2. The Fundamental Equation

1
2. The Fundamental Equation

• Closed Systems
• We require numerical values for thermodynamic
  properties to calculate heat and work (and later
  composition) effects
• Combining the 1st and 2nd Laws leads to a
  fundamental equation relating measurable
  quantities (PVT, Cp, etc) to thermodynamic
  properties (U,S)
• Consider n moles of a fluid in a closed system
• If we carry out a given process, how do the
  system properties change?
• 1st law
• dnU dQ dW
• when a reversible volume change against an
  external pressure is the only form of work
• dWrev - P dnV (2.13)

2
The Fundamental Equation

• When a process is conducted reversibly, the 2nd
  law gives
• dQrev T dnS (5.12)
• Therefore, for a reversible process wherein only
  PV work is expended,
• dnU T dnS - P dnV (6.1)
• This is the fundamental equation for a closed
  system
• must be satisfied for any change a closed system
  undergoes as it shifts from one equilibrium state
  to another
• defined on the basis of a reversible process,
  does it apply to irreversible (real-world)
  processes?

3
Fundamental Eqn and Irreversible Processes

• The fundamental equation
• dnU T dnS - P dnV
• applies to closed systems shifting from one
  equilibrium state to another, irrespective of
  path.
• Note that the terms TdnS and PdnV can be
  identified with the heat absorbed and work
  expended only for the reversible path.
• dQ dW dnU TdnS - PdnV
• whenever we have an irreversible process (AB),
  we find
• dQ lt TdnS AND dW lt PdnV
• the sum yields the expected change of dnU
• Given our focus on fluid phase equilibrium, the
  lost ability to interpret the meaning of TdnS and
  PdnV is of secondary importance.

4
Auxiliary Functions

• The whole of the physical knowledge of
  thermodynamics (for closed systems) is embodied
  in P,V,T,U,S as related by the fundamental
  equation, 6.1
• IT IS ONLY A MATTER OF CONVENIENCE that we
  define auxiliary functions of these primary
  thermodynamic properties.
• Enthalpy H º U PV 2.5
• Helmholtz Energy A º U - TS 6.2
• Gibbs Energy G º H - TS 6.3
• U PV - TS
• All of these quantities are combinations of
  previous functions of state and are therefore
  state functions as well.
• Their utility depends on the particular system
  and process under investigation

5
Differential Expressions for Auxiliary Properties

• The auxiliary equations, when differentiated,
  generate more useful property relationships
• dnU TdnS - PdnV U(S,V)
• dnH TdnS nVdP H(S,P)
• dnA -PdnV - nSdT A(V,T)
• dnG nVdP - nSdT G(P,T) (6.7-6.10)
• Given that pressure and temperature are process
  factors under our control, Gibbs energy is
  particularly well suited to fluid phase
  equilibrium design problems.

6
Defining Maxwells Equations

• The fundamental equations can be expressed as
• from which the following relationships are
  derived

7
Maxwells Equations

• The fundamental property relations are exact
  differentials, meaning that for
• defined as
• 6.11
• then we have,
• 6.12
• When applied to equations 6.7-6.10 for molar
  properties, we derive Maxwells relations
• 6.13-6.16

8
Maxwells Equations - Example 1

• We can immediately apply Maxwells relations to
  derive quantities that we require in later
  lectures. These are the influence of T and P on
  enthalpy and entropy.
• Enthalpy Dependence on T,P-closed system
• Given that HH(T,P)
• The final expression, including the pressure
  dependence is
• 6.20
• Which for an ideal gas reduces to
• 6.22

9
Maxwells Equations - Example 2

• Entropy Dependence on T,P-closed system
• Given that SS(T,P)
• The final expression, including the pressure
  dependence is
• 6.21
• Which for an ideal gas reduces to
• 6.23

10
3. Gibbs Energy and Equilibrium SVNA 14.1

• Suppose our vessel remains at equilibrium with
  its surroundings as changes that require heat
  exchange (q) occur
• A. Charge 1 mole of material containing k
  components at P,T
• B. Watch the system approach equilibrium from
  its initial state
• How does the Gibbs energy vary as it approaches
  an equilibrium state?
• An equilibrium state ( phases, composition of
  each phase) is that which minimizes the total
  Gibbs energy at a given P,T.

11
Gibbs Energy and Equilibrium

• Our system (vessel) was charged with material
  quickly, which exposed the contents to conditions
  (P,T) that were not equilibrium values.
• Our derivation proved that all systems move
  towards equilibrium at a given P,T by minimizing
  the total Gibbs energy of the system.
• 14.3
• for any change the system undergoes at P,T
• This result is interesting to ponder, but
  difficult to apply.
• As engineers, we are concerned with the actual
  state of the system at equilibrium ( phases,
  composition of each phase)
• How do we transform equation 14.3 into something
  we can use?

