Physical Chemistry from a Different Angle

In this first chapter, we will be introduced briefly to the field of matter dynamics. This field is concerned in the most general sense with the transformations of substances and the physical principles underlying the changes of matter. As a consequence, we have to review some important basic concepts necessary for describing such processes like substance, content formula and amount of substance, as well as homogeneous and heterogeneous mixture and the corresponding measures of composition. But in this context, the physical state of a sample is also of great importance. Therefore, we will learn how we can characterize it qualitatively by the different states of aggregation as well as quantitatively by state variables. In the last section, a classification of transformations of substances into chemical reactions, phase transitions, and redistribution processes as well as their description with the help of conversion formulas is given. The temporal course of such a transformation can be expressed by the extent of conversion ξ. Additionally, we will take a short look at the basic problem of measuring quantities and metricizing concepts in this chapter.

Energy is a quantity that not only plays a dominant role in the most diverse areas of the sciences, technology, and economy, but is omnipresent in the everyday world around us. For example, we pay for it with every bill for electricity, gas, and heating oil that arrives at our homes. But we are also confronted more and more with questions about how we can save energy in order to cover our current and future demands. At the beginning of the chapter, the conventional indirect way of defining energy is briefly presented. A much simpler way to introduce this quantity is characterizing it by its typical and easily observable properties using everyday experiences. This phenomenological description may be supported by a direct measuring procedure, a method normally used for the quantification of basic concepts such as length, time, or mass. Subsequently, the law of conservation of energy and different manifestations of energy like that in a stretched spring, a body in motion, etc., are discussed. In this context, important quantities such as pressure and momentum are introduced via the concept of energy.

The chemical potential μ is used as a measure of the general tendency of a substance to transform. Only a few properties are necessary for a complete phenomenological description of this new quantity. They are easy to grasp and can be illustrated by everyday examples. It is possible to derive quantitative scales of μ values (initially at standard conditions) by using these properties, and after choosing a convenient reference level. A first application in chemistry is predicting whether or not reactions are possible by comparing the sum of potentials of the initial and the final states. This is illustrated by numerous experimental examples. The quantitative description can be simplified by defining a “chemical drive” A as the difference of these sums. In this context, a positive value of A means that the reaction proceeds spontaneously in the forward direction. As a last point, a direct and an indirect measuring procedure for the chemical potential are proposed.

The concept of mass action, its relation to the concentration dependence of the chemical potential (mass action equation), and subsequently, its relation to the chemical drive are discussed. An important application in the case of chemical equilibria is the derivation of the “mass action law.” But we also examine some further consequences such as the solubility of ionic solids and gases in liquids, preferably in water. The former leads us to the concept of solubility product, the latter to Henry’s law. With the help of Henry’s law, we can, for example, estimate the oxygen content in bodies of water, a parameter of prime importance for biological processes. Another relevant application results in Nernst’s distribution law which governs the distribution of a solute between two immiscible liquids. Distribution equilibria play a significant role in separating the substances in a mixture by the process of extraction or by partition chromatography. The last section of this chapter illustrates how the concept of mass action can be visualized with the help of potential diagrams.

The concept of mass action can be applied to any transformation of substances. In the case of matter dynamics, it does not matter how we imagine the process in question working at the molecular level: Whether it is by formation or cleavage of chemical bonds, rearranging crystal lattices, migration of particles, transfer of electrons or whole groups of atoms from one type of particle onto the other, etc. In this chapter we will concentrate upon one important example for chemical transformations, namely acid–base reactions, in order to demonstrate that the chemical potential is well suited to describing very specialized and differentiated fields. Acid–base reactions are central to chemistry and its applications; for their quantitative description we introduce the “proton potential” μ _p as a measure of the strength of an acid–base pair. The level equation and the protonation equation are used to describe the behavior of weak acid–base pairs. Subsequently, one important application for acid–base equilibria, the analytic method called acid–base titration, is presented. Finally, the mode of reaction of buffers and indicators is discussed. Buffers play also a significant role in living organisms because even small shifts in the proton potential can there result in disease and death.

