{H}}}{\mathsf{L}\_{\mathsf{H}}^{-1}\mathsf{L}\_{\mathsf{H}}}\right)$$

The above equation is used to calculate the value of pKa of ligand.

When a ligand is titrated with a solution containing Mn+ ion and an equivalent amount of H+ , the resulting solution will have an equilibrium mixture of H<sup>+</sup> , OH�, HL, L�, Mn+, M(n�1)+L, M(n�2)+L2, etc. By measuring the pH values with the help of pH meter and knowing the concentrations of Mn+, H+ , HL, etc., one can calculate the stepwise stability constants K1, K2, K3, etc. During the evaluation of equilibrium constants, the concentrations of Mn+ and L� are varied, and such variations in the concentration will lead to changes in the ionic strength of the solutions. In order to maintain the constant ionic strength, a large excess of an ionic salt is added to the reaction mixture. The presence of large excess of ionic salt will compensate any changes in the ionic strength of the solution. The ionic salts that are added for such purpose should not react with M2+ or L�, and commonly used salts include KNO3 and NaClO4, due the low affinity of NO3 � and ClO4 � ions for most of the M2+ ions. For example, KNO3 was added in excess during the binding study of the ligand p-aminobenzoic acid with Ni, Mg, and Co metal ions. The p-aminobenzoic acid has two coordination sites such as amino and carboxylate groups and has a pKa value of 5.9153. The stability constant values obtained for Ni, Mg, and Co complexes are depicted in **Table 1**.

The term nA is similar to n and is defined as the average number of protons bound to the ligand which are not coordinated to the metal center. PL gives the free ligand exponent. All the three terms n, nA, and PL can be calculated with the help of

nA <sup>¼</sup> *<sup>γ</sup>* � ð Þ V2 � V1 <sup>N</sup> <sup>þ</sup> <sup>ℇ</sup><sup>0</sup> � �

<sup>n</sup> <sup>¼</sup> ð Þ V3 � V2 <sup>N</sup> <sup>þ</sup> <sup>ℇ</sup><sup>0</sup> � �

where N is the normality of base used; V0 is the initial volume of the solution; V1, V2, and V3 are the volume of base consumed during the (A), (A + L), and

Step 3: Determination of formation curves: by plotting formation function (n)

The value formation constants corresponding to formation of protonated ligand are obtained by plotting nA against pH. Similarly, the stepwise stability constants for the formation of metal complexes are obtained from the formation curve

The thermodynamic and kinetic stability of coordination compounds along with the various factors affecting the stability of metal complexes have been discussed in

Authors acknowledge National Institute of Technology Kurukshetra, Haryana,

this chapter. Stability constant and its determination have also been listed.

tion of ligand; ℇ<sup>0</sup> is the initial concentration of acid; γ is the number of titrable or

H<sup>þ</sup> ½ � K2 <sup>þ</sup> <sup>H</sup><sup>þ</sup> ½ �<sup>2</sup> K1K2 T0L � <sup>T</sup>0M � �<sup>n</sup>

PL ¼ log 1 þ

(A + L + M) titrations, respectively, at same pH value; T<sup>0</sup>

against PL and nA against pH for a HL (protonated ligand) system.

8 < : ð Þ V0 <sup>þ</sup> V1 T0L (25)

9 = ;

L is the initial concentra-

(27)

ð Þ V0 <sup>þ</sup> V2 nAT0M (26)

<sup>X</sup>ð Þ V0 <sup>þ</sup> V3 V0

following equations

*Stability of Metal Complexes*

*DOI: http://dx.doi.org/10.5772/intechopen.90894*

replaceable protons.

**5. Conclusions**

**Acknowledgements**

India, for its support.

**Conflict of interest**

**37**

There is no conflict of interest.

resulted by plotting n against PL.

The stability constant values for Ni shows the trend 2:3 < 1:5 < 1:1 < 1:2, while the trend for Co is 1:2 < 2:3 < 1:1 < 1:5 and for Mg it is 1:5 < 1:2 < 1:1 < 2:3. The values obtained from the above study indicates that 1:2 complex of Ni complex is more stable, whereas Co complex is stable in 1:5 ratio and that of Mg is more stable in the ratio of 2:3.

#### **4.4 Irving and Rossotti method**

The Irving and Rossotti method for the determination of stability constant is also based on the principle of potentiometric method [21]. Using this method, the formation curve of metal complex can directly be calculated with the help of pH meter. Another major advantage of this method over the Bjerrum's method is that the calculation is simple and does not require hydrogen ion concentration. Moreover, this method can be used for types of ligands that are conjugate to weak acids. The calculation of stability constant using this method involves the following steps.

Step 1: The following solutions were titrated separately against base solution


Step 2: Calculation of formation functions n, nA, and PL using the values used/ obtained from above three titrations

The term formation function "n," also called as ligand number, is defined as the average number of ligands attached per metal center and is calculated using the following equation

