3.012 Fundamentals of Materials Science Fall 2003
Today :
L AST TIME .............................................................................................................................................................................................. 2
S TATISTICAL M ECHANICS AND M ODELS OF M ATERIALS ........................................................................................................................ 3
Heat and the microscopic state of materials ...................................................................................................................................... 3
A simple model: the Debye solid ........................................................................................................................................................ 4
Microstates and the first postulate of statistical mechanics ............................................................................................................... 4
The second fundamental postulate of statistical mechanics 1 ............................................................................................................. 6
T HE MICROSCOPIC DEFINITION OF ENTROPY ........................................................................................................................................... 7
Testing the microscopic definition of entropy .................................................................................................................................... 7
R EFERENCES ........................................................................................................................................................................................... 9
Reading :
Supplementary Reading :
Planning Notes:
-
Averaging and ensembles
2 postulates of stat mech definition of entropy at the molecular level entropy quantifies available states other sources of entropy first problem: ideal gas
HOMEWORK PROBLEMS:
Lecture 21 – Connectiing molecular events to thermodynamics 1 of 9 4/16/20

3.012 Fundamentals of Materials Science
Fall 2003
Lecture 21 – Connectiing molecular events to thermodynamics 2 of 9 4/16/20

3.012 Fundamentals of Materials Science Fall 2003
 We have so far utilized macroscopic thermodynamics (what is usually referred to as ‘classical’ thermodynamics to understand the behavior of materials as a function of their environment (T, P, composition, reactions, etc.).
Classical thermodynamics does not rely on a theory of the molecular behavior of matter- it only explains macroscopic phenomena (involving ~Avoga dro’s numbers of atoms) based on the empirical laws of thermodynamics. The functional form of free energies/chemical potentials, entropies, and enthalpies is not predicted by classical thermodynamics, because these depend on the molecular details of materials- their structure and bonding. In a practical sense, this doesn’t limit the power of classical thermodynamics, since all of these thermodynamic functions can be empirically measured- for example, empirical tables of the dependence of free energy on temperature are available for many systems. o A particular challenge for new students of thermodynamics is to understand the ‘physical meaning’ of entropy. We have alluded to entropy as a measure of disorder- one statement of the second law is ‘the disorder of the universe must increase in any spontaneous process.’ However, there is little in our discussion so far that clearly connects increasing disorder to increasing entropy. There are hints: One hint comes from our examination of reversible and irreversible processes: drop a cube of sugar into a glass and it will dissolve, never to reform- this is clearly an increase in the disorder of the universe, and because it is a spontaneous process, the entropy must be increasing. Measurements of the molar entropy of gases show that it is typically greater than that of liquids, which in turn is greater than that of the solid state- again in congruence with an intuitive understanding that disorder increases in the order solid – liquid – gas.

In some cases we may wish to understand how the molecular structure of a material influences its thermodynamic properties, or predict how new materials may behave thermodynamically. This requires the development of molecular scale models for the thermodynamic behavior of materials- such models are the realm of statistical mechanics. We will now change gears for a few lectures and examine the fundamentals of statistical mechanics, how entropy relates to disorder at a molecular level, and how simple models of materials are formulated.

We have shown in our discussions of bonding so far that the energies (internal energy) of electrons in atoms and molecules are quantized. In turn, the energy of many degrees of freedom available to atoms and molecules that we will introduce may be quantized, such as rotations and vibrations. Thermal energy (heat) transferred to a molecule does not change the nature of the available energy levels, but it does change which energy levels are occupied by the electrons:

Within a macroscopic material sample (made up of Avogadro’s number of atoms), the total internal energy is still theoretically quantized, though the energy levels may be very close together (so close that we cannot really measure the quantization). At finite temperature, each atom’s electrons are excited to some distribution among the available energy levels; with increasing thermal energy in the system, the electrons are able to access higher and higher energy levels. One of the main objectives of statistical mechanics is to predict how the energy levels are occupied for a given model of how a system responds to thermal energy.
Lecture 21 – Connectiing molecular events to thermodynamics 3 of 9 4/16/20

