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## Chemical Equilibrium—Part 2: Gibbs Energy

In a previous section, we began a description of chemical equilibrium in the context of forward and reverse rates. Three key ideas were presented:

1. At equilibrium, the concentrations of reactants and products in a reversible reaction are not changing in time.
2. A reversible reaction at equilibrium is not static—reactants and products continue to interconvert at equilibrium, but the rates of the forward and reverse reactions are the same.
3. We were NOT going to fall into a common student trap of assuming that chemical equilibrium means that the concentrations of reactants and products are equal at equilibrium.

Here we extend our discussion and put the concept of equilibrium into the context of Gibbs energy, also reinforcing the Energy Story exercise of considering the "Before/Start" and "After/End" states of a reaction (including the inherent passage of time). Figure 1. Reaction coordinate diagram for a generic exergonic reversible reaction. Equations relating Gibbs energy and the equilibrium constant: R = 8.314 J mol-1 K-1 or 0.008314 kJ mol-1 K-1; T is temperature in Kelvin. Attribution: Marc T. Facciotti (original work)

The figure above shows a commonly cited relationship between ∆G° and Keq:

[ ∆G^o = -RTln K_{eq}.]

Here, G° indicates the Gibbs energy under standard conditions (e.g., 1 atmosphere of pressure, 298 K). This equation describes the change in Gibbs energy for reactants converting to products in a reaction that is at equilibrium. The value of ∆G° can therefore be thought of as being intrinsic to the reactants and products themselves. ∆G° is like a potential energy difference between reactants and products. With this concept as a basis, one can also consider a reaction where the "starting" state is somewhere out of equilibrium. In this case, there may be an additional “potential” associated with the out-of-equilibrium starting state. This “added” component contributes to the ∆G of a reaction and can be effectively added to the expression for Gibbs energy as follows:

[∆G = ∆G° + RTln Q, ]

where (Q) is called the reaction quotient. From the standpoint of BIS2A, we will use a simple (a bit incomplete but functional) definition for

[Q = dfrac{[Products]_{st}}{[Reactants]_{st}} ]

at a defined non-equilibrium condition, st. One can extend this idea and calculate the Gibbs energy difference between two non-equilibrium states, provided they are properly defined and thus compute Gibbs energy changes between specifically defined out-of-equilibrium states. This last point is often relevant in reactions found in biological systems as these reactions are often found in multi-step pathways that effectively keep individual reactions in an out-of-equilibrium state.

This takes us to a point of confusion for some. In many biology books, the discussion of equilibrium includes not only the discussion of forward and reverse reaction rates, but also a statement that ∆G = 0 at equilibrium. This can be confusing because these very discussions often follow discussions of non-zero ∆G° values in the context of equilibrium (∆G° = -RTlnKeq). The nuance to point out is that ∆G° is referring to the Gibbs energy potential inherent in the chemical transformation between reactants and products alone. This is different from considering the progress of the reaction from an out-of-equilibrium state that is described by

[∆G = ∆G^o + RT ln Q.]

This expression can be expanded as follows:

[∆G = -RTln K_{eq} + RTln Q]

to bring the nuance into clearer focus. In this case note that as Q approaches Keq that the reaction ∆G becomes closer to zero, ultimately reaching zero when Q = Keq. This means that the Gibbs energy of the reaction (∆G) reaches zero at equilibrium not that the potential difference between substrates and products (∆G°) reaches zero.

## Polymorphism in the Pharmaceutical Industry: Solid Form and Drug Development Rolf Hilfiker is vice president and head of the department Solid-State Development at Solvias AG in Kaiseraugst, Switzerland. He obtained his PhD in physical chemistry from Basel University (Switzerland) and then did postdoctoral work at Stony Brook University (New York). He returned to Basel University as a research fellow and then moved to Ciba-Geigy (now Novartis) in Basel. In 1997 he became head of the Stability & Kinetics group at Novartis. In 1999 he participated in a management buyout to form Solvias AG. He has taught physical chemistry in New York and Basel, as well as numerous courses in solid-state development in Europe, Asia, and the US.

Markus von Raumer is director and group leader of Preformulation and Preclinical Galenics at Idorsia Pharmaceuticals Ltd. in Allschwil, Switzerland. He obtained his PhD in physical chemistry from the University of Fribourg (Switzerland) and then did postdoctoral work in the field of mass spectrometry at the University of Warwick (UK). In 1997 he became head of laboratory of Physical Chemistry at Novartis in Basel, Switzerland. Two years later he moved to Solvias AG in Basel where he was project manager for Solid-State Development until 2008. He then joined Actelion Pharmaceuticals Ltd. in Allschwil where he built up and led the Material Science lab until 2017.

## 16.4 Free Energy

One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that it requires measurements of the entropy change for the system and the entropy change for the surroundings. An alternative approach involving a new thermodynamic property defined in terms of system properties only was introduced in the late nineteenth century by American mathematician Josiah Willard Gibbs . This new property is called the Gibbs free energy (G) (or simply the free energy), and it is defined in terms of a system’s enthalpy and entropy as the following:

Free energy is a state function, and at constant temperature and pressure, the free energy change (ΔG) may be expressed as the following:

(For simplicity’s sake, the subscript “sys” will be omitted henceforth.)

The relationship between this system property and the spontaneity of a process may be understood by recalling the previously derived second law expression:

The first law requires that qsurr = −qsys, and at constant pressure qsys = ΔH, so this expression may be rewritten as:

Multiplying both sides of this equation by −T, and rearranging yields the following:

Comparing this equation to the previous one for free energy change shows the following relation:

The free energy change is therefore a reliable indicator of the spontaneity of a process, being directly related to the previously identified spontaneity indicator, ΔSuniv. Table 16.3 summarizes the relation between the spontaneity of a process and the arithmetic signs of these indicators.

### What’s “Free” about ΔG?

In addition to indicating spontaneity, the free energy change also provides information regarding the amount of useful work (w) that may be accomplished by a spontaneous process. Although a rigorous treatment of this subject is beyond the scope of an introductory chemistry text, a brief discussion is helpful for gaining a better perspective on this important thermodynamic property.

For this purpose, consider a spontaneous, exothermic process that involves a decrease in entropy. The free energy, as defined by

may be interpreted as representing the difference between the energy produced by the process, ΔH, and the energy lost to the surroundings, TΔS. The difference between the energy produced and the energy lost is the energy available (or “free”) to do useful work by the process, ΔG. If the process somehow could be made to take place under conditions of thermodynamic reversibility, the amount of work that could be done would be maximal:

However, as noted previously in this chapter, such conditions are not realistic. In addition, the technologies used to extract work from a spontaneous process (e.g., automobile engine, steam turbine) are never 100% efficient, and so the work done by these processes is always less than the theoretical maximum. Similar reasoning may be applied to a nonspontaneous process, for which the free energy change represents the minimum amount of work that must be done on the system to carry out the process.

### Calculating Free Energy Change

Free energy is a state function, so its value depends only on the conditions of the initial and final states of the system. A convenient and common approach to the calculation of free energy changes for physical and chemical reactions is by use of widely available compilations of standard state thermodynamic data. One method involves the use of standard enthalpies and entropies to compute standard free energy changes, ΔG° , according to the following relation.

### Example 16.7

#### Solution

The standard change in free energy may be calculated using the following equation:

Using the appendix data to calculate the standard enthalpy and entropy changes yields:

Substitution into the standard free energy equation yields:

#### Check Your Learning

The standard free energy change for a reaction may also be calculated from standard free energy of formation ΔGf° values of the reactants and products involved in the reaction. The standard free energy of formation is the free energy change that accompanies the formation of one mole of a substance from its elements in their standard states. Similar to the standard enthalpy of formation, Δ G f ° Δ G f ° is by definition zero for elemental substances in their standard states. The approach used to calculate Δ G ° Δ G ° for a reaction from Δ G f ° Δ G f ° values is the same as that demonstrated previously for enthalpy and entropy changes. For the reaction

the standard free energy change at room temperature may be calculated as

### Example 16.8

#### Solution

(a) Using free energies of formation:

(b) Using enthalpies and entropies of formation:

Both ways to calculate the standard free energy change at 25 °C give the same numerical value (to three significant figures), and both predict that the process is nonspontaneous (not spontaneous) at room temperature.

