Chemistry 544 Lecture Notes
Advanced Thermodynamics and Statistical Mechanics
Chapter 1. Introduction and definitions
Thermodynamics is the study of stable equilibrium in macroscopic systems. The terms
“macroscopic” and “stable equilibrium” require some definition:
Macroscopic system: For a system constituted of N particles, N >> N .
Fluctuations must be small compared to mean, and this can happen only for large particle
numbers. During the inception of thermodynamics in the 19th century, this criterion was
not understood because systems were assumed to be continuous, i.e. not made of of vast
numbers of exceedingly small particles. Derivation of the inequality used in this
definition will have to await our study of statistical mechanics, which acknowledges the
particulate nature of matter.
Stable equilibrium: Stable equilibrium occurs when all macroscopic observables for a
system are constant, no new mass-energy flow occurs, and the system returns to its initial
state after small perturbation is removed.
In reality this is always an approximation valid only for a finite time because any system
(except maybe the whole universe!) is eventually subject to external influences such that
one or more of the three requirements above are no longer met. In particular,
thermodynamics is not about the type of equilibrium where a system is teetering on the
edge, like a ball poised at the top of a conical hill. Stability implies systems analogous to
a ball at the bottom of a conical lake.
Systems where thermodynamics does not apply:
• Very few signaling molecules in a cell (N too small)
• Critical point (fluctuations no longer ~ N )
• Open flame (mass-energy flow occurs even in steady-state)
•
Systems where thermodynamics applies well:
• A pH 3, mM solution of protein
• A completed reaction in a calorimeter.
Even in cases where thermodynamics does not apply, it can put useful constraints on
possible states of the system. These constraints can be extremely firm, because they are
based on a few very simple assumptions only. Hence Einstein’s claim that the second
law is not likely to be overthrown anytime soon. The power of thermodynamics is often
not in what it says can happen, but in what it says cannot happen!
In these notes, we will discuss thermodynamics on a very formal basis, beginning with
the definition of several concepts that we will manipulate, with the aid of some axioms,
or “postulates of thermodynamics.” Everything follows uniquely from these postulates.
One important thing to keep in mind is that thermodynamics makes statements about
macroscopic variables (those with well-defined values in macroscopic systems) while
making almost no prior assumptions about the system. Thermodynamics cannot be used
to derive a fundamental relation that tells us everything about the macroscopic variables;
for that, we need the Hamiltonian of the system and statistical mechanics.
Thermodynamics can take fundamental relations of a system, and give us the values of
almost any macroscopic variable we want, and connections among the variables, – as
long as the system remains at equilibrium. Thermodynamics can also tell us which
equilibrium state can go to another, even more favorable equilibrium state, but not viceversa. For example, given a single fundamental relationship among variables such as V,
T and N for an ideal gas (it’s not PV = NRT by the way), thermodynamics can tell us
anything we want about the connection among variables such as the above, or U (energy),
or α (thermal expansion coefficient), or any other relevant macroscopic variable. It can
also tell us that a state with P = 1 atm, V = 1 liter, T = 298 K can spontaneously evolve to
the state P = 0.5 atm, V=2 liters, T = 298 K, when a barrier dividing a box in half is
removed, but not vice-versa when the barrier is re-inserted.
Some definitions:
•
Simple system: A spatially and temporally homogeneous system, labeled {S} .
•
Constraints: Constraints or “walls” are used to (arbitrarily) partition a system into
subsystems. The constraints may prevent heat flow (adiabatic) or allow it
(diathermal); they may prevent mass flow (impermeable) or allow it (permeable);
they may prevent volume changes (rigid) or allow them (flexible).
The main task of thermodynamics is to calculate new equilibrium states when a
constraint has been removed, and relate all the macroscopic variables in these states.
•
Composite systems: A system composed of subsystems separated by constraints.
(The subsystems may be simple or composite themselves): {S} = ∑ ⊕ {Sk }
k
•
Closed system: A system all of whose macroscopic observables (e.g. mass-energy,
volume, mole numbers, etc.) are constrained. (i.e. it lies within a diathermal,
inflexible, impermeable, etc. wall).
•
Open system: one that is not closed.
A very important point: The thermodynamics postulates we will soon encounter apply
only to closed systems. If we want to deal with open systems, we have to embed them in
a closed system first, as shown in the figure below. We then derive statements about the
open system of interest by applying thermodynamics to the composite closed system
consisting of “open system” and “environment.”
Internal wall determines
what can flow
Overall system is closed.
The wall surrounding it
is assumed to be impermeable, rigid, etc.
Environment
Open system:
The postulates of thermodynamics apply only to
this overall system directly, but properties of the
open subsystem can be
inferred by making assumptions about the internal wall.
The thermodynamic
postulates do not
apply directly
energy, particles, etc.
may flow between
open system and
environment
•
Reservoirs or baths: A particularly useful composite system consists of a closed
system within which there is an open system coupled to a reservoir or “bath”:
The idea is that the “bath” is so large that its properties are affected very little while the
open system reaches equilibrium. The walls separating the open system and the bath can
be diathermal (heat bath), flexible (constant pressure bath), etc. Although some
chemical transformations can be described using simple closed systems (e.g. bomb
calorimeter), most are better described by an open system. Putting this open system
within a large closed system allows us to still apply the postulated the thermodynamics.
