Thermodynamics,
Systems,
Equilibrium &
Energy
Lecture 1
Plan of the Course
• The Geochemical Toolbox
o
o
o
o
o
Thermodynamics
Kinetics
Aquatic systems
Trace elements & magmatic systems
Isotopes
• Radiogenic
• Stable
• The Big Picture: Cosmochemistry
o Formation of the elements
o Formation of the Earth and the Solar System
• Chemistry of the Earth
Other Info
Text: White: Geochemistry, Wiley-Blackwell,
ISBN 978-0-470-65668-6
• Grading: 40% Problem
Sets (6 to 8),20% Prelim,
40% Final Exam
• http://www.geo.cornell
.edu/geology/classes/
geo455/EAS455.html
• Office: 4112 Snee
• Office Hours: no formal
office hours – drop by
anytime.
Thermodynamics
• Thermodynamics is the study of energy & its
transformations.
o 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.
• We can predict that as the concentration of atmospheric CO2
increases, so does that of the ocean and the calcium carbonate
shells of oysters and skeletons of corals will become more soluble.
o (This at first seems counter-intuitive, and has to do with a
decrease in ocean pH).
• Thermodynamics uses a macroscopic approach.
o We can use it without knowledge of atoms or molecules.
o We will occasionally consider the microscopic viewpoint using statistical
mechanics when our understanding can be enhanced by doing so.
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.
‘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.
o We are free to define the
‘system’ anyway we chose.
However, how we define it
may determine whether we
can successfully apply
thermodynamics.
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.
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).
Fundamental Variables
• Pressure: P
o Force/unit area
• Volume; V
• Temperature: T
• Energy: U
o Work: W
• we are mainly concerned with P-V work: pressure integrated over
volume change
• Work done by a system is negative
o Heat: Q
• Entropy: S – more on that in next lecture
Work, Heat, and Energy
• Work is done by moving a mass, M, over some
distance against a force (eg., gravity)
w º - ò 0 F dx
X
o Where
F=M
dv
dt
o 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);
related to kinetic energy, particularly with respect
to translational motions of molecules in a gas.
System International (S.I.)
Units
• Pressure: Pascals, Pa ( Newton/m2 = kg-sec2/m)
•
o Other: atmospheres ~ bars ≈ 0.1 MPa
Volume: m3
o Other: cm3, liter 1m3 = 106 = 103 l
o (note: liter is the standard unit in aquatic chemistry)
• Temperature: Kelvins (K)
•
o Other; ˚C; 1K = 1˚C, 0˚C = 273.16 K
Energy: Joules, J (kg-m2/sec2)
o Other: calories 1J = 4.184 cal.
• Entropy: J/K
• Mass: mole, N (also mol); amount, in grams, of an
element equal to its atomic weight (e.g., 1 mol C =
12g or 0.012 kg).
Extensive vs. Intensive
variables
•
•
o
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:
V =V / N
Density: V/M
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
o Tells us, for a given number of moles, now 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?
Ideal Gas Law
PV = NRT
• Ideal gas law grew out of Boyle’s (1627-1691) experiments with
gases and was formalized by Émile Clapeyron in 1834.
o
Fine as an approximation, but doesn’t work with real gases.
• An ideal gas is one in which:
o
o
o
The molecules occupy no volume
The only interactions between molecules are elastic collisions.
When might a gas most behave ideally?
• Van der Waals Equation
o
o
P=
RT
a
- 2
V -b V
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 PT-V behavior.
• R: Gas constant: simply converts units. (We’ll see it is the molar
equivalent of Boltzmann’s constant, which has atomic units).
General Equation of State
• Obviously, there is no one solution to the ideal gas
law, but we can imagine a couple of special cases:
• Hold pressure constant
o Isothermal compressibility (β): change in volume with change in pressure
at constant temperature per unit volume: β≡ -1/V(∂V/∂P)T
• Hold temperature constant
o 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 write a general equation relating V, T & P:
dV = ∂V/∂T)PdT + (∂V/∂P)TdP
dV = VαdT + VβdP
o These equations are general and apply to all substances. The difference is
that  and have simple solutions for ideal gases (NR/P and 1/T,
respectively), while they are more complex functions for real substances.
Temperature
•
•
Temperature is a measure of
the average internal kinetic
energy of a system.
How do we measure it?
o
o
o
o
o
•
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.
Use the absolute scale (K) in
all thermodynamic
calculations.
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.
The Three E’s
Energy, Entropy and Enthalpy
The First Law
•
•
•
•
•
Statements:
o
o
o
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.
o
(we are mainly concerned with energy changes, not absolute amounts).
Mathematically:
ΔU = Q +W
The First Law implies that energy change 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).
(Of course, the First Law ultimately fails 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.)
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 gas. 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.
State Functions & Exact
Differentials
• State functions have exact differentials.
o (These are not new, they are the kind you have learned about in calculus).
o This means we can obtain (in principle anyway) an exact solution if we
differentiate them (or integrate the differentials).
• 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).
o 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.
Work
• Is Work done by an ideal gas a state function?
o Work is:
dW = -PdV
o Expanding the dV term, we have
dW
éæ
ö
ê ¶V
= -P êç ÷ dT
êçè ¶T ÷ø P
ë
æ
ö
¶V
+ç ÷
çè ¶P ÷ø
T
ù
dP úú
ú
û
• Substituting for (∂V/∂T)P and (∂V/∂P)T
dW = -NRdT + NRT dP
P
¶NR ¹ ¶(NRT / P)
¶P
¶T