12
4.2 Defining Chemical Potential 10.1

• We have developed the Fundamental Property
  Relation for CLOSED systems i.e. those of
  constant composition
• For a system of n moles
• nG G(T,P,n)
• and (dn 0 for a
  closed system)
• Eq. 6.6
• Consider an OPEN system of k components in which
  material may be taken from or added to the phase
  of interest. The Gibbs energy is now a function
  of composition as well as T,P.
• nG G(T,P,n1, n2, n3,... nk)

13
Fundamental Equation for Open Systems

• Given that
• nG G(T,P,n1, n2, n3,... nk)
• changes in total Gibbs energy for the open system
  follow
• Redefining the partial derivatives in terms of
  their intensive properties gives us the
  fundamental equation for closed systems
• 10.2
• where,
• 10.1
• is the chemical potential of component i in the
  system.

14
Chemical Potential

• Each component in the system has a chemical
  potential defined by equation 10.1
• The chemical potential of each component is a
  measure of the systems Gibbs energy change as
  its amount in the system changes.
• Calculating the chemical potential requires an
  expression for nGG(T,P,n1, n2...nk) to which
  equation 10.1 can be applied
• Note that as nG is a function of T,P and
  composition, the chemical potential is likewise
• We will develop a rigorous expression for perfect
  gas mixtures, as well as adapted expressions for
  non-ideal fluids in future lectures

15
Chemical Potential

• The chemical potential of a substance is an
  intensive property with analogies to temperature
  and pressure.
• Recall that intensive properties are spatially
  uniform under equilibrium conditions.
• Temperature gradients lead to heat conduction to
  achieve thermal equilibrium.
• Pressure gradients lead to fluid flow to achieve
  mechanical equilibrium.
• Differences in mi between phases leads to the
  diffusion of component i (or chemical reaction)
  to achieve chemical equilibrium.

16
3. Relating Chemical Potential to Equilibrium
SVNA 10.2

• We have seen that chemical equilibrium
  establishes a state that minimizes the systems
  Gibbs energy. However, a more useful definition
  of equilibrium is one based on intensive
  properties.
• Thermal Spatial uniformity of T
• Mechanical Spatial uniformity of P
• Chemical Spatial uniformity of mi
• For now, consider a two-phase system of k
  components
• The vessel as a whole (vapliq)
• is closed, as energy may be exchanged
• with its surroundings but material
• cannot.
• Each phase, however, is an open
• system, as it may exchange matter
• with the other phase.

17
Gibbs Energy Changes for a Closed System

• For the total vessel contents (vapourliquid
  phases), we can write the fundamental equation
  for a closed system. Recall,
• (6.6) (A)
• where
• n is the total number of moles of material
  mole
• G is the total molar Gibbs energy J/mole
• V is the molar Volume of the total system
  m3/mole
• P represents the system pressure Pa
• S is the total molar Entropy J/moleK
• T represents the system temperature K
• Note that because the composition of the entire
  system (vap liq) cannot change, only changes in
  pressure and temperature can influence the Gibbs
  energy of the whole system.
• Composition is invariant, so no chemical
  potential terms are included in the closed system
  expression.

18
Gibbs Energy Changes for an Open System

• Each phase can exchange not only energy, but
  material with the other. Therefore the vapour
  phase and the liquid phase are individual open
  systems.
• For the vapour phase (superscript v refers to
  vapour)
• (10.2) (B)
• For the liquid phase (superscript l refers to
  liquid)
• (10.2) (C)
• These equations detail how the Gibbs energy of
  each phase is affected by changes in pressure,
  temperature, and composition.

19
Back to the Overall System

• The change in the Gibbs energy of the whole,
  two-phase system is the sum of the vapour and
  liquid changes.
• For the whole system (vap liq), the sum of
  equations B and C yields the total Gibbs energy
  change
• (BC)
• According to this equation, the Gibbs energy of
  the overall system is affected by changes in T, P
  and composition.
• If we are interested in constant temperature and
  pressure processes (dTdP0), this relation
  simplifies to

20
Relating Chemical Potential to Equilibrium

• We now have the tools needed to translate our
  Gibbs energy criterion for equilibrium into one
  based on chemical potential.
• Recall that chemical equilibrium is a state that
  minimizes the total Gibbs energy of the system.
  At a given T,P equilibrium exists when
• Applying this criterion to Equation (BC)
• (D)
• Note that for the two-phase system, conservation
  of mass requires that any matter lost from one
  phase is gained by the other
• Therefore, Equation D becomes
• (E)

21
Relating Chemical Potential to Equilibrium

• Changes in the number of moles (dnv) are
  arbitrary and not necessarily zero. For Equation
  E to be satisfied always
• for all components, i
• Or in other terms,
• for all components, i
• Functional definition of chemical equilibrium
  between phases
• Each substance has an equal value of its chemical
  potential in all phases into which it can freely
  pass
• For a system of p phases, equilibrium exists at a
  given P and T if
• 10.6
• Chemical equilibrium calculations require
  expressions for mi as a function of T,P, and
  composition