Basic Thermodynamics

Thermodynamics is concerned with the understanding and interpretation of the properties of matter in so far as they are affected by changes in temperature and pressure. We study thermodynamics because we would like to know how changes in temperature and pressure affect the state and properties of minerals within planetary interiors. The response of planetary forming minerals to such changes in temperature is governed by thermodynamics, which must be understood since it can been used to link microscopic (atomistic level) processes to macroscopic (global, bulk) processes.

Thermodynamics itself is a construct; a set of self-consistent propositions encapsulated in the four laws of thermodynamics, which all obey the conservation of energy.

The Fundamentals:

The athermal, zero pressure, total energy, i.e., the sum of all different kinds of energies, contained within the boundaries of a system is called the internal energy of the system, denoted U. In classical thermodynamics, the internal energy is defined by the first law (see below).

Microscopically (atomistically), U consists of the kinetic and potential energy of all atoms within the system. Because potential energies always contain arbitrary additive constants, U is not determined as an absolute value, but relative to some reference state (e.g., elements) and only differences in U have any significance (unless something is known about the zero point energy, which arises as a consequence of the intrinsic quantum mechanical nature of matter).

Macroscopically, ΔU may only be found by observing the amounts of energy added to the system across the boundary, and how the system is affected by such changes in its internal energy.

Zeroth Law of Thermodynamics:

If two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

This introduces the concept of temperature, i.e., hotness and coldness. If you put a hot brick on top of a cold brick, and another cold brick on top of the hot brick, and you then allow them to equilibrate, they will reach the same temperature.

First Law of Thermodynamics:

If the state of an otherwise isolated system is changed by the performance of work, the amount of work needed depends solely on the change accomplished, and not on the means by which the work is performed (i.e., mechanical work or heating), nor on the intermediate stages through which the system passes between its initial and final stages.

This is, essentially, just the law of conservation of energy, i.e., the total energy of any isolated system remains constant, whatever happens to it, and the independence of the route taken means such a change is conservative. In other words, all the first law is really saying is that, for example, if you raise a vat of water to a particular temperature by either heating over fire, or by mechanical work via paddle wheels or a whisk, the change in internal energy shall be the same, and the amount of heat you give it will be reflected in the increase of its internal energy.

For an infinitesimal change of the system, the energy which has entered (or been absorbed by) the system as heat is dQ, and that which has entered as mechanical work is dW. Thus the total energy which enters the system in an infinitesimal change is dQ+dW.

So, for any system undergoing any change whatsoever, we find from the conservation of energy,

 Eq. 1

where dU is the change in internal energy, dQ is the heat absorbed, and dW is the work done.

For example, the work done in expanding a gas against a hydrostatic pressure is given by:

 Eq. 2

Entropy:

Entropy is a concept devised in order to obtain something resembling the amount of heat in a body. If heat, dQ, is absorbed in a reversible way (i.e., via small changes which can be reversed) at an absolute temperature T, then the integral dQ/T is independent of the path taken, and evidently increases as the body is heated. This integral from a fixed point (usually absolute zero) is called the entropy, S. It is determined by the state of a system, but measures in a certain way only heat energy, not mechanical energy (or work).

Entropy may also be thought of as a measure of the randomness or disorder in the atomic arrangement of a system in a certain state. This configurational entropy increases when a body is heated and the atomic random motion increases, or by any mechanical method of increasing atomic motion. For example, a projectile which vapourises on hitting a target will have the same ΔS as the same projectile being heated until it vapourises.

Second Law of Thermodynamics:

No process is possible whose only result is the abstraction of heat from a system and the performance of an equivalent amount of work.

There is no such thing as a 100% efficient system; it is impossible to turn all the heat absorbed into mechanical work, and that which is not used in mechanical work generates entropy. This defines entropy as a mathematical construct which only remains constant in a perfectly efficient (but hypothetical) closed thermodynamic cycle.

The second law of thermodynamics defines the heat absorbed thus:

 Eq. 3

where dS is the change in entropy, and is therefore given by:

 Eq. 4

The change in entropy increases by this amount for a reversible process, and by a larger amount for an irreversible process (a complete and irretrievable departure from equilibrium, such as diffusion, explosions, etc., all of which are one-way process unless there is a significant amount of external intervention). This is because in the latter case the greater change in entropy is due to the nature of irreversibility, e.g., bonds being broken generating a dissociation energy. ΔS is always greater than zero with respect to the whole universe, so entropy itself of such an isolated system is always increasing.

Thus for the change in internal energy we have:

 Eq. 5

Third Law of Thermodynamics:

The entropy of a substance approaches zero as its temperature approaches absolute zero.i.e.,

 Eq. 6

Microscopically, this can be thought of in terms of the "slowing down" of atomic motions as the temperature of the atoms approach absolute zero.

