The early thermal history of the Earth is a matter of some speculation. Current scientific consensus is that planet Earth formed by accretion of material with the same composition as chondritic meteorites. Accretion, a process that generated heat as colliding material gave up kinetic energy, led to differentiation of the planetary constituents into concentric layers. When the temperature of the early Earth reached the melting point of iron, the dense iron, accompanied by other siderophile elements such as nickel and sulfur, sank towards the center of the planet to form a liquid core. Meanwhile lighter elements rose to form an outer layer, the primitive mantle. Further differentiation took place later, creating a chemically different thin crust atop the mantle. Only the outer core is now molten, surrounding a solid inner core of iron that solidified out of the core fluid. Lighter elements left behind in the core rise through the corefluid and result in a composition-driven convection in the outer core, which is in addition to thermal convection. Although the short-term behavior of the mantle is like that of a solid, allowing the passage of seismic shear waves, its long-term behavior is characterized by plasticflow, so heat transport by convection or advection is possible. In the solid lithosphere and inner core heat is transported dominantly by thermal conduction.
The physical states of the Earth’s mantle and core are well understood, but the variation of temperature with depth is not well known. Direct access is impossible and it is very difficult in laboratory experiments to achieve the temperatures and pressures in the Earth’s deep interior. Consequently, some important thermody- namic parameters are inadequately known. Points on the melting-point curve can be determined from experiments at high temperature and pressure. Convection ensures that the temperature profile in the mantle and outer core is close to the adiabatic temperature curve, which can be calculated. From these considerations an approximate temperature profile in the Earth’s interior can be estimated (Fig. 6.1). The temperatures in the mantle and outer core are close to the adiabatic curve, little temperature change occurs in the solid inner core, and comparatively rapid change occurs in the asthenosphere and lithosphere.
6.1 Energy and entropy
Analysis of the thermal conditions in the Earth is based upon the First and Second Laws of Thermodynamics. The First Law is an application of the conservation of energy to a thermodynamic system. It states that energy cannot be created or destroyed in a closed system, but can only be transformed from one form to another. In an open system, extra terms must be considered to allow for the transfer of energy into or out the system (e.g., by theflow of matter). The total energy, Q, of a closed system consists of its internal energy, U, and the work, W, done in any external transfer of energy to the surroundings. The energy balance is expressed by the equation
dQ¼ dU þ dW (6:1)
Heat added to (or removed from) a closed system is used to increase the internal energy and to perform external work. For example, the gas molecules in a heated balloon are more energetic, and, if it is able to expand, the volume, V, increases. The external work dW due to the change in volume at constant pressure, P, is
dW¼ P dV (6:2)
and so from the First Law of Thermodynamics the energy equation is
dQ¼ dU þ P dV (6:3) Asthenosphere (partial melting) Lithosphere 1000 2000 3000 4000 0 5000 6000 0 1000 2000 3000 4000 5000 T emper ature ( °C) Depth (km) MANTLE (solid silicate) OUTER CORE (liquid iron alloy)
INNER CORE (solid iron alloy) geotherm solidus 670 400
Fig. 6.1. Models of the adiabatic temperature profile (geotherm, solid curve) and the melting-point curve (solidus, dashed curve) in the Earth’s interior. Data sources: tables in appendix G of Stacey and Davis (2008); for mantle solidus, Stacey (1992), appendix G.
The Second Law of Thermodynamics asserts that the energy of an isolated system tends to become uniformly distributed with the passage of time. The concept of entropy, S, is used as a measure of the microscopic disorder in a system at a particular temperature. The change dS in the entropy of a system caused by a change in energy dQ at a temperature T is defined as
dS¼dQ
T (6:4)
On substituting this into the energy equation we get
T dS¼ dU þ P dV (6:5)
This important relation, uniting the First and Second Laws, is the central equation of thermodynamics. It is important in the analysis of thermal condi- tions inside the Earth, because it defines adiabatic conditions.
An adiabatic thermodynamic process is one in which heat cannot enter or leave the system, i.e., dQ = 0. The entropy of an adiabatic reaction remains constant, because dS = dQ/T = 0. The adiabatic temperature gradient in the Earth serves as an important reference for estimates of the actual temperature gradient and for determining how heat is transferred.
6.2 Thermodynamic potentials and Maxwell’s relations
The thermodynamic state of a system can be expressed with the aid of scalar functions called thermodynamic potentials. These are the internal energy, U, the enthalpy, H, the Helmholtz energy, A, and the Gibbs free energy, G. Each potential consists of a particular combination of the physical parameters pres- sure, temperature, volume, and entropy.6.2.1 Thermodynamic potentials
Internal energy (U) has been described and defined above. A change in internal energy at constant temperature and pressure is related to changes in volume and entropy by
dU¼ T dS P dV (6:6)
Enthalpy (H) is a measure of the total energy of a system; it is a combination of the internal energy and the product of the pressure and volume:
H¼ U þ PV (6:7)
dH¼ dU þ P dV þ V dP (6:8) The conservation of energy, expressed in (6.5), allows us to reduce this to
dH¼ T dS þ V dP (6:9)
The Helmholtz energy (A) is defined from the relationship between the thermodynamic properties of macroscopic materials and their behavior on a microscopic level through statistical mechanics. It is a measure of the work obtainable from a closed thermodynamic system at constant temperature and constant volume, and is defined as
A¼ U TS (6:10)
Taking the differentials of both sides gives
dA¼ dU T dS S dT (6:11)
Using (6.5), this becomes
dA¼ P dV S dT (6:12)
TheGibbs energy (G) is defined in a similar way to the Helmholtz energy, but for constant pressure and temperature. It represents the maximum amount of energy obtainable from a closed system (i.e., one isolated from its surroundings) without increasing its volume, and is defined as
G¼ A þ PV (6:13)
The differentials give the equation
dG¼ dA þ P dV þ V dP (6:14)
Combining this with (6.12) gives
dG¼ V dP S dT (6:15)
6.2.2 Maxwell’s thermodynamic relations
Maxwell’s relations are a set of partial differential equations derived from the definitions of the thermodynamic potentials that relate the parameters S, V, T, and P. The relations depend on the mathematical equality between the second derivatives of these parameters. This follows because the order of differentia- tion of a function F(x, y) of two variables x and y is not important:
∂ ∂x ∂F ∂y x ¼ ∂2F ∂x ∂y¼ ∂2F ∂y ∂x¼ ∂ ∂y ∂F ∂x y
Maxwell’s thermodynamic relations are derived in Box 6.1by applying this condition to the different thermodynamic potentials. Summarized, they are