#### Check Your Learning

(a) 140.8 kJ/mol, nonspontaneous

(b) 141.5 kJ/mol, nonspontaneous

### Free Energy Changes for Coupled Reactions

The use of free energies of formation to compute free energy changes for reactions as described above is possible because ΔG is a state function, and the approach is analogous to the use of Hess’ Law in computing enthalpy changes (see the chapter on thermochemistry). Consider the vaporization of water as an example:

An equation representing this process may be derived by adding the formation reactions for the two phases of water (necessarily reversing the reaction for the liquid phase). The free energy change for the sum reaction is the sum of free energy changes for the two added reactions:

This approach may also be used in cases where a nonspontaneous reaction is enabled by coupling it to a spontaneous reaction. For example, the production of elemental zinc from zinc sulfide is thermodynamically unfavorable, as indicated by a positive value for ΔG°:

The industrial process for production of zinc from sulfidic ores involves coupling this decomposition reaction to the thermodynamically favorable oxidation of sulfur:

The coupled reaction exhibits a negative free energy change and is spontaneous:

This process is typically carried out at elevated temperatures, so this result obtained using standard free energy values is just an estimate. The gist of the calculation, however, holds true.

### Example 16.9

#### Solution

The coupled reaction exhibits a positive free energy change and is thus nonspontaneous.

### Temperature Dependence of Spontaneity

As was previously demonstrated in this chapter’s section on entropy, the spontaneity of a process may depend upon the temperature of the system. Phase transitions, for example, will proceed spontaneously in one direction or the other depending upon the temperature of the substance in question. Likewise, some chemical reactions can also exhibit temperature dependent spontaneities. To illustrate this concept, the equation relating free energy change to the enthalpy and entropy changes for the process is considered:

The spontaneity of a process, as reflected in the arithmetic sign of its free energy change, is then determined by the signs of the enthalpy and entropy changes and, in some cases, the absolute temperature. Since T is the absolute (kelvin) temperature, it can only have positive values. Four possibilities therefore exist with regard to the signs of the enthalpy and entropy changes:

1. Both ΔH and ΔS are positive. This condition describes an endothermic process that involves an increase in system entropy. In this case, ΔG will be negative if the magnitude of the TΔS term is greater than ΔH. If the TΔS term is less than ΔH, the free energy change will be positive. Such a process is spontaneous at high temperatures and nonspontaneous at low temperatures.
2. Both ΔH and ΔS are negative. This condition describes an exothermic process that involves a decrease in system entropy. In this case, ΔG will be negative if the magnitude of the TΔS term is less than ΔH. If the TΔS term’s magnitude is greater than ΔH, the free energy change will be positive. Such a process is spontaneous at low temperatures and nonspontaneous at high temperatures.
3. ΔH is positive and ΔS is negative. This condition describes an endothermic process that involves a decrease in system entropy. In this case, ΔG will be positive regardless of the temperature. Such a process is nonspontaneous at all temperatures.
4. ΔH is negative and ΔS is positive. This condition describes an exothermic process that involves an increase in system entropy. In this case, ΔG will be negative regardless of the temperature. Such a process is spontaneous at all temperatures.

These four scenarios are summarized in Figure 16.12.

### Example 16.10

#### Predicting the Temperature Dependence of Spontaneity

How does the spontaneity of this process depend upon temperature?

#### Check Your Learning

How does the spontaneity of this process depend upon temperature?

ΔH and ΔS are negative the reaction is spontaneous at low temperatures.

When considering the conclusions drawn regarding the temperature dependence of spontaneity, it is important to keep in mind what the terms “high” and “low” mean. Since these terms are adjectives, the temperatures in question are deemed high or low relative to some reference temperature. A process that is nonspontaneous at one temperature but spontaneous at another will necessarily undergo a change in “spontaneity” (as reflected by its ΔG) as temperature varies. This is clearly illustrated by a graphical presentation of the free energy change equation, in which ΔG is plotted on the y axis versus T on the x axis:

Such a plot is shown in Figure 16.13. A process whose enthalpy and entropy changes are of the same arithmetic sign will exhibit a temperature-dependent spontaneity as depicted by the two yellow lines in the plot. Each line crosses from one spontaneity domain (positive or negative ΔG) to the other at a temperature that is characteristic of the process in question. This temperature is represented by the x-intercept of the line, that is, the value of T for which ΔG is zero:

So, saying a process is spontaneous at “high” or “low” temperatures means the temperature is above or below, respectively, that temperature at which ΔG for the process is zero. As noted earlier, the condition of ΔG = 0 describes a system at equilibrium.

### Example 16.11

#### Solution

When this process is at equilibrium, ΔG = 0, so the following is true:

Using the standard thermodynamic data from Appendix G,

The accepted value for water’s normal boiling point is 373.2 K (100.0 °C), and so this calculation is in reasonable agreement. Note that the values for enthalpy and entropy changes data used were derived from standard data at 298 K (Appendix G). If desired, you could obtain more accurate results by using enthalpy and entropy changes determined at (or at least closer to) the actual boiling point.

#### Check Your Learning

313 K (accepted value 319 K)

### Free Energy and Equilibrium

The free energy change for a process may be viewed as a measure of its driving force. A negative value for ΔG represents a driving force for the process in the forward direction, while a positive value represents a driving force for the process in the reverse direction. When ΔG is zero, the forward and reverse driving forces are equal, and the process occurs in both directions at the same rate (the system is at equilibrium).

In the chapter on equilibrium the reaction quotient, Q, was introduced as a convenient measure of the status of an equilibrium system. Recall that Q is the numerical value of the mass action expression for the system, and that you may use its value to identify the direction in which a reaction will proceed in order to achieve equilibrium. When Q is lesser than the equilibrium constant, K, the reaction will proceed in the forward direction until equilibrium is reached and Q = K. Conversely, if Q > K, the process will proceed in the reverse direction until equilibrium is achieved.

The free energy change for a process taking place with reactants and products present under nonstandard conditions (pressures other than 1 bar concentrations other than 1 M) is related to the standard free energy change according to this equation:

R is the gas constant (8.314 J/K mol), T is the kelvin or absolute temperature, and Q is the reaction quotient. For gas phase equilibria, the pressure-based reaction quotient, QP, is used. The concentration-based reaction quotient, QC, is used for condensed phase equilibria. This equation may be used to predict the spontaneity for a process under any given set of conditions as illustrated in Example 16.12.

### Example 16.12

#### Solution

Since the computed value for ΔG is positive, the reaction is nonspontaneous under these conditions.

#### Check Your Learning

For a system at equilibrium, Q = K and ΔG = 0, and the previous equation may be written as

This form of the equation provides a useful link between these two essential thermodynamic properties, and it can be used to derive equilibrium constants from standard free energy changes and vice versa. The relations between standard free energy changes and equilibrium constants are summarized in Table 16.4.

K ΔG° Composition of an Equilibrium Mixture
> 1 < 0 Products are more abundant
< 1 > 0 Reactants are more abundant
= 1 = 0 Reactants and products are comparably abundant

### Example 16.13

#### Solution

The standard free energy change for this reaction is first computed using standard free energies of formation for its reactants and products:

The equilibrium constant for the reaction may then be derived from its standard free energy change:

This result is in reasonable agreement with the value provided in Appendix J.

#### Check Your Learning

To further illustrate the relation between these two essential thermodynamic concepts, consider the observation that reactions spontaneously proceed in a direction that ultimately establishes equilibrium. As may be shown by plotting the free energy versus the extent of the reaction (for example, as reflected in the value of Q), equilibrium is established when the system’s free energy is minimized (Figure 16.14). If a system consists of reactants and products in nonequilibrium amounts (QK), the reaction will proceed spontaneously in the direction necessary to establish equilibrium.

