# Statistical Mechanics (Unit 2).docx

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Massey University

123

123 .201

John Harrison

Spring

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123201 Notes
Statistical Mechanics
relating the thermodynamic properties to behaviour at an atomic level
The Quantum Structure of Matter
There are a variety of ways that a molecule can store energy:
Translational
Rotational
Vibrational
Electrical
The Quantum States are the certain values that are allowed for each type of energy (as all
the types of energy above are quantised)
εi= the energy of quantum state i
∑ ni= the number of molecules in quantum
state i
∑ ∑
Differential:
The total energy can be altered by either changing the population in each level (di) or
changing the energy of the levels themselves (dε) – often it is hard to change the energy
i
levels, so the population of each energy level is changed
The entropy is a measure of the spread of the occupied levels
Probability and Entropy
Basic Probability Concepts
n A represent the number in the category A
N = total number of possible outcomes
Independent Events occur when the outcomes in each event does not depend on the
outcome of any other events.
Mutually Exclusive Events is when one outcome prevents that value from being obtained as
an outcome of any of the other events.
Multiplicity of Events is a weighting for each type of event. There can be multiple ways of
the same outcome occurring.
Correlated Events is when the outcome of one event influences the outcome of another
1 123201 Notes
Relationship between Probability and Entropy
S = Entropy
k = Boltzmann’s Constant
W = probability of finding the molecules in a particular configuration
or microstates
R = ideal gas constant and A = Avogadro’s Number
A maximum for S is equivalent to the most probable state (and largest W)
There can be a significant different
between the distribution of energy in
the most probable state and the next
probable.
In this example there is more than 500
ways of distributing the energy in the
most probable state.
Lattice Models
take advantage of the quantum mechanical nature of matter, where not all values
for all quantities are possible
A Lattice Model for Gas Pressure
Arranging N objects into M sites:
( )
( )
The probability that the molecules are concentrated in one area is very unlikely, with the
most probable arrangement that sites are filled over the whole space.
2 123201 Notes
A Lattice Model for Diffusion
W = number of permutation possibilities
The most even distribution of particles corresponds to the largest number of
possible permuations
The system will evolve towards the state of maximum multiplicity (most even
distribution of particles)
relates to the Second Law of Thermodynamics
A Lattice Model for a Polymer Chain
This is used for predicting the polymer behaviour when it is fixed at one end to a surface
In this example the red polymer unit remains fixed to the surface. The
possible configurations (R:B:Y) are:
(1:2:3), (1:2:2), (1:1:2), (1:1:1)
However not all the configurations have the same weight, multiple ways of arranging the
units can arise in certain configurations
The average distance from the surface
polymer is equal to the weighted average of
the furthest distance:
Average Length = (1x3 +2x2+2x2+2x1)/7 =
1.86
This proves that the most likely state for a polymer is the partially contracted state (this is a
statistical interpretation of elasticity).
3 123201 Notes
A Lattice Model for Heat Flow
The tendency is for the energy to be distributed between two evenly sized systems
When the systems are not evenly sized, the most likely case is the one with the
highest multiplicity. This corresponds to the situation where the fraction of
molecules in the excited states are the same for the two systems (both left and right)
The ratio between systems is related to the temperature of the system – hence this is a
statistical interpretation of the zeroth law of thermodynamics. (This does not imply that
both systems have the same internal energy as they may have different heat capacities).
Entropy and Constraints
When all possible states are equally likely the principle maximum entropy takes the form of
a uniform distribution over all these possible states.
p = probabilities
∑ n = different outcomes

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