It is easy for anyone to judge by word of mouth that a certain system is either stable or unstable. However in geoscience those words will not be of significance until they explain the stability of a system in concepts of thermodynamics. Thermodynamics plays an important role in understanding the system stability of geological processes. Example in Mineral physics we can calculate the stabilities of various mineral assemblages at a particular set of temperature and pressure conditions. Changes in mineral state may cause change in internal energy, for example when aragonite changes to calcite at 25°C and 1 atm. So there is an open relationship between geodynamics and thermodynamics, when the major energy to our system is from heat and work done. Of course there is much I want you to share with you on this topic but simply the time and space are limited. I will try to let you know about this through other posts. But how brilliant today have I written this piece of post? It is about the basics of Maxwell's thermodynamic equation, which describes the fundamental properties of the thermodynamic system. Let's move on.
Before going direct to derive this equation, let's grasp some concepts related to this topic,
What is thermodynamics?
Thermodynamics can be defined as a set of mathematical models and concepts that describe the way changes in physical conditions affect the state of equilibrium of a system.
Physical conditions can be pressure, temperature.
Or simply you can define it as the science that deals with interaction between energy (heat) and work.
Gibbs Free Energy can be defined as the energy in excess of the internal energy. If we denote it by G. Then the Gibbs free energy (G) can be given by,
G = E + PV - TS............(1)
Where S - entropy, P - Pressure, V - volume, T - temperature, E - Internal energy
If we do some bit of differentiation of equation 1
dG = dE + VdP + Pdv - TdS - SdT..............(2)
When we recall 2nd and 1st Laws of thermodynamics, we will find that,
dE = dq - Pdv.............(3) This is statement of 1st Law of thermodynamics, where dq - heat energy
dS = dq/T............(4) This is the statement of the 2nd Law of thermodynamics.
Okay! I have recalled these statements because I don't want to start from scratch, there are plenty of textbooks on this niche that you can consult yourself and find how they are being derived.
We have to make the subject dq from equation 4 and put it into equation 3,
dq = TdS, then dE = TdS - PdV...........(5)
By substituting equation 5 into equation 2
dG = TdS - PdV + VdP + PdV - TdS - SdT
dG = TdS - PdV + VdP + PdV - TdS - SdT
dG = VdP - SdT...........(6)
If our system is at equilibrium means P and T are both constant
Then, dG = 0
Then at constant T, (dG/dP)T = V..........(7)
Also if we keep P constant, (dG/dT)p = -S..........(8)
If we take the total differential of E such as DE, from equation 5, dE = TdS - PdV
DE = (∂E/∂S)v ∂S + (∂E/∂V)s ∂V ................(9)
If you compare equations 5 and 9, we will find that,
(∂E/∂S)v = T, and (∂E/∂V)s = -P
If we take second partial differential for above equations,
∂2E/∂V∂S = ∂2E/∂S∂V.............(10)
Then, from equation 10,
(∂T/∂V)s = - (∂P/∂S)T..............(11) is the Maxwell's Thermodynamic relations
What does equations 6, 7, and 8 tell us?
Implications laid behind these equations is that,
Equation 6, imply to us that Gibbs free energy is a function of Pressure (P) and Temperature (T) .
Equation 7, implies to us that phases with small volume are formed at higher pressure.
Equation 8, implies to us that phases with high entropy such as high disorder are formed at higher temperature.
All in all in mineral physics one mineral can have variation of stability due to different entropy values at the same physical conditions, for example we can find graphite at 25°C to have higher entropy than diamond, simply because graphite has larger volume than diamond at that temperature. A large volume allows more disorder of carbon atoms and this gives a higher entropy.
Thank you so much!
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