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Chapter 1 Basic Concepts

1.1.2 Elements, Compounds and Mixtures

1.2.2 A review of some commonly used measurements

1.2.3 Accuracy and precision

1.3.2 Isotopes, radioactivity and the types of radiation

1.4 The concepts of stoichiometry: calculations of quantity in chemistry

1.4.2 AvogadroΥ¦s number and the concept of the mole

1.4.3 Formulae and molecular mass

1.4.4 Mass percent composition

1.4.5 Empirical and molecular formulae

1.4.6 Writing and Balancing Chemical Equations

1.4.7 Balancing Equations: A systematic approach

1.4.9 Concentration of solutions

Chapter 2. Atoms, Periodicity and Chemical Bonding

2.3 The Bohr Model of the Atom

2.4 An introduction to atomic orbitals

2.5 Electron configurations in atoms

2.7 An introduction to bonding. How atoms become molecules.

2.7.4 Formaloxidation states

2.7.5 Polarisation: covalent or ionic bonding?

2.7.7 Shapes of molecules ΥV the VSEPR approach

2.8 Covalent bonding ΥV atomic and molecular orbitals

2.9.1 Dipole-dipole interactions

2.9.2 Dispersion (London) Forces

2.9.4 Biological implications of hydrogen bonding

Chapter 3. An Introduction to the Chemistry of Carbon

3.3 Classification of organic molecules

3.3.1 Nomenclature (naming) of organic compounds

3.3.2 Systematic Nomenclature

3.3.3 Introduction to the Functional Groups concept

3.3.4 Naming of aliphatic compounds containing functional groups

3.4 The structure of organic molecules

3.4.1 Structural features of organic chemistry

3.4.2 Introduction to isomerism

3.4.3 Structural/constitutional isomerism

3.4.4 Introduction to stereoisomerism

3.4.6 Introduction to configurational isomerism

3.4.7 Geometrical isomerism

3.4.8 Symmetry, chirality and optical isomerism

3.4.9 Why is shape important? ΥV some examples.

4.1.2 Energy: heat, work and the first law of thermodynamics

4.2.2 Heat capacity, C and specific heat capacity, c

4.2.3 Endothermic and Exothermic Processes

4.3 The First Law of Thermodynamics ΥV introducing the concept of work

4.3.2 Energy in the chemistry context

4.3.3 The Concept of Enthalpy

4.3.4 Examples of enthalpy changes in biological processes

4.3.5 The Determination of Enthalpies: HessΥ¦s Law

4.4 Spontaneous processes, entropy and free energy

4.4.1 The 2 nd Law of thermodynamics.

4.4.2 Free energy and ATP: Coupling of reactions

4.4.3 Biological example: Thermodynamic rationale of micelle behaviour

Chapter 5 Equilibria: How far does a reaction go?

