Equilibrium Criteria

1
Equilibrium Criteria

• Overview
• This chapter addresses the conditions required
  for a system to be at equilibrium
• Postulate I states that equilibrium states exist
  and Postulate II states that systems will
  approach that equilibrium position
• In Chapter 4 we learned that for any isolated
  system (or the entire universe) the total entropy
  must increase or remain the same, i.e., the
  system maximizes its entropy
• In this chapter, we examine how to determine if a
  system has reached a stable equilibrium state
  under various constraints

Quasi-Static Process
Reversible Process
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Classification of Equilibrium States
Equilibrium States The figure below illustrates
the four classes of equilibrium states

• Stable following any perturbation (push to the
  right or left) the system returns to its original
  position
• Metastable the system reverts to the original
  position after a small perturbation, however a
  large perturbation could displace the system to a
  lower potential energy
• Unstable any minor perturbation will displace
  the system to a new state
• Neutral the system can be altered by any
  perturbation without a change in the systems
  potential energy

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Classification of Equilibrium States

• The Real World
• Most real systems are metastable (e.g., all
  organic compounds containing carbon and hydrogen
  atoms could attain a more stable state in the
  presence of oxygen by reacting to form CO2 and
  H2O)
• In many cases the barriers to transition to the
  more stable state are large enough to prevent the
  change from occurring within a reasonable time
  period
• Obtaining the rate at which these transitions
  occur requires a kinetic analysis
• To ascertain the stability of a given state, we
  consider only minor (virtual) perturbations to a
  system
• If minor perturbations leave the system
  unchanged, we say the original state is a stable
  equilibrium state

4
Classification of Equilibrium States

• Numerous Possible Perturbations
• In general, a system can be perturbed through a
  variety of mechanisms
• For example, in our previous analogy a push to
  the right or left can be considered different
  perturbation types
• Consider the figure below the system is
  metastable with respect to a push to the left and
  unstable with respect to a push to the right
• Therefore, we must classify the stability of a
  system with respect to particular kinds of
  perturbations

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Extrema Principles

• Entropy Maximum
• In Chapter 4 we found that for a process
  occurring in an isolated system the total entropy
  change must be zero or positive
• Therefore, for a system with constant total
  energy E, constant total volume V, and constant
  total mass M
• From this analysis, we can make the following
  conclusionfor an isolated system to be at
  stable equilibrium, the entropymust have a
  maximum value withrespect to any allowed
  variations

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Extrema Principles

• Energy Minimum
• One can show that a totally equivalent set of
  criteria invoking the total energy E rather than
  total entropy S can be used to define the
  conditions for a stable equilibrium state
• For a system with constant total entropy S,
  constant total volume V, and constant total mass
  M
• From this analysis, we can make the following
  conclusionfor a stable equilibrium system at
  constant S, V, and M, the total internal energy
  must be a minimum

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Extrema Principles

• Equilibrium Criteria
• To find a stable equilibrium state (entropy
  maximum), we vary a given parameter z1 (S is a
  function of z1,z2,,zn2) until theentropy is
  maximized, i.e., (?S/?z1) 0 and (?2S/?z12) lt 0
• The system is now in a stable equilibrium state
  with respect to variations of z1

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Extrema Principles

• Small Perturbations
• We now analyze small perturbations about some
  original state
• We propose a small perturbation is which z1 is
  altered by dz1, z2 by dz2, etc., and find the
  change in entropy by expanding S in a Taylor
  series about the conditions of the original
  state
• with

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Extrema Principles

• Small Perturbations
• If DS represents the entropy change from the
  original assumed equilibrium state to the
  perturbed state, and if S is a maximum in the
  equilibrium state of an isolated system of
  constant E, V, and M, then for all possible
  variations originating from a stable equilibrium
  state
• If S is a smoothly varying function of zi, the
  necessary and sufficient conditions for a maximum
  in S are
• Criterion for equilibrium
• Criterion for stability

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Extrema Principles

• Energy Route
• An analogous set of conditions can be stipulated
  for constantS, V, and M
• Here the appropriate potential is the internal
  energy
• If U is a smoothly varying function of zi, the
  necessary and sufficient conditions for a minimum
  in U are
• Criterion for equilibrium
• Criterion for stability
• Restraints for a composite system

