Statistical mechanics, entropy and free energy of reaction
Chapter 1. From mechanics to statistical mechanics
Mechanics is very powerful. Equations such as F=ma or H=i/t allow us to calculate how
the state of a system evolves. Both equations conserve ener
Chapter 2. Postulates of statistical mechanics and entropy
a. Postulates and the ergodic approximation
Thermodynamics puts constraints on the behavior of macroscopic systems
without talking about the underlying microscopic properties. It does not provide
Chapter 3. Entropy, temperature, and the microcanonical partition function: how to
calculate results with statistical mechanics.
The goal of equilibrium statistical mechanics is to calculate the density (probability) eq
so we can evaluate average observab
Chapter 4: Going from microcanonical to canonical ensemble, from energy to
temperature.
All calculations in statistical mechanics can be done in the microcanonical ensemble,
where all copies of the system are in microstates of the same energy with wellde
Chapter 5. Chemical equilibrium and the free energy.
1) Entropy of the closed system and free energy of the open system (reaction)
In chapter 2 we derived the second law of thermodynamics from statistical mechanics:
S(t>0) > S(t=0),
Or
S > 0
for a spontan
Nonequilibrium statistical mechanics
Chapter 19. Fluctuation and dissipation
1. Form of the equations of motion of the probability as a function of time
So far, we have derived equations to calculate the equilibrium density
operator/probability distributi
Chapter 20. The fluctuationdissipation theorem
When chemical reactions reach macroscopic equilibrium, they continue microscopically:
individual molecules still exchange between reactant and product, even though there is no
net reaction. One would expect
2. Classical dynamics and electrodynamics
Interactions in chemistry are almost exclusively electromagnetic in nature, with gravity, weak
and strong interactions seldom of interest. Classical electrodynamics or its quantized versions
are therefore required
3. Density matrices
3.1 General properties
An observable's expectation value as a function of time is given by
i
i
Ht
< (t)   (t) >= < (0)  e+ Hte  (0) >
A
A
OS
*
*
OH
S
H
S
H
31
i
where e h Ht is the timeevolution operator. As shown by the righ
Appendix A: Mathematical tools
The following pages provide a brief summary of the mathematical material used in the
lectures of Advanced Physical Chemistry II. No rigorous proofs are provided, although the proof
of statements is generally either outlined,
The postulates of thermodynamics:
P0:
Simple systems have equilibrium states that are fully characterized by a unique set of
extensive state functions cfw_U,Xi, where U is the internal energy (energy for short) and the
Xi are other required extensive stat
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 defin
Some statistics used in statistical mechanics
When discussing systems i and observables A , natural questions to characterize them
are:
What is the average A(t) = A over an ensemble of macroscopic
e
systems? Its time average A = A(t) t ?

What is the ave
50Minute Exam 1 Solutions in blue
Chem 544
1.
Consider the model system for a gas of identical particles in a 2D box shown in the figure below: 4x6 overall
size of box, with a fixed partition in the middle (x=0) and identical particles as shown:
a. Writ
Mitre 1: #086033 is given by
The partition function of a single diatomic molecule in an external field with if; "
/\
99.0%,. peerieg C462": 8 _
q :2 mm sinhgw nitrate sinhx (ex ~ 3") / 29 sinhx 2 coshx m (6* + e") / 2
trot: k?
a. What is the partition
Homework #5 solution set
Chem 544 Fall 2014
Problem #1
Review the Jacobi identities from homework #1; well be using them! Remember, the
Jacobians are determinants of derivatives, so the properties of determinants in the math
notes can be used for proof
Homework set #4 solutions
Chem 544
TA: Yuanxi Fu
Problem #1
The fundamental equation for a twocomponent system is given by
VU 3/ 2
S = An + nR ln 5/ 2 n1 R ln(n1 / n) n2 Rln(n2 / n)
n
,
where n = n1 + n2, A is constant and R = 8.31 J/mole/K are consta
Hour Exam III
Chem 544 Fall 2014
Problem #1
The Euler formula: A = UTS = a + n
We use the positive sign for a because the system tends to minimize its surface area. Therefore,
in order for it to expand, the environment has to do work on it.
dA = SdT+ da
Homework #1
Solution sets
TA: Yuanxi Fu
Problem #1
1
Problem #2
Problem #3
Problem #4
2
Problem #5
g k =
!
1
2
e!
! /!
e!"# dx
!
Examine the argument of the exponent first and complete the square
=
1
2
x!
1
ia! k ! a! k !
+ ikx = ! (x +
)
a!
a
Homework Set #3 solutions
Chem 544
TA: Yuanxi Fu
Problem 1
In class we showed that dSV =dqV,n/T=dU/T at constant volume.
a. Now that you know that enthalpy is the Legendre transform H=U+PV, show similarly
that dS =dqP,n/T=dH/T at constant pressure instead
HWK#7solution sets
Chem 544, fall 2014/12/4
Problem #1
Show that the following relations for state functions hold:
a. In analogy to (lnZ/)N,V=U and (2 lnZ/ )N,V=2U proved in class, what are
(ln/)N,V and (2 ln/()N,V, where =ePV is the grand canonical parti
Chapter 14. Noninteracting particles
In real systems, particles interact: they have a potential energy that depends on the
relative positions and/or velocities of particles. However, noninteracting systems (e.g.
ideal gas, ideal solution, ideal metal) are
Chapter 15. Chemical equilibrium
A remaining goal not achieved by thermodynamics is the computation of the equilibrium
K from molecular information. At constant T, P, the most natural ensemble to discuss
chemical equilibrium is the Gibbs ensemble
ln
(T
1
Homework #2
Solution sets
TA: Yuanxi Fu
1. Do the lattice gas model problem in the box on page 3 of Chapter 2 of the Survey
notes, but with a 10x10x3 m room at 1 atm and 298 K, neglecting any distinction
between N2, O2etc for simplicity. What is the fra
Homework #6 Solution sets
CHEM 544, Fall 2014
Problem #1
In class, we derived the classical partition function for an ideal gas, and for a
singlecomponent nonideal gas interacting by only a pairwise potential energy of the
form
Pairwise means that V i
The postulates of statistical mechanics:
P1: Extension of microscopic laws of motion
Liouville von Neumann dynamics applies to the density operator i of any finite closed
system i specified by its timeindependent Hamiltonian and its constraints (extensiv