5.2 Developing the idea of equilibrium: the equilibrium constant

5.2.1 Calculation of equilibrium constants and concentrations

5.3 Equilibrium and energetics

5.3.2 The reaction quotient

5.3.3 Calculating equilibrium constants in the gas phase, using partial pressures Kp

5.4 The relationship between ƒ´G ƒá and K.

5.4.1 A more detailed look at reaction quotient Q and equilibrium constant, K.

5.5 Disturbing an equilibrium

5.5.1 Statement of Le ChatelierΥ¦s Principle

5.5.2 Le ChatelierΥ¦s principle and the effect of temperature on equilibria.

5.5.3 Examples involving Le ChatelierΥ¦s principle

5.6 Energetics and equilibria in the biological context.

5.6.1 Calculating ƒ´G Υò from experimentally determined compositions (via K values)

5.6.2 Calculating equilibrium compositions from ƒ´G Υò

5.6.3 Macromolecule-ligand interactions.

5.7 Revisiting coupled reactions

Chapter 6 Aqueous Equilibria

6.1.1 Why is this important in biology?

6.1.2 The importance of pH and pH control

6.2 Self ionisation of water

6.3.1 What do the terms acid and base mean?

6.3.3 Properties of bases6.3.4 Strong acids and strong bases

6.4.1 Behaviour of weak acids

6.4.2 Behaviour of weak bases

6.5 Dissociation of acids and bases - conjugate acids and bases

6.6 Acids and bases in aqueous solution ΥV the concept of pH

6.6.2 What happens when acids are dissolved in water?

6.6.3 What happens when the water equilibrium is disturbed

6.6.4 Calculating pH values for acids

6.7 The control of pH - buffer solutions

6.7.2 Theoretical aspects of buffers

6.7.3 General Strategy for making buffer solutions

6.10 Introducing solubility

6.10.1 Insoluble ionic compounds. The concept of solubility product.

6.10.2 The common ion effect

Chapter 7 Biomolecules and biopolymers

7.2.1 Fats, oils and fatty acids

7.2.3 Uses of fats - micelles

7.3.2 Carbohydrate stereochemistry

7.3.3 Cyclisation in sugars

7.3.4 Di- and polysaccharides

7.4 Amino acids, peptides and proteins

7.4.2 Acid-base behaviour of amino acids: zwitterions

7.4.3 The isoelectric point

7.4.4 The stereochemistry of amino acids

7.4.5 Peptides and proteins

7.4.6 Primary, secondary, tertiary and quaternary structures

7.4.7 Denaturing of proteins

7.5.2 Primary structure of nucleic acids

7.5.3 Secondary structure in nucleic acids

7.5.4 Structural features of RNA

Chapter 8 Reaction mechanisms

8.2 Organic Reaction Types

8.2.2 Elimination reactions

8.2.3 Substitution reactions

8.2.4 Isomerisation reactions

8.2.5 Oxidation and reduction

8.3.4 Carbocations and carbanions types and key points

8.4 Electronegativity and bond polarity

8.5.1 Electrophilic additions to alkenes and alkynes

8.5.2 Addition of HBr to unsymmetrical alkenes

8.5.3 Addition of other electrophiles to alkenes

8.5.4 Electrophilic addition in biology

8.5.5 Electrophilic addition without subsequent nucleophilic addition loss of H +

8.5.6 Addition of HBr to conjugated dienes

8.6 Substitution and elimination reactions

8.6.1 Nucleophilic substitution at a saturated carbon atom

8.6.2 Bimolecular nucleophilic substitution SN2

8.6.3 Unimolecularnucleophilic substitution SN1

8.6.4 Determining which mechanism is followed

8.7.1 Bimolecular elimination, E2

8.7.2 Unimolecular elimination, E1

8.8 Biological example of an SN2 reaction

8.9 Reaction mechanisms of carbonyl compounds

8.9.2 Structure of the carbonyl group, C=O

8.10 Reactions of aldehydes and ketones

8.10.1 Reaction of aldehydes and ketones with Υ§hydrideΥ¨

8.10.2 Hydration of aldehydes and ketones

8.10.3 Hemiacetal formation

8.10.4 Acetal (ketal) formation

8.10.5 Formation of SchiffΥ¦s bases and imines

8.10.6 Oxidation of aldehydes and ketones

8.11 Carboxylic acid derivatives

8.11.2 Acid catalysed hydrolysis of esters

8.11.3 Base (:OH - ) induced hydrolysis of esters

8.12 Enolisation and enolisation reactions

8.12.1 Enols as carbon nucleophiles

8.13 Reactions resulting from enolisation

8.13.2 Crossed aldol reactions / condensations

8.13.3 Claisen condensations

8.14 Reaction mechanisms in biological reactions: synthesis of steroids

8.15 Summary of mechanisms of carbonyl reactions under different conditions

Chapter 9. Chemical kinetics

9.2 Rates, rate laws and rate constants

9.2.2 Rates and concentration

9.2.3 Units of the rate constant

9.2.4 Determination of rate laws and rate constants

9.3 Temperature dependence of reaction rates and rate constants

9.4.1 Deducing reaction mechanisms

9.4.2 A more comprehensive look at complex reaction mechanisms

9.5 Kinetics of enzyme catalysed reactions

9.5.1 Catalysts and catalysis

9.5.3 Single-substrate enzyme reactions

9.5.4 Analysis of enzyme kinetic data

9.6.1 Mechanisms of inhibition

Chapter 10 Bioenergetics and Bioelectrochemistry

10.2 Electrochemical cells

10.2.1 Cells and cell nomenclature

10.2.3 Measurement of cell voltage

10.2.4 Free energy relationship

10.2.5 Determination of the reaction taking place in a cell

10.2.6 Effect of concentration

10.3 Sensors and reference electrodes

10.3.1 The silver electrode

10.3.2 The calomel electrode

10.4.1 Biochemical/biological standard state

10.4.2. Biological membranes

10.4.3 The thermodynamics of membrane transport

Chapter 11. The role of elements other than carbon

11.2 Phosphorus and phosphate esters

11.2.1Phosphoric acid and phosphate esters

11.3 Metals in the chemistry of biology

11.4 Transition metals and their role in biological systems

11.4.1 Introduction to ligands in biological systems.

11.4.2 Introduction to transition metals

11.4.4Examples of transition metals in biological systems

11.5 The alkali and alkaline-earth metals

11.5.2 Solid state structures

11.5.3 Coordination chemistry of group 1 and group 2 metals

11.5.4 Ions of alkali and alkaline-earth metal ions in biology

Chapter 12 Metabolism

12.2.1 Introduction to glycolysis

12.2.2 The glycolysis pathway

12.3 Analysis of the mechanism of glycolysis

12.4 What now? Where does the pyruvate go?

12.4.1 Conversion of pyruvate into lactate

12.4.2 Conversion of pyruvate into ethanol

12.4.3 Conversion of pyruvate into acetyl-coenzyme-A

12.5.1 Introduction and overview

12.6 Analysis of the mechanism of the TCA cycle

12.7 Summary of outcomes of the glycolysis and TCA cycles

Chapter 13 Structural Methods

13.2.2 Analysis of a mass spectrum

13.2.3 Isotopes: complicating factors or diagnostic tools?

13.2.4 Fragmentation pathways involving functional groups

13.3 Introduction to electromagnetic radiation

13.3.1 Background principles

13.4 Ultraviolet and visible (UV-vis) spectroscopy

13.4.2 Measurement of the spectrum

13.4.3 Using UV-vis Spectra for characterising compounds

13.4.5 Using UV-visible spectra for measuring concentrations of biologically important compounds

## A Complete Introduction to MODERN NMR Spectroscopy, Written by Roger S. Macomber J . 1 What is NMR Spectroscopy? 1 .2 Properties of Electromagnetic Radiation 1 .3 Interaction of Radiation with Matter: The Classical Picture 1 .4 Uncertainty and the Question of Time Scale Chapter Summary Review Problems

Chapter 2 MAGNETIC PROPERTIES OF NUCLEI

2.1 The Structure of an Atom 2.2 The Nucleus in a Magnetic Field 2.3 Nuclear Energy Levels and Relaxation Times 2.4 The Rotating Frame of Reference 2.5 Relaxation Mechanisms and Correlation Times Chapter Summary Additional Resources Review Problems

Chapter 3 OBTAINING AN NMR SPECTRUM

3.1 Electricity and Magnetism 3.2 The NMR Magnet 3.3 Signal Generation the Old Way: The Continuous- Wave (CW) Experiment 3.4 The Modern Pulsed Mode for Signal Acquisition 3.5 Line Widths, Lineshape, and Sampling Considerations 3.6 Measurement of Relaxation Times Chapter Summary Additional Resources Review Problems

Chapter 4 A LITTLE BIT OF SYMMETRY

4.1 Symmetry Operations and Distinguishability 4.2 Conformations and Their Symmetry 4.3 Homotopic, Enantiotopic, and Diastereotopic Nuclei 4.4 Accidental Equivalence Chapter Summary Additional Resources Review Problems

Chapter 5 THE H AND 13 C NMR SPECTRA OF TOLUENE

5.1 The ‘H NMR Spectrum of Toluene at 80 MHz 5.2 The Chemical Shift Scale 5.3 The 250- and 400-MHz ‘H NMR Spectra of Toluene 5 .4 The 1 3 C NMR Spectrum of Toluene at 20. 1, 62.9, and 1 00.6 MHz 5.5 Data Acquisition Parameters Chapter Summary Review Problems

Chapter 6 CORRELATING PROTON CHEMICAL SHIFTS WITH MOLECULAR STRUCTURE

6.1 Shielding and Deshielding 6.2 Chemical Shifts of Hydrogens Attached to Tetrahedral Carbon 6.3 Vinyl and Formyl Hydrogen Chemical Shifts 6.4 Magnetic Anisotropy 6.5 Aromatic Hydrogen Chemical Shift Correlations 6.6 Hydrogen Attached to Elements Other than Carbon Chapter Summary References Additional Resources Review Problems

Chapter 7 CHEMICAL SHIFT CORRELATIONS FOR 13 C AND OTHER ELEMENTS

7.1 l3 C Chemical Shifts Revisited 7.2 Tetrahedral (sp 2 Hybridized) Carbons 7.3 Heterocyclic Structures 7.4 Trigonal Carbons 7.5 Triple Bonded Carbons 7.6 Carbonyl Carbons 7.7 Miscellaneous Unsaturated Carbons 7.8 Summary of l3 C Chemical Shifts 7.9 Chemical Shifts of Other Elements Chapter Summary References Review Problems

Chapter 8 FIRST-ORDER (WEAK) SPIN-SPIN COUPLING

8.1 Unexpected Lines in an NMR Spectrum 8.2 The ‘H Spectrum of Diethyl Ether 8.3 Homonuclear ‘H Coupling: The Simplified Picture 8.4 The Spin-Spin Coupling Checklist 8.5 Thew+lRule 8.6 Heteronuclear Spin-Spin Coupling 8.7 Review Examples Chapter Summary Review Problems

Chapter 9 FACTORS THAT INFLUENCE THE SIGN AND MAGNITUDE OF SECOND-ORDER (STRONG) COUPLING EFFECTS

9.1 Nuclear Spin Energy Diagrams and the Sign of J 9.2 Factors that Influence /: Preliminary Considerations 9.3 One-Bond Coupling Constants 9.4 Two-Bond (Geminal) Coupling Constants 9.5 Three-Bond (Vicinal) Coupling Constants 9.6 Long-Range Coupling Constants 9.7 Magnetic Equivalence 9.8 Pople Spin System Notation 9.9 Slanting Multiplets and Second-Order (Strong Coupling) Effects 9.10 Calculated Spectra 9.11 The AX -> AB -> A 2 Continuum 9. 12 More About the ABX System: Deceptive Simplicity and Virtual Coupling Chapter Summary References Review Problems

Chapter 10 THE STUDY OF DYNAMIC PROCESSES BY NMR

10.1 Reversible and Irreversible Dynamic Processes 10.2 Reversible Intramolecular Processes Involving Rotation Around Bonds 10.3 Simple Two-Site Intramolecular Exchange 10.4 Reversible Intramolecular Chemical Processes 10.5 Reversible Intermolecular Chemical Processes 10.6 Reversible Intermolecular Complexation 1 0.7 Other Examples of Reversible Complexation: Chemical Shift Reagents Chapter Summary References Review Problems

Chapter 11 ELECTRON PARAMAGNETIC RESONANCE SPECTROSCOPY AND CHEMICALLY INDUCED DYNAMIC NUCLEAR POLARIZATION

11.1 Electron Paramagnetic Resonance 11.2 Free Radicals 11.3 The £ Factor 11 .4 Sensitivity Considerations 11 .5 Hyperfine Coupling and the a Value 11.6 A Typical EPR Spectrum 11 .7 CIDNP: Mysterious Behavior of NMR Spectrometers 11.8 The Net Effect 11.9 The Multiple! Effect 11.10 The Radical-Pair Theory of The Net Effect 11.11 The Radical-Pair Theory of the Multiplet Effect 11.12 A Few Final Words about CIDNP Chapter Summary References Review Problems