Transformations of substances like chemical reactions, phase transitions, distribution in space, etc., are often accompanied by very striking side effects, such as glowing and flashing, fizzling and cracking, bubbling and rising of smoke. These side effects which make chemistry so fascinating are primarily based upon changes of volume that can cause violent explosions and implosions, exchange and generation of entropy, which is responsible for effects like glowing and heating up, and energy exchange that we use in muscles, motors, and batteries. The goal of this chapter is to understand and quantitatively describe these phenomena, and to sensibly make use of them. For this purpose, the so-called partial molar properties such as the (partial) molar volume or the (partial) molar entropy of a dissolved substance are introduced. For describing the changes of volume and entropy associated with transformations, we will use the quantities molar reaction volume and molar reaction entropy. The special role of entropy makes a further differentiation into latent, generated, and exchanged reaction entropy necessary. We will also learn how the chemical drive of a reaction, the corresponding exchange of energy, and eventually the generated entropy are interrelated. In closing, this relationship will be used for determining the chemical drive with the help of a calorimeter.

If one changes from dilute (ideal) gases to real gases with higher density, the interaction between the particles and the phenomenon of condensation cannot be neglected any longer. The consideration of such effects results in the van der Waals equation, a modification of the general gas law. A closer look at the process of condensation leads us to the critical phenomena, meaning the unusual physical properties displayed by substances near their critical points. If we are interested to know how the phase transition liquid ⇄ gaseous can be influenced by factors such as temperature and pressure, we can use the T and p dependence of the chemical potential for calculating the boiling pressure curve (vapor pressure curve) of a given pure substance. This curve illustrates how the vapor pressure of the substance varies with temperature and is an example of a so-called phase boundary. The other phase transitions can also be represented in a p(T) diagram in the form of phase boundaries, producing a complete phase diagram. Such a diagram is a kind of “map” which shows the conditions of temperature and pressure at which a certain phase is most stable and illustrates the ranges of existence of stable phases.

So far, the discussion of the chemical potential has concentrated primarily on chemical reactions and phase transitions. But another property of substances is also of great importance: their tendency to spread out or disperse in space. The phenomenon of diffusion will be explained in this context. The subject area of this chapter also includes the discussion of the effect of a small amount of solute on certain properties of a solution. The properties we have in mind are the lowering of vapor pressure of the solvent, the elevation of its boiling point, the lowering of its freezing point, and last but not least the origin of osmotic pressure. These phenomena are found everywhere, in households and in nature but also in engineering. In everyday life, a prime example for freezing-point depression is the melting effect of road salt. Or have you ever asked yourself why juice is drawn out of sugared strawberries but cherries swell up and burst after a long rain? Then have a look at Sect. 12.4 dealing with osmosis. For a quantitative discussion of all these phenomena, we first have to learn about indirect mass action and the corresponding colligative lowering of chemical potential.

In chemistry but also in everyday life, we are very often confronted with mixtures, be they homogeneous or heterogeneous. Think for example of hard liquor, basically a homogeneous mixture of ethanol and water, but also of fog, a heterogeneous mixture of air and minuscule water droplets. First, we concentrate on mixtures made up of two liquid components. We discuss the behavior of the chemical potential of one component in such mixtures and the reason for spontaneous mixing or demixing. For an adequate quantitative description, the concept of chemical potential has to be extended on substances in real solutions by introducing an extra potential \( \overset{+}{\mu }. \)

The average chemical potential of a mixture depends not only upon the composition but also upon the temperature (and pressure). These dependencies and the fact that the phase with the lowest chemical potential at a given temperature (or pressure) will be stable can be used to construct the phase diagrams of different mixtures. First, we will discuss the temperature–composition diagrams of two liquid phases. With the help of these diagrams, we can judge under which conditions the two liquids are mutually miscible and under which they are not; the diagrams are therefore also called miscibility diagrams. Liquid–solid phase diagrams are used to identify the regions of temperature and composition at which solids and liquids exist in a two-component system. Such diagrams are of great commercial and industrial relevance; they play an important role in metallurgy but also in the manufacture of ceramics and semiconductors. In the last section, the phase diagrams of binary mixtures of two volatile components are discussed. This kind of diagram is important for understanding distillation, one of the most significant processes used in chemical laboratories and industry for separating liquid mixtures. It has been in use since ancient times to extract essential oils such as attar of roses. An important industrial application is distilling of petroleum in oil refineries that produce the heavy and light gasoline used to fuel engines.

