1. Thermodynamics, Systems, Equilibrium & Energy
Lecture 1

2. Why geochemistry? Questions geochemistry can address:
How and when did the matter in the cosmos form?
How and when did the Earth form?
How and when did the continents form?
How do magmas form and why their diversity (basalts to granites)?
Metamorphism transforms rocks through heat and pressure – but specifically what temperatures and pressures?
Our atmosphere consists mainly of N2 and O2. Was it always so?
When did life on Earth begin?
Why does calcite precipitate from surface waters and dissolve in deep waters?
How do gold deposits form?
How do oil deposits form?
How cold were the “Ice Ages” and what caused them?
What did our ancestors eat?

3. We’ll address many of these questions in this course
To do so, we’ll need the tools of geochemistry.
In the first part of this course, we’ll acquire the tools we need.
Those tools all have their basis in physical chemistry (and ultimately physics and math), which is where we’ll begin.

4. Plan of the Course The Geochemical Toolbox
Thermodynamics
Kinetics
Aquatic systems
Trace elements & magmatic systems
Isotopes
Radiogenic
Stable
The Big Picture: Cosmochemistry
Formation of the elements
Formation of the Earth and the Solar System
Chemistry of the Earth

5. Other Info
Grading: 40% Problem Sets (6 to 8),20% Prelim, 40% Final Exam
Office: 4112 Snee
Office Hours: no formal office hours – drop by anytime.
Text: White: Geochemistry, Wiley-Blackwell, ISBN

6. Thermodynamics
Thermodynamics is the study of energy & its transformations.
Chemical changes involve energy; by “following the energy”, we can predict the ‘equilibrium’ state of a system and therefore the outcome of reactions.
For example, we can predict the minerals that will crystallize from a cooling magma or from the composition of minerals in metamorphic rocks, infer the temperature and pressure of metamorphism
We can predict that as the concentration of atmospheric CO2 increases, the pH of the ocean rises and the calcium carbonate shells of oysters and skeletons of corals will become more soluble.
From the abundance of certain metals in sedimentary rocks, we can infer that oxygen was absent from the oceans in the early Earth.
Thermodynamics uses a macroscopic approach.
We can use it without knowledge of atoms or molecules.
We will occasionally consider the microscopic viewpoint using statistical mechanics when our understanding can be enhanced by doing so.

7. Thermodynamics and Kinetics
The equilibrium state of a system is independent of any previous state. So, for example, if we do a partial melting experiment with rock, it should not matter if we start with a solid and partially melt it or with a melt and partially crystallize it, or whether we partially dissolve calcium carbonate or partially precipitate it.
Kinetics is the study of rates and mechanisms of reaction. Kinetics concerns itself with the pathway to equilibrium; thermodynamics does not. Very often, equilibrium in the Earth is not achieved, or achieved only very slowly, which naturally limits the usefulness of thermodynamics.

8. ‘The System’ Four Kinds of Systems
A thermodynamic system is the part of the universe we are considering. Everything else is referred to as the surroundings.
We are free to define the ‘system’ anyway we chose. However, how we define it may determine whether or not we can successfully apply thermodynamics.

9. Equilibrium
The equilibrium state is the one toward which a system will change in the absence of constraints.
It is time invariant on the macroscopic scale, but not necessarily on the microscopic one.

10. Equilibrium & Thermodynamics
Conundrum: strictly speaking, we can apply thermodynamics only to the equilibrium state. If a reaction is proceeding, then the system is out of equilibrium and thermodynamic analysis cannot be applied!
Solution: we imagine reversible processes in which systems are only infinitesimally out of equilibrium. In contrast, natural processes can proceed only in one direction and are irreversible.
We might also imagine local equilibrium where even if the whole system (e.g. the ocean or magma and crystals) is out of equilibrium, we can imagine a part of it is in equilibrium (rim of the crystal).