Chapter 12 DOUBLE-RESONANCE TECHNIQUES AND COMPLEX PULSE SEQUENCES

12.1 What is Double Resonance? 12.2 Heteronuclear Spin Decoupling 12.3 Polarization Transfer and the Nuclear Overhauser Effect 12.4 Gated and Inverse Gated Decoupling 12.5 Off-Resonance Decoupling 12.6 Homonuclear Spin Decoupling 12.7 Homonuclear Difference NOE: The Test for Proximity 12.8 Other Homonuclear Double-Resonance Techniques 12.9 Complex Pulse Sequences 12.10 The 7-Modulated Spin Echo and the APT Experiment 12. 1 1 More About Polarization Transfer 12.12 Distortionless Enhancement by Polarization Transfer Chapter Summary References Additional Resources Review Problems

Chapter 13 TWO-DIMENSIONAL NUCLEAR MAGNETIC RESONANCE

13. 1 What is 2D NMR Spectroscopy? 13.2 2D Heteroscalar Shift-Correlated Spectra 13.3 2D Homonuclear Shift-Correlated Spectra 13.4 NOE Spectroscopy (NOESY) 13.5 Hetero- and Homonuclear 2D ./-Resolved Spectra 13.6 ID and 2D INADEQUATE 13.7 2D NMR Spectra of Systems Undergoing Exchange Chapter Summary References Additional Resources Review Problems

Chapter 14 NMR STUDIES OF BIOLOGICALLY IMPORTANT MOLECULES

14. 1 Introduction 14.2 NMR Line Widths of Biopolymers 14.3 Exchangeable and Nonexchangeable Protons 14.4 Chemical Exchange 14.5 The Effects of pH on the NMR Spectra of Biomolecules 14.6 NMR Studies of Proteins 14.7 NMR Studies of Nucleic Acids 14.8 Lipids and Biological Membranes 14.9 Carbohydrates Chapter Summary References Additional Resources Review Problems

Chapter 15 SOLID-STATE NMR SPECTROSCOPY

15.1 Why Study Materials in the Solid State? 15.2 Why is NMR of Solids Different from NMR of Fluids? 15.3 Chemical Shifts in Solids 15.4 Spin-Spin Coupling 15.5 Quadrupole Coupling 297 15.6 Overcoming Long T Cross Polarization Chapter Summary Additional Resources Review Problems

Chapter 16 NMR IN MEDICINE AND BIOLOGY: NMR IMAGING

16.1 A Window into Anatomy and Physiology 16.2 Biomedical NMR 16.3 Pictures with NMR: Magnetic Resonance Imaging 16.4 Image Contrast 16.5 Higher Dimensional Imaging 16.6 Chemical Shift Imaging 16.7 NMR Movies: Echo Planar Imaging 16.8 NMR Microscopy 16.9 In Vivo NMR Spectroscopy 16.10 Nonmedical Applications of MRI Chapter Summary Additional Resources

Appendix 1 ANSWERS TO REVIEW PROBLEMS

Appendix 2 PERIODIC TABLE

A Complete Introduction to MODERN NMR Spectroscopy by Roger S. Macomber

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## Contents

Jannik Bjerrum developed the first general method for the determination of stability constants of metal-ammine complexes in 1941.  The reasons why this occurred at such a late date, nearly 50 years after Alfred Werner had proposed the correct structures for coordination complexes, have been summarised by Beck and Nagypál.  The key to Bjerrum's method was the use of the then recently developed glass electrode and pH meter to determine the concentration of hydrogen ions in solution. Bjerrum recognised that the formation of a metal complex with a ligand was a kind of acid–base equilibrium: there is competition for the ligand, L, between the metal ion, M n+ , and the hydrogen ion, H + . This means that there are two simultaneous equilibria that have to be considered. In what follows electrical charges are omitted for the sake of generality. The two equilibria are

Hence by following the hydrogen ion concentration during a titration of a mixture of M and HL with base, and knowing the acid dissociation constant of HL, the stability constant for the formation of ML could be determined. Bjerrum went on to determine the stability constants for systems in which many complexes may be formed.

The following twenty years saw a veritable explosion in the number of stability constants that were determined. Relationships, such as the Irving-Williams series were discovered. The calculations were done by hand using the so-called graphical methods. The mathematics underlying the methods used in this period are summarised by Rossotti and Rossotti.  The next key development was the use of a computer program, LETAGROP   to do the calculations. This permitted the examination of systems too complicated to be evaluated by means of hand-calculations. Subsequently, computer programs capable of handling complex equilibria in general, such as SCOGS  and MINIQUAD  were developed so that today the determination of stability constants has almost become a "routine" operation. Values of thousands of stability constants can be found in two commercial databases.  

The formation of a complex between a metal ion, M, and a ligand, L, is in fact usually a substitution reaction. For example, in aqueous solutions, metal ions will be present as aquo ions, so the reaction for the formation of the first complex could be written as

The equilibrium constant for this reaction is given by

[L] should be read as "the concentration of L" and likewise for the other terms in square brackets. The expression can be greatly simplified by removing those terms which are constant. The number of water molecules attached to each metal ion is constant. In dilute solutions the concentration of water is effectively constant. The expression becomes

Following this simplification a general definition can be given, for the general equilibrium

The definition can easily be extended to include any number of reagents. The reagents need not always be a metal and a ligand but can be any species which form a complex. Stability constants defined in this way, are association constants. This can lead to some confusion as pKa values are dissociation constants. In general purpose computer programs it is customary to define all constants as association constants. The relationship between the two types of constant is given in association and dissociation constants.

### Stepwise and cumulative constants Edit

A cumulative or overall constant, given the symbol β, is the constant for the formation of a complex from reagents. For example, the cumulative constant for the formation of ML2 is given by

The stepwise constants, K1 and K2 refer to the formation of the complexes one step at a time.

A cumulative constant can always be expressed as the product of stepwise constants. Conversely, any stepwise constant can be expressed as a quotient of two or more overall constants. There is no agreed notation for stepwise constants, though a symbol such as K L
ML is sometimes found in the literature. It is good practice to specify each stability constant explicitly, as illustrated above.

### Hydrolysis products Edit

The formation of a hydroxo complex is a typical example of a hydrolysis reaction. A hydrolysis reaction is one in which a substrate reacts with water, splitting a water molecule into hydroxide and hydrogen ions. In this case the hydroxide ion then forms a complex with the substrate.

In water the concentration of hydroxide is related to the concentration of hydrogen ions by the self-ionization constant, Kw.

The expression for hydroxide concentration is substituted into the formation constant expression

In general, for the reaction

In the older literature the value of log K is usually cited for an hydrolysis constant. The log β * value is usually cited for an hydrolysed complex with the generic chemical formula MpLq(OH)r.

### Acid–base complexes Edit

A Lewis acid, A, and a Lewis base, B, can be considered to form a complex AB.

There are three major theories relating to the strength of Lewis acids and bases and the interactions between them.

1. Hard and soft acid–base theory (HSAB).  This is used mainly for qualitative purposes.
2. Drago and Wayland proposed a two-parameter equation which predicts the standard enthalpy of formation of a very large number of adducts quite accurately. −ΔH ⊖ (A − B) = EAEB + CACB. Values of the E and C parameters are available. 
3. Guttmann donor numbers: for bases the number is derived from the enthalpy of reaction of the base with antimony pentachloride in 1,2-Dichloroethane as solvent. For acids, an acceptor number is derived from the enthalpy of reaction of the acid with triphenylphosphine oxide. 

The thermodynamics of metal ion complex formation provides much significant information.  In particular it is useful in distinguishing between enthalpic and entropic effects. Enthalpic effects depend on bond strengths and entropic effects have to do with changes in the order/disorder of the solution as a whole. The chelate effect, below, is best explained in terms of thermodynamics.

An equilibrium constant is related to the standard Gibbs free energy change for the reaction

R is the gas constant and T is the absolute temperature. At 25 °C, ΔG ⊖ = (−5.708 kJ mol −1 ) ⋅ log β . Free energy is made up of an enthalpy term and an entropy term.