In this chapter, we will discuss how the chemical and physical properties of substances at interfaces differ from those in the bulk. For quantitative description, quantities like surface tension and surface energy have to be introduced. With the help of these quantities, phenomena known from everyday life like the lotus effect can be explained. However, perhaps you are more interested to learn how detergents clean? Then have a look at Sect. 16.​3 which deals with the adsorption on liquid surfaces. The next section covers the adsorption on solid surfaces and the variation of the extent of coverage with pressure or concentration of the substance to be adsorbed. Langmuir’s isotherm, the simplest description of such an adsorption process, is deduced by kinetic interpretation of the adsorption equilibrium. Alternatively, it can be derived by introducing the chemical potential of free and occupied sites and considering the equilibrium condition. In the last part of the chapter, some important applications such as surface measurement and adsorption chromatography are discussed.

The branch of matter dynamics called chemical kinetics will be the topic of the next four chapters. Chemical kinetics is concerned with the temporal course of chemical reactions, meaning one investigates how fast the reactants are consumed or the products are formed. The goal of such investigations is to provide the means for predicting the rate of processes and to find the influencing factors that promote a desired reaction or inhibit an undesired one. In this introductory chapter we will first get to know the fundamental quantities conversion rate and rate density as well as different methods for measuring them in slow and fast reactions. In the last part of the chapter, it will be shown how the dependence of the rate density on the concentrations of reactants (and products) can be summarized by mathematical expressions called rate laws. Subsequently, the relatively simple rate laws of different types of reactions taking place in only one single step will be discussed.

Kinetic measurements show that the simple rate laws known from the last chapter are often not sufficient for a correct description of the temporal course of a reaction or the composition of a reaction mixture. Many reactions take place by mechanisms that involve several elementary steps. Three fundamental types of composite reactions are discussed in this chapter: opposing or equilibrium reactions, parallel reactions, and consecutive reactions. Composite reactions not only play a large role in industrial applications (e.g., heterogeneous catalysis) but are also very important in nature (e.g., enzyme reactions).

Everyday experience demonstrates that, most of the time, the rate of a chemical reaction will increase with a rise in temperature. Food, for example, will spoil outside on a hot summer day much faster than it would in a refrigerator. A simple but remarkably accurate relationship for the temperature dependence of reaction rates was empirically found by the Swedish chemist Svante Arrhenius in 1889. The interpretation of the parameters in the Arrhenius equation leads to the development of the idea that when reactants convert into products, they must go through an activated state that requires a characteristic energy. This was the basis of two of the most important theories of reaction rates, collision theory and transition state theory. Collision theory, which only suffices for simple gas phase reactions, essentially views reactants as if they were particles with a certain kinetic energy. Reactions can only occur if two molecules collide with a minimum energy necessary for rearranging the bonds. Matter dynamic considerations play no role here. In transition state theory, a more comprehensive theory that can, in principle, be applied to every possible type of reaction, the rate coefficient is expressed in terms of a difference in chemical potentials between the reactants and a kind of “transition substance” (“ensemble” of all activated complexes), a so-called “potential barrier.” For a deeper understanding, the transition state can be interpreted on a molecular level with the help of potential energy surfaces and the “motion” of molecules through these surfaces.

Reactions can not only be accelerated by raising their temperatures, but also by addition of small amounts of a substance, a so-called catalyst, which is not consumed during the process. An everyday example of a catalyst is the exhaust gas catalytic converter in motor vehicles with gasoline engines, which eliminates combustion pollutants by accelerating subsequent reactions. But why do reactions proceed faster with a catalyst than without a catalyst? The catalyst lowers the reaction resistance by opening up more easily accessible bypasses with smaller activation thresholds. Enzymes, the vitally essential biological catalysts, and the kinetics of their reactions with structurally suitable substrates are discussed in detail. An enzyme can be compared to a lock into which only the proper key (substrate) can fit (key–lock principle). This is where the “key” for the exceedingly high substrate specificity of an enzyme lies. The chapter ends with the discussion of the technically important heterogeneous catalysis.