3.012 Fundamentals of Materials Science Fall 2003

Our introduction to the connection of thermodynamic quantities to microscopic behavior begins by considering a simple model of a solid- models of materials are the starting point for statistical mechanics calculations. Suppose for definiteness we have a sample of X, where the bonding between atoms is X . As a model for how this material behaves in response to temperature, we propose that the most important degree of freedom available to the atoms to respond to thermal energy is vibration of the atoms about their at rest positions. o The bonding between atoms creates a potential energy well in which the atoms are centered at their at rest position. Oscillations of the atoms about the at rest position can be induced by thermal energy in the material.
 ANIMATION OF THE HARMONIC OSCILLATOR MODEL . o The potential is called harmonic potential because of its shape- similar to the potential of a spring in classical mechanics. o Because the oscillation of the atoms is constrained by bonding to center about their at rest positions in the crystal lattice, the energy of vibration for each atom is quantized: ( WILL NICOLA HAVE ALREADY
SHOWN THIS? SHOULD WE DERIVE IT?
)
(Eqn 1)
 The spacing between each energy level is:
  h

 …where h is Planck’s constant ( h = 6.62x10
-27 gm cm 2 sec -2 ) and

is the frequency of the atomic vibrations.
 Let’s for simplicity imagine a solid composed of only 3 atoms (we’ll see how the result generalizes to the case of
N
Av
atoms). The total energy of the solid is the sum of the individual energies of each oscillator. Suppose the total energy of the system is 2

.
You can see that there are multiple ways the 3 atoms can vibrate at that total energy. Using the energy level diagram, where each filled circle represents the vibration state of one of the atoms:
Lecture 21 – Connectiing molecular events to thermodynamics 4 of 9 4/16/20

3.012 Fundamentals of Materials Science Fall 2003 o Each of the single unique arrangements of energy distributed among the 3 atoms is called a microstate .
The total number of allowed microstates is a parameter we will refer to again and again; we give it the symbol

. For the system above,

= 6. The collection of all

microstates for a given system is called its ensemble . (For the case of a system with fixed (E,V,N) it is referred to as the microcanonical ensemble ).
 Usually we don’t care which atom is in a given energy level, only how many atoms are at a given energy level.
Thus we can write the energy level diagram more compactly as: o We have collected the individual microstates now into 2 groups characterized by the number of atoms in each energy level (n
0
, n
1
, and n
2
). We will call the set of microstates that has a given set of occupation numbers a state (as opposed to microstate). The total number of distinguishable states is W .
 There are
 and

2
1
= 3 different microstates in the j=1 state (characterized by n0 = 2, n1 = 0, n2 = 1)
= 3 microstates in j=2 state (n
0
= 1, n
1
= 2, n
2
= 0).
 For our overly-simplified 3-atom model with a low total energy of E total
= 2 , the number of distinct arrangements is small. However, for a material containing a mole of atoms at room temperature, the number of possible ways to occupy the available energy levels is enormous. Thus, rather than writing diagrams of all the possible microstates, we become concerned with the probability of finding a certain set of microstates j in the ensemble that have a given distribution of the atoms among the energy levels. For the 3-atom system with E total
= 2 , there are two different distributions (j = 1, 2), that will each be found in the ensemble with some probability, p j
.
The first postulate of statistical mechanics

The first postulate of statistical mechanics tells us the probability of each of these arrangements being found in the ensemble: o In an ensemble of

microstates (

very large) at fixed total (E,V,N), each of the allowed microstates is equally probable.
 Note that an ensemble at fixed total energy, volume and number of atoms represents an isolated thermodynamic system in classical thermodynamics.
Lecture 21 – Connectiing molecular events to thermodynamics 5 of 9 4/16/20

3.012 Fundamentals of Materials Science Fall 2003
 This postulate is often called the principle of equal a priori probabilities . It says that if the microstates have the same energy, volume, and number of particles, then they occur with equal frequency in the ensemble. o This postulate tells us what the p j
’s for the two states in our 3-atom E
 We have a total of total
=2 system are: the probability for each state is:
(Eqn 2) p
1
 p
2

1
W

1
2
 Each state has 1 chance in W to be encountered in the ensemble.

At finite temperatu res, the vibrations distribute to energy levels above the ground state (“0” energy). The total it occupies).
1
The second postulate connects ensemble averages to measured thermodynamic quantities

Statistical mechanics makes one postulate that connects the energy-level model of a material to its macroscopic thermodynamic properties: o In the limit of
  
, the ensemble average of a thermodynamic parameter X is equal to the time average value of the measurable macroscopic thermodynamic value of X in the real system.