The standard enthalpy change can be determined by calorimetry or by using the Van 't Hoff equation, though the calorimetric method is preferable. When both the standard enthalpy change and stability constant have been determined, the standard entropy change is easily calculated from the equation above.

The fact that stepwise formation constants of complexes of the type MLn decrease in magnitude as n increases may be partly explained in terms of the entropy factor. Take the case of the formation of octahedral complexes.

For the first step m = 6, n = 1 and the ligand can go into one of 6 sites. For the second step m = 5 and the second ligand can go into one of only 5 sites. This means that there is more randomness in the first step than the second one ΔS ⊖ is more positive, so ΔG ⊖ is more negative and K 1 > K 2 >K_<2>> . The ratio of the stepwise stability constants can be calculated on this basis, but experimental ratios are not exactly the same because ΔH ⊖ is not necessarily the same for each step.  Exceptions to this rule are discussed below, in #chelate effect and #Geometrical factors.

### Ionic strength dependence Edit

The thermodynamic equilibrium constant, K ⊖ , for the equilibrium

where is the activity of the chemical species ML etc. K ⊖ is dimensionless since activity is dimensionless. Activities of the products are placed in the numerator, activities of the reactants are placed in the denominator. See activity coefficient for a derivation of this expression.

Since activity is the product of concentration and activity coefficient (γ) the definition could also be written as

where [ML] represents the concentration of ML and Γ is a quotient of activity coefficients. This expression can be generalized as

To avoid the complications involved in using activities, stability constants are determined, where possible, in a medium consisting of a solution of a background electrolyte at high ionic strength, that is, under conditions in which Γ can be assumed to be always constant.  For example, the medium might be a solution of 0.1 mol dm −3 sodium nitrate or 3 mol dm −3 sodium perchlorate. When Γ is constant it may be ignored and the general expression in theory, above, is obtained.

All published stability constant values refer to the specific ionic medium used in their determination and different values are obtained with different conditions, as illustrated for the complex CuL (L = glycinate). Furthermore, stability constant values depend on the specific electrolyte used as the value of Γ is different for different electrolytes, even at the same ionic strength. There does not need to be any chemical interaction between the species in equilibrium and the background electrolyte, but such interactions might occur in particular cases. For example, phosphates form weak complexes with alkali metals, so, when determining stability constants involving phosphates, such as ATP, the background electrolyte used will be, for example, a tetralkylammonium salt. Another example involves iron(III) which forms weak complexes with halide and other anions, but not with perchlorate ions.

When published constants refer to an ionic strength other than the one required for a particular application, they may be adjusted by means of specific ion theory (SIT) and other theories. 

### Temperature dependence Edit

All equilibrium constants vary with temperature according to the Van 't Hoff equation 

R is the gas constant and T is the thermodynamic temperature. Thus, for exothermic reactions, where the standard enthalpy change, ΔH ⊖ , is negative, K decreases with temperature, but for endothermic reactions, where ΔH ⊖ is positive, K increases with temperature.

### The chelate effect Edit

Consider the two equilibria, in aqueous solution, between the copper(II) ion, Cu 2+ and ethylenediamine (en) on the one hand and methylamine, MeNH2 on the other.

In the first reaction the bidentate ligand ethylene diamine forms a chelate complex with the copper ion. Chelation results in the formation of a five-membered ring. In the second reaction the bidentate ligand is replaced by two monodentate methylamine ligands of approximately the same donor power, meaning that the enthalpy of formation of Cu–N bonds is approximately the same in the two reactions. Under conditions of equal copper concentrations and when then concentration of methylamine is twice the concentration of ethylenediamine, the concentration of the bidentate complex will be greater than the concentration of the complex with 2 monodentate ligands. The effect increases with the number of chelate rings so the concentration of the EDTA complex, which has six chelate rings, is much higher than a corresponding complex with two monodentate nitrogen donor ligands and four monodentate carboxylate ligands. Thus, the phenomenon of the chelate effect is a firmly established empirical fact: under comparable conditions, the concentration of a chelate complex will be higher than the concentration of an analogous complex with monodentate ligands.

The thermodynamic approach to explaining the chelate effect considers the equilibrium constant for the reaction: the larger the equilibrium constant, the higher the concentration of the complex.

When the analytical concentration of methylamine is twice that of ethylenediamine and the concentration of copper is the same in both reactions, the concentration [Cu(en)] 2+ is much higher than the concentration [Cu(MeNH2)2] 2+ because β11β12.

The difference between the two stability constants is mainly due to the difference in the standard entropy change, ΔS ⊖ . In the reaction with the chelating ligand there are two particles on the left and one on the right, whereas in equation with the monodentate ligand there are three particles on the left and one on the right. This means that less entropy of disorder is lost when the chelate complex is formed than when the complex with monodentate ligands is formed. This is one of the factors contributing to the entropy difference. Other factors include solvation changes and ring formation. Some experimental data to illustrate the effect are shown in the following table. 

Equilibrium log β ΔG ⊖ /kJ mol −1 ΔH ⊖ /kJ mol −1 TΔS ⊖ /kJ mol −1
Cd 2+ + 4 MeNH2 ⇌ Cd(MeNH
2 ) 2+
4
6.55 −37.4 −57.3 19.9
Cd 2+ + 2 en ⇌ Cd(en) 2+
2
10.62 −60.67 −56.48 −4.19

These data show that the standard enthalpy changes are indeed approximately equal for the two reactions and that the main reason why the chelate complex is so much more stable is that the standard entropy term is much less unfavourable, indeed, it is favourable in this instance. In general it is difficult to account precisely for thermodynamic values in terms of changes in solution at the molecular level, but it is clear that the chelate effect is predominantly an effect of entropy. Other explanations, including that of Schwarzenbach,  are discussed in Greenwood and Earnshaw. 

The chelate effect increases as the number of chelate rings increases. For example, the complex [Ni(dien)2)] 2+ is more stable than the complex [Ni(en)3)] 2+ both complexes are octahedral with six nitrogen atoms around the nickel ion, but dien (diethylenetriamine, 1,4,7-triazaheptane) is a tridentate ligand and en is bidentate. The number of chelate rings is one less than the number of donor atoms in the ligand. EDTA (ethylenediaminetetracetic acid) has six donor atoms so it forms very strong complexes with five chelate rings. Ligands such as DTPA, which have eight donor atoms are used to form complexes with large metal ions such as lanthanide or actinide ions which usually form 8- or 9-coordinate complexes. 5-membered and 6-membered chelate rings give the most stable complexes. 4-membered rings are subject to internal strain because of the small inter-bond angle is the ring. The chelate effect is also reduced with 7- and 8- membered rings, because the larger rings are less rigid, so less entropy is lost in forming them.

#### Deprotonation of aliphatic –OH groups Edit

Removal of a proton from an aliphatic –OH group is difficult to achieve in aqueous solution because the energy required for this process is rather large. Thus, ionization of aliphatic –OH groups occurs in aqueous solution only in special circumstances. One such circumstance is found with compounds containing the H2N–C–C–OH substructure. For example, compounds containing the 2-aminoethanol substructure can form metal–chelate complexes with the deprotonated form, H2N–C–C–O − . The chelate effect supplies the extra energy needed to break the –OH bond.

An important example occurs with the molecule tris. This molecule should be used with caution as a buffering agent as it will form chelate complexes with ions such as Fe 3+ and Cu 2+ .

### The macrocyclic effect Edit

It was found that the stability of the complex of copper(II) with the macrocyclic ligand cyclam (1,4,8,11-tetraazacyclotetradecane) was much greater than expected in comparison to the stability of the complex with the corresponding open-chain amine.  This phenomenon was named "the macrocyclic effect" and it was also interpreted as an entropy effect. However, later studies suggested that both enthalpy and entropy factors were involved. 

An important difference between macrocyclic ligands and open-chain (chelating) ligands is that they have selectivity for metal ions, based on the size of the cavity into which the metal ion is inserted when a complex is formed. For example, the crown ether 18-crown-6 forms much stronger complexes with the potassium ion, K + than with the smaller sodium ion, Na + . 

