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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)

Differential:

The total energy can be altered by either changing the population in each level (dni) or

changing the energy of the levels themselves (dεi) – often it is hard to change the energy

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

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

εi = the energy of quantum state i

ni = the number of molecules in quantum

state i

nA = represent the number in the category A

N = total number of possible outcomes

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123201 Notes

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Relationship between Probability and Entropy

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.

S = Entropy

k = Boltzmann’s Constant

W = probability of finding the molecules in a particular configuration

or microstates

R = ideal gas constant and NA = Avogadro’s Number

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123201 Notes

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A Lattice Model for Diffusion

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)

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).

W = number of permutation possibilities

relates to the Second Law of Thermodynamics

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