and the effect of nuclear binding energy on mass was too small for nineteenth-century chemists to detect.8 * It is often convenient to think of the electrons orbiting the nucleus much as the planets orbit the Sun. This analogy has its limitations. The electron's position cannot be precisely specified as can a planet's. In quantum mechanics, the Schrödinger wave function, ψ (or more precisely, ψ2) determines the probability of the electron being located in a given region about the atom. As an example of failure of the classical physical description of the atom, consider an electron in the 1s orbital. Both quantum number specifying angular momentum, l and m, are equal to 0, and hence the electron has 0 angular momentum, and therefore cannot be in an orbit in the classical sense.
Chapter 2 Energy, entropy, and fundamental thermodynamic concepts
2.1 THE THERMODYNAMIC PERSPECTIVE
We defined geochemistry as the application of chemical knowledge and techniques to solve geologic problems. It is appropriate, then, to begin our study of geochemistry with a review of physical chemistry. Our initial focus will be on thermodynamics. Strictly defined, thermodynamics is the study of energy and its transformations. Chemical reactions and changes of states of matter inevitably involve energy changes. By using thermodynamics to follow the energy, we will find that we can predict the outcome of chemical reactions, and hence the state of matter in the Earth. In principle, at least, we can use thermodynamics to predict at what temperature a rock will melt and the composition of that melt, and we can predict the sequence of minerals that will crystallize to form an igneous rock from the melt. We can predict the new minerals that will form when that igneous rock undergoes metamorphism, and we can predict the minerals and the composition of the solution that forms when that metamorphic rock weathers and the nature of minerals that will ultimately precipitate from that solution. Thus, thermodynamics allows us to understand (in the sense that we defined understanding in Chapter 1) a great variety of geologic processes.
Thermodynamics embodies a macroscopic viewpoint, that is, it concerns itself with the properties of a system, such as temperature, volume, and heat capacity, and it does not concern itself with how these properties are reflected in the internal arrangement of atoms. The microscopic viewpoint, which is concerned with transformations on the atomic and subatomic levels, is the realm of statistical mechanics and quantum mechanics. In our treatment, we will focus mainly on the macroscopic (thermodynamic) viewpoint, but we will occasionally consider the microscopic (statistical mechanical) viewpoint when our understanding can be enhanced by doing so. More detailed treatments of geochemical thermodynamics can be found in Anderson and Crerar (1993), Nordstrom and Munoz (1986), and Fletcher (1993).
In principle, thermodynamics is only usefully applied to systems at equilibrium. If an equilibrium system is perturbed, thermodynamics can predict the new equilibrium state, but cannot predict how, how fast, or indeed whether the equilibrium state will be achieved. (The field of irreversible thermodynamics, which we will not treat in this book, attempts to apply thermodynamics to nonequilibrium states. However, we will see in Chapter 5 that thermodynamics, through the principle of detailed balancing and transition state theory, can help us predict reaction rates.)
Kinetics is the study of rates and mechanisms of reaction. Whereas thermodynamics is concerned with the ultimate equilibrium state and not concerned with the pathway to equilibrium, kinetics concerns itself with the pathway to equilibrium. Very often, equilibrium in the Earth is not achieved, or achieved only very slowly, which naturally limits the usefulness of thermodynamics. Kinetics helps us to understand how equilibrium is achieved and why it is occasionally not achieved. Thus, these two fields are closely related, and together form the basis of much of geochemistry. We will treat kinetics in Chapter 5.
2.2 THERMODYNAMIC SYSTEMS AND EQUILIBRIUM
We now need to define a few terms. We begin with the term system, which we have already used. A thermodynamic system is simply that part of the universe we are considering. Everything else is referred to as the surroundings. A thermodynamic system is defined at the convenience of the observer in a manner so that thermodynamics may be applied. While we are free to choose the boundaries of a system, our choice must nevertheless be a careful one as the success or failure of thermodynamics in describing the system will depend on how we have defined its boundaries. Thermodynamics often allows us this sort of freedom of definition. This can certainly be frustrating, particularly for someone exposed to thermodynamics for the first time (and often even the second or third time). But this freedom allows us to apply thermodynamics successfully to a much broader range of problems than otherwise.
A system may be related to its environment in a number of ways. An isolated system can exchange neither energy (heat or work) nor matter with its surroundings. A truly isolated system does not exist in nature, so this is strictly a theoretical concept. An adiabatic system can exchange energy in the form of work, but not heat or matter, with its surroundings, that is to say it has thermally insulating boundaries. Though a truly adiabatic system is probably also a fiction, heat transport in many geologic systems is sufficiently slow that they may be considered adiabatic. Closed systems may exchange energy, in the form of both heat and work, with their surrounding but cannot exchange matter. An open system may exchange both matter and energy across it boundaries. The various possible relationships of a system to its environment are illustrated in Figure 2.1.
Figure 2.1 Systems in relation to their surroundings. The ball represents mass exchange, the arrow represents energy exchange.
Depending on how they behave over time, systems are said to be either in transient or time-invariant states. Transient states are those that change with time. Time-independent states may be either static or dynamic. A dynamic time-independent state, or steady-state, is one whose thermodynamic and chemical characteristics do not change with time despite internal changes or exchanges of mass and energy with its surroundings. As we will see, the ocean is a good example of a steady-state system. Despite a constant influx of water and salts from rivers and loss of salts and water to sediments and the atmosphere, its composition does not change with time (at least on geologically short time-scales). Thus, a steady-state system may also be an open system. We could define a static system as one in which nothing is happening. For example, an igneous rock or a flask of seawater (or some other solution) is static in the macroscopic perspective. From the statistical mechanical viewpoint, however, there is a constant reshuffling of atoms and electrons, but with no net changes. Thus, static states are generally also dynamic states when viewed on a sufficiently fine scale.
Let's now consider one of the most important concepts in physical chemistry, that of equilibrium. One of the characteristics of the equilibrium state is that it is static from a macroscopic perspective, that is, it does not change measurably with time. Thus, the equilibrium state is always time-invariant. However, while a reaction A→B may appear to have reached static equilibrium on a macroscopic scale, this reaction may still proceed on a microscopic scale but with the rate of reaction A→B being the same as that of B→A. Indeed, as we shall see in Chapter 5, a kinetic definition of equilibrium is that the forward and reverse