In hemoglobin an iron(II) ion is complexed by a macrocyclic porphyrin ring. The article hemoglobin incorrectly states that oxyhemoglogin contains iron(III). It is now known that the iron(II) in hemoglobin is a low-spin complex, whereas in oxyhemoglobin it is a high-spin complex. The low-spin Fe 2+ ion fits snugly into the cavity of the porhyrin ring, but high-spin iron(II) is significantly larger and the iron atom is forced out of the plane of the macrocyclic ligand.  This effect contributes the ability of hemoglobin to bind oxygen reversibly under biological conditions. In Vitamin B12 a cobalt(II) ion is held in a corrin ring. Chlorophyll is a macrocyclic complex of magnesium(II).

 Cyclam Porphine, the simplest porphyrin.
 Structures of common crown ethers: 12-crown-4, 15-crown-5, 18-crown-6, dibenzo-18-crown-6, and diaza-18-crown-6

### Geometrical factors Edit

Successive stepwise formation constants Kn in a series such as MLn (n = 1, 2, . ) usually decrease as n increases. Exceptions to this rule occur when the geometry of the MLn complexes is not the same for all members of the series. The classic example is the formation of the diamminesilver(I) complex [Ag(NH3)2] + in aqueous solution.

In this case, K2 > K1. The reason for this is that, in aqueous solution, the ion written as Ag + actually exists as the four-coordinate tetrahedral aqua species [Ag(H2O)4] + . The first step is then a substitution reaction involving the displacement of a bound water molecule by ammonia forming the tetrahedral complex [Ag(NH3)(H2O)3] + . In the second step, all the aqua ligands are lost and a linear, two-coordinate product [H3N–Ag–NH3] + is formed. Examination of the thermodynamic data  shows that the difference in entropy change is the main contributor to the difference in stability constants for the two complexation reactions.

equilibrium ΔH ⊖ /kJ mol −1 ΔS ⊖ /J K −1 mol −1
Ag + + NH3 ⇌ [Ag(NH3)] + −21.4 8.66
[Ag(NH3)] + + NH3 ⇌ [Ag(NH3)2] + −35.2 −61.26

Other examples exist where the change is from octahedral to tetrahedral, as in the formation of [CoCl4] 2− from [Co(H2O)6] 2+ .

### Classification of metal ions Edit

Ahrland, Chatt and Davies proposed that metal ions could be described as class A if they formed stronger complexes with ligands whose donor atoms are nitrogen, oxygen or fluorine than with ligands whose donor atoms are phosphorus, sulfur or chlorine and class B if the reverse is true.  For example, Ni 2+ forms stronger complexes with amines than with phosphines, but Pd 2+ forms stronger complexes with phosphines than with amines. Later, Pearson proposed the theory of hard and soft acids and bases (HSAB theory).  In this classification, class A metals are hard acids and class B metals are soft acids. Some ions, such as copper(I), are classed as borderline. Hard acids form stronger complexes with hard bases than with soft bases. In general terms hard–hard interactions are predominantly electrostatic in nature whereas soft–soft interactions are predominantly covalent in nature. The HSAB theory, though useful, is only semi-quantitative. 

The hardness of a metal ion increases with oxidation state. An example of this effect is given by the fact that Fe 2+ tends to form stronger complexes with N-donor ligands than with O-donor ligands, but the opposite is true for Fe 3+ .

### Effect of ionic radius Edit

The Irving–Williams series refers to high-spin, octahedral, divalent metal ion of the first transition series. It places the stabilities of complexes in the order

This order was found to hold for a wide variety of ligands.  There are three strands to the explanation of the series.

1. The ionic radius is expected to decrease regularly for Mn 2+ to Zn 2+ . This would be the normal periodic trend and would account for the general increase in stability.
2. The crystal field stabilisation energy (CFSE) increases from zero for manganese(II) to a maximum at nickel(II). This makes the complexes increasingly stable. CFSE returns to zero for zinc(II).
3. Although the CFSE for copper(II) is less than for nickel(II), octahedral copper(II) complexes are subject to the Jahn–Teller effect which results in a complex having extra stability.

Another example of the effect of ionic radius the steady increase in stability of complexes with a given ligand along the series of trivalent lanthanide ions, an effect of the well-known lanthanide contraction.

Stability constant values are exploited in a wide variety of applications. Chelation therapy is used in the treatment of various metal-related illnesses, such as iron overload in β-thalassemia sufferers who have been given blood transfusions. The ideal ligand binds to the target metal ion and not to others, but this degree of selectivity is very hard to achieve. The synthetic drug deferiprone achieves selectivity by having two oxygen donor atoms so that it binds to Fe 3+ in preference to any of the other divalent ions that are present in the human body, such as Mg 2+ , Ca 2+ and Zn 2+ . Treatment of poisoning by ions such as Pb 2+ and Cd 2+ is much more difficult since these are both divalent ions and selectivity is harder to accomplish.  Excess copper in Wilson's disease can be removed by penicillamine or Triethylene tetramine (TETA). DTPA has been approved by the U.S. Food and Drug Administration for treatment of plutonium poisoning.

DTPA is also used as a complexing agent for gadolinium in MRI contrast enhancement. The requirement in this case is that the complex be very strong, as Gd 3+ is very toxic. The large stability constant of the octadentate ligand ensures that the concentration of free Gd 3+ is almost negligible, certainly well below toxicity threshold.  In addition the ligand occupies only 8 of the 9 coordination sites on the gadolinium ion. The ninth site is occupied by a water molecule which exchanges rapidly with the fluid surrounding it and it is this mechanism that makes the paramagnetic complex into a contrast reagent.

EDTA forms such strong complexes with most divalent cations that it finds many uses. For example, it is often present in washing powder to act as a water softener by sequestering calcium and magnesium ions.

The selectivity of macrocyclic ligands can be used as a basis for the construction of an ion selective electrode. For example, potassium selective electrodes are available that make use of the naturally occurring macrocyclic antibiotic valinomycin.

An ion-exchange resin such as chelex 100, which contains chelating ligands bound to a polymer, can be used in water softeners and in chromatographic separation techniques. In solvent extraction the formation of electrically-neutral complexes allows cations to be extracted into organic solvents. For example, in nuclear fuel reprocessing uranium(VI) and plutonium(VI) are extracted into kerosene as the complexes [MO2(TBP)2(NO3)2] (TBP = tri-n-butyl phosphate). In phase-transfer catalysis, a substance which is insoluble in an organic solvent can be made soluble by addition of a suitable ligand. For example, potassium permanganate oxidations can be achieved by adding a catalytic quantity of a crown ether and a small amount of organic solvent to the aqueous reaction mixture, so that the oxidation reaction occurs in the organic phase.

In all these examples, the ligand is chosen on the basis of the stability constants of the complexes formed. For example, TBP is used in nuclear fuel reprocessing because (among other reasons) it forms a complex strong enough for solvent extraction to take place, but weak enough that the complex can be destroyed by nitric acid to recover the uranyl cation as nitrato complexes, such as [UO2(NO3)4] 2− back in the aqueous phase.

### Supramolecular complexes Edit

Supramolecular complexes are held together by hydrogen bonding, hydrophobic forces, van der Waals forces, π-π interactions, and electrostatic effects, all of which can be described as noncovalent bonding. Applications include molecular recognition, host–guest chemistry and anion sensors.

A typical application in molecular recognition involved the determination of formation constants for complexes formed between a tripodal substituted urea molecule and various saccharides.  The study was carried out using a non-aqueous solvent and NMR chemical shift measurements. The object was to examine the selectivity with respect to the saccharides.

An example of the use of supramolecular complexes in the development of chemosensors is provided by the use of transition-metal ensembles to sense for ATP. 

Anion complexation can be achieved by encapsulating the anion in a suitable cage. Selectivity can be engineered by designing the shape of the cage. For example, dicarboxylate anions could be encapsulated in the ellipsoidal cavity in a large macrocyclic structure containing two metal ions. 

