Chapter 6 Statistical Thermodynamics

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Chapter 6Statistical Thermodynamics

• Notes on
• Thermodynamics in Materials Science
• by
• Robert T. DeHoff
• (McGraw-Hill, 1993).

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Combinatorial Analysis

• Consider a system of N particles that are allowed
  to occupy r states.
• Microstate --- The description of the system that
  provides the state of each particle.
• Number of possible microstates
  

Macrostate --- The description of how many
particles, ni, are in each of the r
states. Number of microstates in a macrostate
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Combinatorial Analysis
Microstates
Distribution
N3, r3
Macrostates
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Assumptions

• Consider all particles to be identical.
• The net value of a macroscopic property depends
  on the number of particles (ni) in each state
  (i). Exchanging the specific identity of the
  particles in a state does not change the value of
  the property.
• On average the fraction of time each particle
  spends in any energy state is the same.

Hypothesis
Probability of a macrostate is equal to the
fraction of time the system of particles spends
in that macrostate
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Probability of Macrostates

• Hypothesis
• Fraction of time in a macrostate
• probability of that macrostate.

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Probability of Macrostates
Sharp distribution --- Most probable state and/or
those near it are observed most of the time.
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Plot probability as function of fractional range.
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Plot probability as function of fractional range.
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Problem 6.4
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Boltzman Hypothesis

• where
• S is the entropy.
• W is the of microstates in a macrostate.
• The Boltzman constant, k R/NO.
• NO is Avogardos number.
• R is the ideal gas constant.

Provides a sharp extremum. Range is compressed by
assuming logarithmic relation. Average energy of
particles is fixed.
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Find Conditions for Equilibrium

• Find an expression for change in entropy of the
  system.
• Determine the constraints.
• Apply the constraints and the extremum criterion
  

• Solve the remaining equations for the
  conditions for equilibrium.

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Find an expression for dS(ni)

• Substitute for W

Expand
Note the Stirling approximation
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Find an expression for dS(ni)
Note
and
Rearranging
Taking the derivative
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Isolation Constraints
Consider an isolated system.
Closed system ---
Rigid system ---
Insulated system ---
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Constrained Maximum Entropy

• Apply Lagrange multipliers to constraints add
  to condition for entropy maximum.

Substitute for entropy and constraints
Rearrange, raise to power of e to yield r
equations
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Constrained Maximum Entropy
Apply
and
Solve for a
Define
Yielding
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Constrained Maximum Entropy
Compare phenomenological statistical
expressions for dS to evaluate a b
By analogy.
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Constrained Maximum Entropy
Complete expression equilibrium distribution of
particles over energy levels
All equilibrium thermodynamic functions can be
derived if the partition function is known.
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Thermodynamic Functions in Terms of Partition
Function
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Thermodynamic Functions in Terms of Partition
Function
Deduce F from S
Apply
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Thermodynamic Functions in Terms of Partition
Function
Apply
Apply
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Monatomic Gas Model

• Assumptions
• All particles are identical.
• Volume lx x ly x lz
• Energy of the system is not quantized is equal
  to S kinetic energies of the particles.

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Thermodynamic Propertiesof Ideal Monatomic Gases
Apply
Apply
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Thermodynamic Propertiesof Ideal Monatomic Gases
Apply
Apply
Equipartition of energy
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Einsteins Model of a Crystal

• Consider a simple cubic crystal --- 6 nearest
  neighbors, 1 atom 3 bonds per unit cell.
• Hypothesis --- Energy of crystal is the sum of
  the energies of its bonds. The atoms vibrate
  around equilibrium positions as if bound by
  vibrating springs. Only certain vibrational
  frequencies are allowed in coupled springs. The
  energies (ei) of the bonds are proportional to
  their vibrational frequencies (n).
• ei (i 1/2) hn
• where h Plancks constant.
• The adjustable parameter n is set by assuming an
  Einstein temperature qE hn/k

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Einsteins Model of a Crystal
Evaluate the partition function
Factor
Approximate as infinite series
Substitute for infinite series
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Einsteins Model of a Crystal
Take ln of both sides
For simple cubic
Apply
Apply
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Einsteins Model of a Crystal
Apply
Apply