Diffusion can play an important role for the kinetics of chemical reactions in solutions. We use that as an opportunity to discuss this process of molecular motion more closely. The migration velocity is determined by a gradient of chemical potential and therefore eventually by a concentration gradient. This leads us to the quantitative description with the help of Fick’s law of diffusion. But not only matter can be transported from one place to another but also some other properties such as entropy or momentum. Entropy conduction is determined by the migration of entropy down a temperature gradient and viscosity by a migration of linear momentum down a velocity gradient. In order to carve out the commonalities and differences of the transport phenomena discussed, they are summed up in the last section and compared with the transport of electric charge, because the latter is the best known of these phenomena.

Initially, the terms Galvani potential for the electric potential in the bulk of a phase, electrochemical potential \( \tilde{\mu} \), and electron potential μ _e are introduced to characterize processes in which charge-carrying species are involved. The electrochemical potentials can be used to determine the Galvani potential difference between two phases in equilibrium, as an especially simple example between two different metals. The formation of an electric double layer at the interface of both metals as well as the corresponding Galvani potential difference, the so-called contact voltage, will be presented. More important for practical use like that in galvanic cells is, however, the Galvani potential difference between a metal electrode and an electrolyte solution. The electrochemical potentials and their possible composition dependence are used to describe the underlying charge transfer reaction and to derive Nernst’s equation. This type of charge transfer reaction can be regarded from a formal point of view as a special kind of a so-called redox reaction. Redox reactions in which electrons are transferred from one species to another are together with the proton transfer typical of acid–base reactions central to chemistry and its applications. Subsequently, different types of half-cells such as redox electrodes, gas electrodes, as well as film electrodes and the corresponding Galvani potential differences are discussed. The Galvani potential differences across liquid–liquid interfaces and membranes will be the topic of the last section. Such membrane voltages described by Donnan’s equation play an important role in biological membranes, for example, for information transfer in nerve cells.

In the last chapter, we learned a lot about Galvani potential differences across different individual interfaces and the usefulness of these potential differences, but we did not get to know how they can be measured. The problem is that it is impossible to measure the Galvani potential difference across a single interface in a half-cell directly, because an electrolyte’s contact to the conductors of an electric measuring device requires a second electrode and this produces a new interface with an additional Galvani potential difference. The way out of the dilemma is the use of always the same reference half-cell, the standard hydrogen electrode, so that the measured voltage of the galvanic cell is only determined by the measuring half-cell. In this way, we obtain the so-called redox potential of a half-cell which represents, just like the electron potential, a measure of the strengths of reducing or oxidizing agents. The redox potentials under standard conditions are often compiled according to their values in a table, the electrochemical series. The combination of two arbitrary half-cells results in a galvanic cell. The reversible cell voltage of such a cell, meaning the cell voltage in equilibrium, can be described by Nernst’s equation and used to predict the chemical drive, the equilibrium constant, and other thermodynamic properties of chemical reactions. Subsequently, some technically important galvanic cells will be discussed, which yield usable energy due to the spontaneous chemical reactions running inside them. In closing, the technique of potentiometry and the corresponding potentiometric titration is presented. This electroanalytical method uses the concentration dependence of the reversible cell voltage for quantitative analysis of ions.

In addition to the terms discussed so far a number of other quantities and functions are used in thermodynamics without which textbooks that follow the conventional concept cannot manage. Because knowing these additional terms is essential for understanding traditional textbooks and the corresponding data collections, we will deal with the most important of them in this chapter and establish the relations to the concept chosen in this book. The major subsidiary terms are the four energetic quantities inner energy U, enthalpy H, Helmholtz energy A, and Gibbs energy G. The same quantity can serve different purposes depending on the variables chosen. The function U(S, V, …) characterizes the system under consideration. It is almost never explicitly stated (the abstractness of the variable S can be considered the underlying cause for this), but its differential plays a central role for all derivations. The functions U(T, V, …) and H(T, p, …) serve to describe the heat exchanged between system and surroundings under different experimental conditions (the first at constant volume, the second at constant pressure). The functions A(T, V, …) and G(T, p, …) play a similar role. Both are used to calculate the energy released during the considered process. This enables us to predict whether or not the process may run spontaneously. In the last section, we will discuss quantities such as activity, fugacity, etc. These quantities are used for describing deviations from what is considered ideal behavior of dissolved substances and gases.