The method developed by Bjerrum is still the main method in use today, though the precision of the measurements has greatly increased. Most commonly, a solution containing the metal ion and the ligand in a medium of high ionic strength is first acidified to the point where the ligand is fully protonated. This solution is then titrated, often by means of a computer-controlled auto-titrator, with a solution of CO2-free base. The concentration, or activity, of the hydrogen ion is monitored by means of a glass electrode. The data set used for the calculation has three components: a statement defining the nature of the chemical species that will be present, called the model of the system, details concerning the concentrations of the reagents used in the titration, and finally the experimental measurements in the form of titre and pH (or emf) pairs.

It is not always possible to use a glass electrode. If that is the case, the titration can be monitored by other types of measurement. Ultraviolet–visible spectroscopy, fluorescence spectroscopy and NMR spectroscopy are the most commonly used alternatives. Current practice is to take absorbance or fluorescence measurements at a range of wavelengths and to fit these data simultaneously. Various NMR chemical shifts can also be fitted together.

The chemical model will include values of the protonation constants of the ligand, which will have been determined in separate experiments, a value for log Kw and estimates of the unknown stability constants of the complexes formed. These estimates are necessary because the calculation uses a non-linear least-squares algorithm. The estimates are usually obtained by reference to a chemically similar system. The stability constant databases   can be very useful in finding published stability constant values for related complexes.

In some simple cases the calculations can be done in a spreadsheet.  Otherwise, the calculations are performed with the aid of a general-purpose computer programs. The most frequently used programs are:

• Potentiometric and/or spectrophotometric data: PSEQUAD 
• Potentiometric data: HYPERQUAD,  BEST, ReactLab pH PRO
• Spectrophotometric data: HypSpec, SQUAD,  SPECFIT, ReactLab EQUILIBRIA. 
• NMR data HypNMR, WINEQNMR2

In biochemistry, formation constants of adducts may be obtained from Isothermal titration calorimetry (ITC) measurements. This technique yields both the stability constant and the standard enthalpy change for the equilibrium.  It is mostly limited, by availability of software, to complexes of 1:1 stoichiometry.

The following references are for critical reviews of published stability constants for various classes of ligands. All these reviews are published by IUPAC and the full text is available, free of charge, in pdf format.

## Abstract

A new 2-(9-anthrylmethylamino)ethyl-appended cyclen, L 3 (1-(2-(9-anthrylmethylamino)ethyl)-1,4,7,10-tetraazacyclododecane) (cyclen = 1,4,7,10-tetraazacyclododecane), was synthesized and characterized for a new Zn 2+ chelation-enhanced fluorophore, in comparison with previously reported 9-anthrylmethylcyclen L 1 (1-(9-anthrylmethyl)-1,4,7,10-tetraazacyclododecane) and dansylamide cyclen L 2 . L 3 showed protonation constants log Kai of 10.57 ± 0.02, 9.10 ± 0.02, 7.15 ± 0.02, <2, and <2. The log Ka3 value of 7.15 was assigned to the pendant 2-(9-anthrylmethylamino)ethyl on the basis of the pH-dependent 1 H NMR and fluorescence spectroscopic measurements. The potentiometric pH titration study indicated extremely stable 1:1 Zn 2+ −L 3 complexation with a stability constant log Ks(ZnL 3 ) (where Ks(ZnL 3 ) = [ZnL 3 ]/[Zn 2+ ][L 3 ] (M - 1 )) of 17.6 at 25 °C with I = 0.1 (NaNO3), which is translated into the much smaller apparent dissociation constant Kd (=[Zn 2+ ]free[L 3 ]free/[ZnL 3 ]) of 2 × 10 - 11 M with respect to 5 × 10 - 8 M for L 1 at pH 7.4. The quantum yield (Φ = 0.14) in the fluorescent emission of L 3 increased to Φ = 0.44 upon complexation with zinc(II) ion at pH 7.4 (excitation at 368 nm). The fluorescence of 5 μM L 3 at pH 7.4 linearly increased with a 0.1−5 μM concentration of zinc(II). By comparison, the fluorescent emission of the free ligand L 1 decreased upon binding to Zn 2+ (from Φ = 0.27 to Φ = 0.19) at pH 7.4 (excitation at 368 nm). The Zn 2+ complexation with L 3 occurred more rapidly (the second-order rate constant k2 is 4.6 × 10 2 M - 1 s - 1 ) at pH 7.4 than that with L 1 (k2 = 5.6 × 10 M - 1 s - 1 ) and L 2 (k2 = 1.4 × 10 2 M - 1 s - 1 ). With an additionally inserted ethylamine in the pendant group, the macrocyclic ligand L 3 is a more effective and practical zinc(II) fluorophore than L 1 .

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### Extended Data Fig. 1 Screen to identify modulators of LYPLAL1 activity.

(a) ABPP-based fluorescence polarization assay (Fluopol-ABPP) amenable for high-throughput screening. Purified mLYPLAL1 is incubated with compounds prior to addition of the FP-rhodamine (FP-Rh) probe. FP-biotin probe (a non-fluorescent FP probe, 25 μM) serves as a control for inhibition, while the PPARγ ligand rosiglitazone (25 μM) is used as an inactive control. Data are shown as mean ± s.d., n = 12, where n represents independent samples. (b) A 16,000-compound screen (single point, 10 μM) identified hits that reduced the fluorescence polarization signal of FP-rhodamine labeling of mLYPLAL1 (putative inhibitors) and some that increased it (potential activators). Percent mLYPLAL1 activity was calculated relative to DMSO (100% activity) and FP-biotin (0% activity) wells. 4 is shown as a red dot.

### Extended Data Fig. 2 Validation of primary HTS hits.

Candidate LYPLAL1 inhibitors and activators were analyzed by gel-based ABPP. Hit picks (10 μM) were incubated with 100 nM of purified mLYPLAL1 for 1.5 h at 37 °C prior to the addition of FP-rhodamine probe (1 μM). After 1 h at 37 °C, reactions were quenched, separated by SDS–PAGE and in-gel fluorescence scanned. LYPLAL1 activity was calculated relative to DMSO (100%). Designations above lanes correspond to compound location on the plate. Red asterisk denotes 4. Representative results from two independent experiments similar results were obtained in both experiments. Uncropped gels are shown in Supplementary Fig. 6.

### Extended Data Fig. 3 Derivatized thermal melt curves for compound-treated mLYPLAL1.

Temperature-dependent fluorescence shifts in purified mLYPLAL1 incubated with increasing concentrations of 4 and 12. Increased protein flexibility is noted upon compound interaction. In contrast, an inactive compound, 80, does not shift the derivatized thermal melt curve of mLYPLAL1. Representative results from three independent experiments similar results were obtained in all experiments.

### Extended Data Fig. 4 Evolution of the global Root Mean Square Deviation (RMSD) during Molecular Dynamics (MD) trajectories.

(a) WT hLYPLAL1 and R80 mutants, and (b) WT hLYPLAL1 with chlorobenzene as cosolvent. The RMSD was calculated considering only the backbone atoms and using the starting frames as the reference structures. Results from 10 independent simulations.

## Problem:

Starting with the change in free energy at constant temperature: D G° = D H° - T D S°, and with the relation between D G and equilibrium constant, K: D G° = -RT lnK, derive a linear equation that expresses lnK as a function of 1/T (a linear equation is of the form y = mx + b).

(b) The equilbrium constants for the conversion of 3-phosphoglycerate to 2-phosphoglycerate at pH 7 is provided:

Calculate D G° at 25°C.

(c) What is the concentration of each isomer of phosphoglycerate if 0.150 M phosphoglycerate reaches equilbrium at 25°C.

## Watch the video: Chemical Equilibrium Lab (September 2022).

1. Ata'halne'

the Response, a sign of the wit)

2. Jaran

go crazy

3. Garai

I beg your pardon, this does not suit me. Who else can breathe?

4. Peredur

It is simply matchless :)