If the nature of the constraints between the open system and the bath is well understood,
much information about the open system can be obtained once equilibrium has been
reached.
•
Extensive variables X are additive for a composite system:
{S} = ∑ ⊕ {Sk } ⇒ X = ∑ X k
k
Examples:
Internal energy U = ∑U k
(1.1)
k
(should be mass-energy in general).
k
Volume
V = ∑Vk
k
Magnetization M = ∑ M k
k
Mole number n = ∑ n k
k
Area
A = ∑ Ak
k
Dipole D = ∑ Dk
k
In the case where the composite system is an “ensemble” or a scaled version of a system
{S}, {S} → λ{S} ⇒ X → λX . For λ {S} to correspond to an ensemble of systems {S} ,
λ has to be a natural number.
•
Intensive variables Iij are scale invariant for a simple system. Let Ni and Nj be two
extensive variables, then
⎛ ∂ Ni ⎞
⎛ ∂ Ni ⎞
⎛ ∂λ N i ⎞ ⎛ ∂ N i ⎞
I ij = ⎜
is intensive: as {S} → λ {S} ⇒ I ij = ⎜
→⎜
⎟
⎟
⎟ =⎜
⎟ = I ij
⎝ ∂ N j ⎠ Nk
⎝ ∂Nj ⎠
⎝ ∂λ N j ⎠ ⎝ ∂ N j ⎠
⎛ ∂U ⎞
⎛ ∂U ⎞
Example: ⎜
= −P, ⎜
=H
⎟
⎝ ∂V ⎠ N k
⎝ ∂ M ⎟⎠
• Perfect and imperfect differentials: State functions
For a macroscopic variable to be well-defined in a given equilibrium state, it must be
unique to that state. To phrase this somewhat tautological statement a bit differently: as
we observe a system and see that it has different equilibrium states, only variables that
always retain the same value for a given state are useful for labeling that state.
− = Xdx + Ydy. Does a function u exist? If u exists, the functions X and Y cannot
Let du
be arbitrary functions:
⎛ ∂x ⎞
⎛ ∂u ⎞
⎛ ∂u ⎞
⎛ ∂u ⎞
du = ⎜ ⎟ dx + ⎜ ⎟ dy ⇒ X = ⎜ ⎟ ,Y = ⎜ ⎟
⎝ ∂x ⎠ y
⎝ ∂x ⎠ y
⎝ ∂y ⎠ x
⎝ ∂y ⎠ x
⎛ ∂X⎞
∂ 2 u ⎛ ∂Y ⎞
⇒⎜
=
=⎜
⎟
⎝ ∂ y ⎟⎠ x ∂ x∂ y ⎝ ∂ x ⎠ y
⎛ ∂X ⎞ ⎛ ∂ Y ⎞
− = du is a perfect differential.
The function u exists if and only if ⎜ ⎟ = ⎜ ⎟ ; then du
⎝
⎠
∂x y
⎝ ∂y ⎠ x
−
Otherwise du is imperfect.
This has important implication for thermodynamics, especially for state functions. Let x
and y describe the state a system. For a closed loop C(t) = {x(t), y(t)} it follows that
dy ⎫
⎧ dx
Δu = 
∫C du = C∫ (Xdx + Ydy} = ∫ dt ⎨⎩ x dt + y dt ⎬⎭
But if u = u(x,y) exists, then
⎛ du ⎞
t
C∫ du = C∫ du = ∫ ⎜⎝ dt ⎟⎠ dt = u |t10 = u(x0 , y0 ) − u(x0 , y0 ) = 0
Thus, perfect differentials belong to the state functions, which are uniquely defined for a
given state. Of course, any perfect differential can be broken up into imperfect
differentials: du = d q + d w. In thermodynamics, the most useful division is into heat
flow and work:
du = d q + d w
The distinction between heat and work is a practical one: the energy put into the system
or removed from the system via manipulation of the few extensive variables or “degrees
of freedom” we can control is called “work.” For example, a force may displace the
− = - F dx; or a pressure may change a volume, so dw
− = - P dV; or an electric
system so dw
−
field may induce a dipole, so dw = - E dP.
(Source IBMResearch Laboratories)
The rest of the energy flow is “heat.”
(Generally, few means << N , usually O(1)). If
we learn to control an additional degree of freedom
that could previously not be manipulated, its
energy flow then becomes “work.”
A
misunderstanding of this fact has led to many
fruitless discussions about “Maxwell’s demons”
and similar things in the thermodynamic literature,
and what they are supposed to be able to do, and
what not. It is in fact perfectly possible in
principle to build a “demon” that will open a door
to let molecules from one chamber into another,
eventually creating a vacuum in the first chamber,
reversing a process of random expansion. One
needs to just look at the STM pictures of individual
atoms arranged into letters by lifting them with a tip to realize that this sort of thing must
be doable. It’s just that the demon will have to cause an increase in entropy elsewhere to
offset such a major decrease in entropy of the system it controls, as required by the
second postulate (or “law”) of thermodynamics. Furthermore, if we have no direct
control over most degrees of freedom, we cannot convert them with complete efficiency
into the few manipulable ones, meaning we cannot turn heat into work with 100%
efficiency. This brings us to the formulation of the postulates of thermodynamics, which
determine exactly what can and cannot be done.