Thermodynamic Potential Energies:

We wish to have a number of expressions for the energy of a system in terms of different variables, for we shall sometimes know the pressure and temperature of a system, but not the volume or entropy; or we might know the volume and temperature but not the pressure and entropy. In most cases, we shall know only a few relevant quantities and will wish to find out many more. This leads to the following definitions two or more thermodynamic quantities:

Internal Energy U

Enthalpy (no TS term) H = U+PV

Helmholtz Free Energy (no PV term) F = U - TS

Gibbs Free Energy (everything) G = H - TS

From the first and second laws of thermodynamics:

 Eq. 5

Therefore differentiation gives:

 Eq. 7

 Eq. 8

 Eq. 9

So the functional variables are: U = U(S,V); H = H(S,P); F = F(T,V); G = G(P,T).

Using Thermodynamic Potential Energies:

How do we use these thermodynamic potentials?

From the first and second laws of thermodynamics, dU and dQ are not obviously measurable. Therefore we need thermodynamic manipulation to get useful properties from what is measurable, i.e., P, V and T. Thermodynamics quantities are more commonly defined in terms of partial differentials. For example:

Internal energy:

 Eq. 10

 Eq. 11

Enthalpy:

 Eq. 12

 Eq. 13

 Eq. 14

Helmholtz free energy - the maximum amount of mechanical work extractable from a system at constant temperature:

 Eq. 15

 Eq. 16

The Gibbs free energy is constant for a reversible process under isothermal isobaric conditions, and decreases for an irreversible one, reaching a minimum consistent with the pressure and temperature at equilibrium. G remains unchanged at equilibrium; this is very important when considering phase changes within planetary interiors, since minimum energy mineral structures can be determined from the Gibbs free energy. First derivatives include:

 Eq. 17

 Eq. 18

Similar partial differentials may be obtained for other thermodynamic quantities.

Maxwell's Relations:

Now, considering the second derivatives; from equations 13 and 14:

 Eq. 19

 Eq. 20

therefore:

 Eq. 21

Similarly, from the second differential of Gibbs free energy:

 Eq. 22

Chain Rule:

The chain rule is the final link in order to obtain relations between all thermodynamic variables.

 Eq. 23

so, for example:

 Eq. 24

Physical Measurables:

a.  Heat Capacity at constant V (from Eq.5):

 Eq. 25

and from the definition of heat capacity, (defined by the heat required to raise unit mass of a material by one degree Kelvin) where the heat capacity multiplied by the rise in temperature = heat absorbed at constant volume:

 Eq. 26

so

 Eq. 27

therefore

 Eq. 28

b.  Bulk Modulus (incompressibility):

 Eq. 29

therefore

 Eq. 30

and

 Eq. 31

N.B. KS effects elastic waves, which propagate quickly through the system so no heat escapes and the process is adiabatic; KT effects finite deformation, where the process is slower, and heat has time to flow out of the system (compare hitting a bell with heating soup).

c.  Thermal Expansion at constant P:

 Eq. 32

therefore

 Eq. 33

and so

 Eq. 34

For a full set of all possible combinations, see Table 1.2 in Poirier.

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Thermoelastic Coupling: thermodynamics-elasticity

The combination of the effect of temperature and the effect of pressures (stress) through thermal expansion, a, and incompressibility, K, is called thermoelastic coupling, and is the most important cross-term in geophysics.

a) The Grüneisen Parameter:

The inter-relation between stress and temperature is dealt with via the Grüneisen parameter, an approximately constant, pressure and temperature independent parameter of the order of 1. Thermodynamically, the thermal Grüneisen parameter is defined by:

 Eq. 65

and after some unpleasant thermodynamics this leads to:

 Eq. 66

Therefore a knowledge of g gives a good handle on many of the important thermoelastic properties of minerals, which shall be used later in the course.

b) The Mie-Grüneisen Equation of State:

When a solid is heated at constant volume, the atoms within it vibrate more vigourously, and this results in a thermal pressure acting from within the system outwards, which, if unresisted, will give rise to thermal expansion. From thermodynamics:

 Eq. 67

Integrating at constant volume and assuming g is a constant (see also, Eq. 26):

 Eq. 68

where U is the internal energy. Therefore:

 Eq. 69

and this is the Mie-Grüneisen equation of state.

The adiabatic temperature gradient (no heat escapes, S is constant) is that caused by adiabatic compression; i.e., the change in temperature throughout the Earth as a result of pressure loading from above.

From thermodynamics we can show:

 Eq. 70

or

 Eq. 71

which on integration (for constant g) gives:

 Eq. 72

Therefore if we know g we can estimate how the temperature varies with density in a planetary interior, assuming it is adiabatically generated.

Another approach may be made by considering another relationship from thermodynamics:

 Eq. 73

but, recalling Eq. 63, the adiabatic temperature gradient is found to be:

 Eq. 74

Therefore, the adiabatic temeprature gradient may be obtained from a knowledge of the seismic parameter (i.e., from PREM).

References

 Poirier. J.-P. (2000) Introduction to the Physics of the Earth's Interior - Chapter 1 only Krauskopf. K.B., and D.K. Bird (1995) Introduction to Geochemistry. Chapters 7 & 8. McGraw-Hill International. Poirier. J.-P. (2000) Introduction to the Physics of the Earth's Interior. Putnis. A. (1992) An Introduction to Mineral Sciences. Chapter 8. Cambridge University Press.