11. Fundamental Variables
Pressure: P
Force/unit area
Volume; V
Temperature: T
Energy: U
Work: W
we are mainly concerned with P-V work: pressure integrated over volume change
Work done by a system is negative
Heat: Q
Entropy: S – more on that in next lecture

12. Work, Heat, and Energy
Work is done by moving a mass, M, over some distance against a force (eg., gravity)
Where
Note that the minus sign occurs because work done by a system is negative, work done on a system is positive.
Heat is thermal energy that results from collective random motion of atoms or molecules in a system (including rotational and vibrational motions); its related to kinetic energy, such as translational motions of molecules in a gas or vibration of atoms in crystals.

13. System International (S.I.) Units
Pressure: Pascals, Pa ( Newton/m2 = kg-sec2/m)
Other: atmospheres ~ bars ≈ 0.1 MPa
Volume: m3
Other: cm3, liter 1m3 = 106 = 103 l
(note: liter is the standard unit in aquatic chemistry)
Temperature: Kelvins (K)
Other; ˚C; 1K = 1˚C, 0˚C = K
Energy: Joules, J (kg-m2/sec2)
Other: calories 1J = cal.
Entropy: J/K
Mass: kilograms (kg) or number of moles, N (mol);
A mole is the amount, in grams, of an element or compound equal to its atomic or molecular weight (e.g., 1 mol C = 12g or kg).

14. Extensive vs. Intensive variables
Extensive variables are ones that depend on the mass of the system, intensive ones do not.
Which of the following are extensive?
Pressure
Volume
Temperature
Energy
Work
Heat
Moles
We can convert extensive variables to intensive ones by dividing on extensive variable by another.
Molar volume:
Density: V/M

15. State Variables and Equations of State
State variables depend only on the state of the system, not on the path taken. Not all the variables listed above are state variables.
Equations of state express the relationship between state variables, e.g.:
PV=NRT
Tells us, for a given number of moles, how temperature, volume, and pressure of an ideal gas relate to each other; i.e., if we heat the gas, what happens to volume and pressure?

16. Ideal Gas Law
PV = NRT
Ideal gas law grew out of Boyle’s ( ) experiments with gases and was formalized by Émile Clapeyron in 1834.
Fine as an approximation, but doesn’t work with real gases.
An ideal gas is one in which:
The molecules occupy no volume
The only interactions between molecules are elastic collisions.
When might a gas most behave ideally?
Van der Waals Equation
The b term corrects for the finite volume of molecules while the a term corrects for their interactions.
This is still often a poor approximation to the behavior of real gases. Two geochemically important gases, water vapor and CO2 show particularly complex P-T-V behavior. 
R: the Gas Constant: simply converts units. (We’ll see it is the molar equivalent of Boltzmann’s constant, which has atomic units). In essence, this just converts units.

17. General Equation of State
There is no one solution to the ideal gas law, but we can imagine a couple of special cases:
Hold temperature constant
Isothermal compressibility (β): change in volume with change in pressure at constant temperature per unit volume: β≡ -1/V(∂V/∂P)T
Hold pressure constant
Coefficient of thermal expansion (α): change in volume with change in temperature at constant pressure per unit volume 1/V(∂V/∂T)P.
We can also write a general equation relating V, T & P:
dV = ∂V/∂T)PdT + (∂V/∂P)TdP
dV = VαdT - VβdP
These equations are general and apply to all substances. The difference is that α and β have simple solutions for ideal gases (NR/P and 1/T, respectively), while they are more complex functions for real substances.
(∂V/∂T)P = NR/P and (∂V/∂P)T = -1/P2

18. A Note on Partial Differential Equations
The ideal gas law tells us that the volume of a mole of gas will depend on 2 variables: temperature and pressure.
Partial differential equations are a way to deal with each of these dependencies separately:
dV = ∂V/∂T)PdT + (∂V/∂P)TdP
The ∂ symbol tells us this is a partial differential
The subscript (e.g., P) says that we are holding pressure constant.
So the term ∂V/∂T)P expresses the change in volume with change in temperature at constant pressure.