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Extrema Principles
Equivalence of Entropy Maximization and Energy
Minimization
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Example 6.1
In the figure below the ball is shown at the
bottom of the well to represent a case of stable
equilibrium. Prove this to be so by considering
a virtual displacement process wherein the ball
moves to a point on the wall, above the bottom.
Develop two proofs, one using entropy
maximization and the other using energy
minimization
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Use of Other Potential Functions to Define
Equilibrium States

• Alternative Equilibrium Conditions
• Up to this point, we have developed criteria for
  equilibrium and stability for systems at constant
  U, V, M or constant S, V, M
• We now develop the criteria for systems under
  different constraints
• To determine these criteria, we examine a system
  connected to large thermal and work reservoirs
  (RT and RP), which hold the temperature and/or
  pressure constant
• The thermal gate is impermeable, rigid, and
  diathermal (thermal reservoir at constant V, M)
• The piston is impermeable, frictionless, and
  adiabatic (workreservoir at constant S, M)
• The entire (global) system is isolated and held
  at a constant total S, V, and M

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Use of Other Potential Functions to Define
Equilibrium States

• Alternative Equilibrium Conditions
• Under the conditions described above, the global
  system must adhere to the energy minimum criteria
• For any process for which the global system is
  originally a stable equilibrium state (S refers
  to the global system)
• We now look at situations in which the system
  interacts with one or more of the reservoirs and
  deduce the criteria for equilibrium

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Use of Other Potential Functions to Define
Equilibrium States

• Piston Unlatched, Thermal Gate Closed (Constant
  S, P, M)
• Applying the First Law to the pressure reservoir
  yields
• Or, using the relationships above
• Combining
• Therefore, for a system maintained at constant S,
  P, and M, the criteria for equilibrium at
  stability are

or
(S, P, M constant)
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Use of Other Potential Functions to Define
Equilibrium States

• Piston Locked, Thermal Gate Open (Constant T, V,
  M)
• Applying the First Law to the thermal reservoir
  yields
• Or, using the relationships above
• Combining
• Therefore, for a system maintained at constant T,
  V, and M, the criteria for equilibrium at
  stability are

or
(T, V, M constant)
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Use of Other Potential Functions to Define
Equilibrium States

• Summary
• If both the thermal gate is open and the piston
  is unlocked (constant T, P, M) the Gibbs energy
  appears as the relevant potential function
• We can now generalize our results for a set of
  constraints other than S, V, M, equilibrium is
  attained by minimizing the partial Lengendre
  transform of the energy U that results from
  switching to the new constraint variables
• In the entropy representation for a set of
  constraints other than U, V, M, equilibrium is
  attained by maximizing the partial Lengendre
  transform of the entropy S that results from
  switching to the new constraint variables

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Use of Other Potential Functions to Define
Equilibrium States
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Application Membrane Equilibrium

• Membrane Equilibrium
• Consider an isolated, complex system containing
  two subsystems, each of which contains a
  nonreacting binary mixture of components A and B
• The criterion of equilibrium for thecomposite
  system is
• We now expand this expression in terms of the
  properties of the two systems

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Application Membrane Equilibrium

• Membrane Equilibrium
• Using the constraints dU dU(1) dU(2) 0
  (same for V, NA, and NB), the above simplifies
  to
• We now obtain the criterion for equilibrium for
  three special cases

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Application Membrane Equilibrium

• The internal boundary is permeable to B only,
  diathermal, and movable
• The additional restraint is
• Equilibrium criteria
• The internal wall is rigid, diathermal, and
  permeable to both A and B
• Now we have
• Equilibrium criteria
• The internal wall is adiabatic, moveable, and
  permeable (A and B)
• In this case dU(1) and dU(2) are not zero since
  the energy of each subsystem can still change
  (e.g., mass transfer, compression)
• Equilibrium criteria

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Application Phase Equilibria

• Phase Equilibria
• A system with more than one phase may be
  considered to be a composite of simple systems
  with phase-separating membranes that are
  moveable, diathermal, and permeable
• The equilibrium criterion for a system containing
  p phases is
• Using the entropy representation of the
  Fundamental Equation
• The constraints from isolation are

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Application Phase Equilibria

• Phase Equilibria
• The equilibrium criterion can be reduced using
  the method of Lagrange undetermined multipliers
  to obtain
• The well known conditions for phase equilibria
  are now readily apparent

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Problem 6.6