Introduction to Thermodynamics

Because we will be using a great deal of thermodynamics in discussing the generation and evolution of basaltic melts in the ocean crust, it seems like a good idea to review some of the basic thermodynamic relationships and definitions.

Definitions:

A System is any portion of the universe isolated for the purpose of considering experimental or natural changes in conditions within it. Examples include a beaker, and experimental charge, or an ocean basin. A closed system is one that changes only by receiving energy from the outside environment or by yielding energy to it. An open system may exchange both matter and energy with it surroundings.

Equilibrium within a system is achieved when the system reaches its lowest energy state consistent with the imposed conditions. If temperature and pressure are specified, the equilibrium configuration possesses the lowest possible Gibbs Free energy (G). Stability is the condition of equilibrium (minimum G).

Phases are physically separate regions of homogeneous chemistry. Possible phases include a liquid phase (multiple liquid phases if the liquids are immiscible), a gas phase, and multiple solid phases. Solid phases may exhibit solid solution, i.e., have a range of compositional variation.

Components of a phase consist of the smallest number of chemically distinct substances needed to specify the bulk composition of the phase.

Phase diagrams depict phase relationships within multi-component systems.

Critical Relationships

Zero'th law of Thermodynamics

Two bodies which are in thermal equilibrium with a third body are in thermal equilibrium with each other.

First law of Thermodynamics

The total energy of an isolated system must remain constant, although there may be changes from one form of energy to another. If we designate the total or internal energy of an isolated or closed system as E, heat as Q and work as W , then the circular integral involving no change in net internal energy is:

If the system is closed to the input of matter, but not of energy, then

Mechanical work is, of course only the product of force times distance, and force is pressure P time surface area, so mechanical work is simply the product of pressure times surface area times distance, or PV. At constant pressure:

Substituting in Eq. 2 yields the most common form of the first law of thermodynamics:

In other words, you can't win -- energy cannot be created, only converted from one form to another.

Enthalpy

The enthalpy or heat content is defined as:

It's clear from this relationship that enthalpy is the sum of two energy terms. Differentiating (5) at constant pressure:

and from (4) dE=dQ-PdV, we have

Heat Capacity

Heat capacity is defined as the heat added to the system divided by the rise in temperature. While the heat capacity is not strictly speaking constant, it remains a nearly constant material property over wide temperature ranges. Heat capacity is either considered at constant pressure:

or constant volume

Heat capacity normally has the units Wkg-1° C -1 and is physically defined as the amount of heat required to raise 1 kg of a material 1 ° C. Closely related is the Specific Heat, which has the same units, and is the ratio of the heat capacity of a substance to that of water at 15 ° C.

Second Law of Thermodynamics

A succinct statement of the second law: "Things break down", or "You can't even break even". Entropy, S is the measure of the disorder in the system and is a single-valued function of the state of the system. Like the internal energy, it is dependent on the mass of the system. Under equilibrium conditions:

meaning that the process is reversible. for a spontaneous, disequilibrium process:

That is, irreversible processes increase the entropy of the system.

Third Law of Thermodynamics

There is such a thing as absolute zero, a temperature below which there is no more entropy in the system. Knowing this entropy allows us to use the second law to calculate the isobaric entropy within a phase at any temperature. From Eq. 8, the definition of isobaric heat capacity, we have:

which can be substituted into Eq. 10 to yield

The entropy at some fixed pressure and any temperature T is then

or

where the second term accounts for the heat of transition for any phase transformation that may have occurred along the way.

Gibbs Free Energy

The energy available to drive reactions in a system is less than the total energy, because some is tied up in entropy and some is tied up in the PV term. The remainng energy is called the Gibbs free energy, and is defined as:

at constant P and T:

However, for the more general case, let us differentiate Eq. 16:

Substituting the expressions for the first and second laws (Eqs. 4 and ??) gives:

Two important relationships from this equation lie in its partial derivatives with respect to pressure and temperature:

and

Reversible equilibrium reactions take place with no change in G, whereas spontaneous reactions involve a decrease in G.

Chemical Potential

The Gibbs free energy is only a function of P, T and the quantities of phases in the system. It is necessary to extend G to also describe compositional changes in the phases. In phases showing chemical variation, it is useful to establish another quantity, the chemical potential m , which is the partial molar free energy of a component in a phase, and is defined as:

where ni are the numbers of moles of each component in each phase in the system. The more general form of the gibbs free energy equation then becomes:

So not only must the Gibbs free energy of both reactant and product in a single component system be equal, at equilibrium (as we saw in the previous section) but the chemical potentials of each component must be equal between the phases. The chemical potentials will be different from each other within each given phase, of course. The minimization of the Gibbs free energy of a system is the basis of almost all petrology and geochemistry.

Gibbs Phase Rule

The variance or degrees of freedom in a system is denoted by F, and it is the number of independent parameters which must be fixed or determined arbitrarily in order to specify completely the state of the system. This includes both the physical conditions (P and T) of the system, but also the compositions of all the phases.

Consider a closed system consisting of p phases and c components, the temperature and pressure are also free parameters. Suppose that all c components occur in each phase. The composition of a phase is then defined by c-1 composition variables (the last being the reminder). With p phases, the total number of unknowns in the system is p(c - 1). In addition there are pressure and temperature for a total of p(c - 1) + 2 unknowns. Obviously, P and T must be the same for all phases at equilibrium. However, we also know that the chemical potentials of all the components must be equal in each phase. So if we determine m i in one phase, we have determined it in all the other phases as well. Therefore there are (p-1) chemical restrictions for each component, for a total of c(p - 1) dependent relationships. The total number of independent variables can be found by subtracting the restrictions from the total unknowns we found above:

which reduces to

Equation 25 is known as the Gibbs Phase Rule, and holds for systems closed to gain or loss of mass. Where physical conditions are fixed, the number of independent variables is reduced. For an isobaric or isothermal system, the the expression becomes c - p + 1.

Consider a system at constant pressure, where there is a reactant that undergoes a phase transformation to a product at Tm as the temperature rises. Before Tm is reached, there is one component, 1 phase and thus one degree of freedom -- the system is said to be univariate. As the temperature rises to Tm, a second phase appears, the product phase, and the number of degrees of freedom drops to 0. This is referred to as an invariant point, and the temperature of the system remains constant until the last of the product phase is consumed, and the number of degrees of freedom returns to 1. A pleasant fact of life on a hot summer day.

More Definitions

Thermal expansion coefficient a is the ratio of the change of length per degree C to the length at 0 ° C. The coefficient of volume expansion for solids (b ) is approximately 3 times the linear expansion coefficient. It is a ratio, and so has no unit, and typical values are around 10-5.

Thermal conductivity k is the time rate of transfer of heat by conduction, through unit thickness, across unit area for unit difference in temperature. It has units of Watts per meter per degree C (Wm-1° C-1). The thermal conductivity of solids varies widely, from 418 Wm-1° C-1 for silver to 2-3 Wm-1° C-1 for rock, to 0.1 Wm-1° C-1 for wood.

Thermal Diffusivity k is a diffusion coefficient for heat. It is the ability heat to move through a material and is defined as k/r Cp, where r is the density and Cp is the heat capacity at constant pressure, defined above.