19. Temperature
Temperature is a measure of the average internal kinetic energy of a system.
How do we measure it?
Since the volume of an ideal gas is a simple function of T at constant P, we can use it to construct a thermometer.
We can arbitrarily define a scale such that
V = V0(1+γτ)
Where τ is our measure of temperature. If so, we might have negative τ.
But note V cannot be negative, so there must be a minimum value of τ: an absolute minimum to temperature.
The absolute minimum of T occurs where the volume of an ideal gas is 0.
Always use the absolute scale (K) in thermodynamic calculations.

20. Zeroth Law
Two bodies in thermal equilibrium have the same temperature.
Two bodies in equilibrium with a third body are in equilibrium with each other.

21. Energy, Entropy and Enthalpy
The Three E’s
Energy, Entropy and Enthalpy

22. The First Law Statements: Energy is conserved in all transformations
Heat and work are equivalent (the sum of the two is always the same).
Change in energy of a system is independent of the path taken.
This might seem obvious to us, but it wasn’t when James Joule proposed it in 1843.
James Joule
Brewer, Scientist

23. The First Law Statements: Mathematically: ΔU = Q +W
Energy is conserved in all transformations
Heat and work are equivalent (the sum of the two is always the same).
Change in energy of a system is independent of the path taken.
Mathematically: ΔU = Q +W
The First Law implies that energy change involved when a system goes from one state to another (e.g., convert gasoline to CO2 + H2O or liquid water to ice) is path independent and thus that energy is a state variable. Heat and work are not.
State variables have exact differentials (heat and work do not).
(we are mainly concerned with energy changes, not absolute amounts).
(Of course, the First Law had to be tweaked because energy can be created out of mass. This turns out to be important because this energy source powers not only the Sun, and hence many processes at the surface of the Earth, it also powers, in part, the Earth’s interior and geologic processes such as plate tectonics. That need not concern us until much later in the course. The revised version of the first law is mass-energy conservation.)

24. State Functions & Path Independence
State functions are path independent and have exact differentials.
Think about an internal combustion engine. Chemical energy is released by burning gasoline. Some of that energy goes into heat and some to work. There is no fixed rule about how much goes to each – it depends on your engine design (engineers work to increase the amount going into work).
Therefore, heat and work cannot be state functions.
However, no matter how you design the engine, the sum of heat and work for a given amount of (fully) combusted gasoline is the same.
Energy is path independent and a state function.

25. State Functions & Path Independence
Consider again:
dV = (∂V/∂T)PdT + (∂V/∂P)TdP
Because V (or U) is a state function, in a transformation of P and T, the order and way in which we raise temperature and pressure does not matter: the final volume or energy will be the same.
Hence we can integrate the two parts of above equation separately.
Caveat: the compressibility and thermal expansion terms are themselves be functions of T and P,

26. State Functions & Exact Differentials
State functions have exact differentials.
(These are not something new to you, they are the kind you have learned about in calculus).
This means we can obtain (in principle anyway) an exact solution if we differentiate or integrate them.
Exact differentials have the property that the cross differentials are equal (in other words, if we differentiate by two separate variables, the order doesn’t matter).
Again, this is what you learn in calculus.
Consider dV = (∂V/∂T)PdT + (∂V/∂P)TdP
If V is a state function, then (∂V2/∂T∂P)= (∂V2/∂P∂T)
This is not true of non-state functions like work and heat.

27. Work (∂V/∂T)P = NR/P and (∂V/∂P)T = -NRT/P2
Is Work done by an ideal gas a state function?
Work is:
dW = -PdV
Expanding the dV term, we have
Substituting for (∂V/∂T)P and (∂V/∂P)T
(∂V/∂T)P = NR/P and (∂V/∂P)T = -NRT/P2