What it means: o The postulate tells us that we can calculate the thermodynamic properties of a material if we have a model that provides us with a way to describe the possible microstates of the system. As we make a measurement on the real system, we might imagine the atoms in the system undergoing thermal fluctuations that allow the system to rapidly fluctuate between the many microstates of our model, making the time average measurement on the real system equivalent to averaging the property in our ensemble of microstates in the model .
o Note that though we drew a ‘mini-ensemble’ for our 3-atom imaginary solid above, the postulate rigorously holds only when the number of microstates possible is very large. Fortunately, this will always be the case in models of real materials containing N
Av
numbers of atoms!
The second postulate applied to a fixed (E, V, N) system

We now have what we need to make thermodynamic calculations for our Debye solid at fixed total energy.
Suppose we fix the total energy of the 3-atom ( N = 3 ) harmonic oscillator solid to a value E = 2 . Applying the postulates to the 3-atom harmonic oscillator solid, we can predict the thermodynamic energy: o The allowed microstates are sketched in the ensemble above. We have a total of

= 6 allowed microstates that fit the constraints. First, applying the first postulate, we know the probability of each of the allowed states is p1 = p2 = 1/2.
o Applying the second postulate, the thermodynamic internal energy is:
(Eqn 3) all states
U

E

 p j
E j


1 j
2  1
 j

1
2 j
2  1

2 put this in real energy units

1
2

This result is trivial since we fixed the total energy E total
= 2, but it gets interesting when we consider the more practical situation of a system held at fixed temperature instead of fixed energy. We’ll tackle this next, after we

Lecture 21 – Connectiing molecular events to thermodynamics 6 of 9 4/16/20

3.012 Fundamentals of Materials Science Fall 2003

Given our simple model for thermal energy in a solid, we now turn to the problem of connecting this model to thermodynamic properties of the solid. The first property we’ll introduce is the microscopic definition of the
 entropy.
The total number of possible microstates of a system is defined by the symbol

. For our 3-atom solid with total energy 2, we found the of number of states was

= 6.

Ludwig Boltzmann sought to connect entropy to the quantum mechanical picture of molecules, and found an answer in the following simple expression that connects the entropy to the probabilities of each state:
(Eqn 4)
W
S
  k
 p

1 j j ln p j o This is equivalent to the famous expression:
(Eqn 5)  S
 k ln W …engraved on Boltzmann’s tomb. o Here S is the entropy of the system, k is Boltzmann’s constant ( k = the ideal gas constant/Avogadro’s number) and

is the number of microstates available to the system.


We can show the equivalence of (Eqn 4) and (Eqn 5) easily for the case of the microcanonical
ensemble (a system with fixed (E,V,N)): The probabilities p j
= 1/W in the microcanonical ensemble, therefore:
(Eqn 6) S
  k
W  j

1 p j ln p j
  k
W  j

1 ln p p j j
  k ln
 
W   k ln

 1

W
1
W


W
  k ln
1
W
 k ln W o
 This simple definition is full of physical meaning for the nature of entropy. Let’s compare the behavior of k ln
 with the required behavior of the classical thermodynamic quantity S .
Entropy is an extensive quantity
 The statistical mechanical definition of entropy is extensive, just like S . To see this, consider how many states are available to a material formed by combining material A having

A material B having

B
microscopic states:
different microscopic states with
(Eqn 7) W combined

W
A
W
B
(Eqn 8) S combined
 k ln W combined
 k ln

W
A
W
B
  k ln W
A
 k ln W
B

S
A

S
B

Etc. see Bent


Entropy increases with E o Like the thermodynamic function entropy, increases with increasing energy in a model system.
Consider our 3-atom Debye solid once again. How many states do we have if we increase E total
to 3

?
Lecture 21 – Connectiing molecular events to thermodynamics 7 of 9 4/16/20

3.012 Fundamentals of Materials Science

The derivative is always positive like T

The derivative increases with E, like T
The third law of thermodynamics

What
Entropy as a measure of disorder

Larger S implies a greater number of available states- which matches with the notion of disorder

Fall 2003
Lecture 21 – Connectiing molecular events to thermodynamics 8 of 9 4/16/20

3.012 Fundamentals of Materials Science Fall 2003
1. Hill, T. L. An Introduction to Statistical Thermodynamics (Dover Publications, Inc., New York, 1986).
Lecture 21 – Connectiing molecular events to thermodynamics 9 of 9 4/16/20