are equal.
The equilibrium state is entirely independent of the manner or pathway in which equilibrium is achieved. Indeed, once equilibrium is achieved, no information about previous states of the system can be recovered from its thermodynamic properties. Thus a flask of CO2 produced by combustion of graphite cannot be distinguished from CO2 produced by combustion of diamond. In achieving a new equilibrium state, all records of past states are destroyed.
Time-invariance is a necessary but not sufficient condition for equilibrium. Many systems exist in metastable states. Diamond at the surface of the Earth is not in an equilibrium state, despite its time-invariance on geologic time-scales. Carbon exists in this metastable state because of kinetic barriers that inhibit transformation to graphite, the equilibrium state of pure carbon at the Earth's surface. Overcoming these kinetic barriers generally requires energy. If diamond is heated sufficiently, it will transform to graphite, or in the presence of sufficient oxygen, to CO2.
The concept of equilibrium versus metastable or unstable (transient) states is illustrated in Figure 2.2 by a ball on a hill. The equilibrium state is when the ball is in the valley at the bottom of the hill, because its gravitational potential energy is minimized in this position. When the ball is on a slope, it is in an unstable, or transient, state and will tend to roll down the hill. However, it may also become trapped in small depressions on the side of the hill, which represent metastable states. The small hill bordering the depression represents a kinetic barrier. This kinetic barrier can only be overcome when the ball acquires enough energy to roll up and over it. Lacking that energy, it will exist in the metastable state indefinitely.
Figure 2.2 States of a system.
In Figure 2.2, the ball is at equilibrium when its (gravitational) potential energy is lowest (i.e., at the bottom of the hill). This is a good definition of equilibrium in this system, but as we will soon see, it is not adequate in all cases. A more general statement would be to say that the equilibrium state is the one toward which a system will change in the absence of constraints. So in this case, if we plane down the bump (remove a constraint), the ball rolls to the bottom of the hill. At the end of this chapter, we will be able to produce a thermodynamic definition of equilibrium based on the Gibbs free energy. We will find that, for a given pressure and temperature, the chemical equilibrium state occurs when the Gibbs free energy of the system is lowest.
Natural processes proceeding at a finite rate are irreversible under a given set of conditions; that is, they will only proceed in one direction. Here we encounter a problem in the application of thermodynamics: if a reaction is proceeding, then the system is out of equilibrium and thermodynamic analysis cannot be applied. This is one of the first of many paradoxes in thermodynamics. This limitation might at first seem fatal, but we get around it by imagining a comparable reversible reaction. Reversibility and local equilibrium are concepts that allow us to “cheat” and apply thermodynamics to nonequilibrium situations. A “reversible” process is an idealized one where the reaction proceeds in sufficiently small steps that it is in equilibrium at any given time (thus allowing the application of thermodynamics).
Local equilibrium embodies the concept that in a closed or open system, which may not be at equilibrium on the whole, small volumes of the system may nonetheless be at equilibrium. There are many such examples. In the example of a mineral crystallizing from magma, only the rim of the crystal may be in equilibrium with the melt. Information about the system may nevertheless be derived from the relationship of this rim to the surrounding magma. Local equilibrium is in a sense the spatial equivalent to the temporal concept of reversibility and allows the application of thermodynamics to real systems, which are invariably nonequilibrium at large scales. Both local equilibrium and reversibility are examples of simplifying assumptions that allow us to treat complex situations. In making such assumptions, some accuracy in the answer may be lost. Knowing when and how to simplify a problem is an important scientific skill.
2.2.1 Fundamental thermodynamic variables
In the next two chapters we will be using a number of variables, or properties, to describe thermodynamic systems. Some of these will be quite familiar to you, others less so. Volume, pressure, energy, heat, work, entropy, and temperature are the most fundamental variables in thermodynamics. As all other thermodynamic variables are derived from them, it is worth our while to consider a few of these properties.
Energy is the capacity to produce change. It is a fundamental property of any system, and it should be familiar from physics. By choosing a suitable reference frame, we can define an absolute energy scale. However, it is changes in energy that are generally of interest to us rather than absolute amounts. Work and heat are two of many forms of energy. Heat, or thermal energy, results from random motions of molecules or atoms in a substance and is closely related to kinetic energy. Work is done by moving a mass, M, through some distance, x = X, against a force F:
(2.1)
where w is work and force is defined as mass times acceleration:
(2.2)
(the minus sign is there because of the convention that work done on a system is positive, work done by a system is negative). This is, of course, Newton's first law. In chemical thermodynamics, pressure–volume work is usually of more interest. Pressure is defined as force per unit area:
(2.3)
Since volume is area times distance, we can substitute eqn. 2.3 and dV = Adx into eqn. 2.1 and obtain:
Thus, work is also done as a result of a volume change in the presence of pressure.
Potential energy is energy possessed by a body by virtue of its position in a force field, such as the gravitational field of the Earth, or an electric field. Chemical energy will be of most interest to us in this book. Chemical energy is a form of potential energy stored in chemical bonds of a substance. Chemical energy arises from the forces involved in the interaction between atoms and electrons. Internal energy, which we denote with the symbol U, is the sum of the potential energy arising from these forces as well as the kinetic energy of the atoms and molecules (i.e., thermal energy) in a substance. It is internal energy that will be of most interest to us.
We will discuss all these fundamental variables in more detail in the next few sections.
2.2.2 Properties of state
Properties or variables of a system that depend only on the present state of the system, and not on the manner in which that state was achieved, are called variables of state or state functions. Extensive properties depend on total size of the system. Mass, volume, and energy are all extensive properties.
