Transcript No Slide Title

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)
CHEE 311
J.S. Parent
1
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?
CHEE 311
J.S. Parent
2
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 (AB), we find
dQ < TdnS AND dW < 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.
CHEE 311
J.S. Parent
3
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  U + PV
2.5
Helmholtz Energy: A  U - TS
6.2
Gibbs Energy:
G  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
CHEE 311
J.S. Parent
4
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.
CHEE 311
J.S. Parent
5
Defining Maxwell’s Equations
The fundamental equations can be expressed as:
 F 
 F 
dF    dx    dy
 x  y
 y  x
from which the following relationships are derived:
 U 
T 
 S  V
 H 
T 
 S P
 A 
P   
 V  T
 G 
V  
 P  T
CHEE 311
 U 
P   
 V S
 H 
V  
 P S
 A 
S   
 T  V
 G 
S   
 T P
J.S. Parent
6
Maxwell’s Equations
The fundamental property relations are exact differentials, meaning
that for:
 F 
 F 
dF    dx    dy
 x  y
 y  x
defined as:
dF  Mdx  Ndy
6.11
then we have,
 M 
 N 
6.12


 
 x  y  y  x
When applied to equations 6.7-6.10 for molar properties, we derive
Maxwell’s relations:
 T 
 P 
    
 V S
 S  V
 P 
 S 

 
 

T
  V  V  T
CHEE 311
 T 
 V 
   
 P S  S P
 V 
 S 


 
 

T
 P
 P  T
J.S. Parent
6.13-6.16
7
Maxwell’s Equations - Example #1
We can immediately apply Maxwell’s 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 H=H(T,P):
 H 
 H 
dH    dT    dP
 T P
 P  T
 H 
 Cp dT    dP
 P  T
The final expression, including the pressure dependence is:
6.20
dP
dH  Cp dT  
Which for an ideal gas reduces to:
dHig  CpdT  
CHEE 311
J.S. Parent
dP
6.22
8
Maxwell’s Equations - Example #2
Entropy Dependence on T,P-closed system
Given that S=S(T,P)
 S 
 S 
dS    dT    dP
 T P
 P  T
 S 
 V 
   dT    dP
 T P
 T P
The final expression, including the pressure dependence is:
dT   V 
dS  
 T P
dP
6.21
Which for an ideal gas reduces to:
dS  
CHEE 311
R
dT  dP
P
J.S. Parent
6.23
9
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?
q
Surroundings @ Tsurr
q
System of k components
p phases
An equilibrium state (# phases, composition of each phase) is that
which minimizes the total Gibbs energy at a given P,T.
CHEE 311
J.S. Parent
10
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
dGsys T,P  0
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?
CHEE 311
J.S. Parent
11
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
 nG 
 nG 
d(nG)=
dP+


 dT
 P  T,n
 T P,n
=nV dP - nS dT
(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)
CHEE 311
J.S. Parent
12
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:
i k  nG 
 nG 
 nG 

d(nG)  
dni
 dT  
 dP   
i 1  ni  T,P,n
 T P,n
 P  T,n
j
Redefining the partial derivatives in terms of their intensive
properties gives us the fundamental equation for closed systems:
i k
d(nG)  nV dP  nS dT    i dni
10.2
i 1
where,
 nG 
 i  

 ni  T,P,n j
10.1
is the chemical potential of component i in the system.
CHEE 311
J.S. Parent
13
Chemical Potential
Each component in the system has a chemical potential defined by
equation 10.1
 nG 
 i  

 ni  T,P,n j
The chemical potential of each component is a measure of the
system’s Gibbs energy change as its amount in the system
changes.
Calculating the chemical potential requires an expression for
nG=G(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
CHEE 311
J.S. Parent
14
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 i between phases leads to the diffusion of
component i (or chemical reaction) to achieve chemical
equilibrium.
CHEE 311
J.S. Parent
15
3. Relating Chemical Potential to Equilibrium
SVNA 10.2
We have seen that chemical equilibrium establishes a state that minimizes
the system’s 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 i
For now, consider a two-phase system of k components:
The vessel as a whole (vap+liq)
is closed, as energy may be exchanged
with its surroundings but material
cannot.
Vap, k components
Liq, k components
Each phase, however, is an open
system, as it may exchange matter
with the other phase.
CHEE 311
J.S. Parent
@ uniform T,P
16
Gibbs Energy Changes for a Closed System
For the total vessel contents (vapour+liquid phases), we can write the
fundamental equation for a closed system. Recall,
(6.6) (A)
d(nG)  (nV )dP  (nS)dT
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.
CHEE 311
J.S. Parent
17
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):
i k
d(n G )  (n V )dP  (n S )dT    iv dniv
v
v
v
v
v
v
(10.2) (B)
i 1
For the liquid phase (superscript l refers to liquid):
i k
d(n G )  (n V )dP  (n S )dT    li dnli
l
l
l
l
l
l
(10.2) (C)
i 1
These equations detail how the Gibbs energy of each phase is affected by
changes in pressure, temperature, and composition.
CHEE 311
J.S. Parent
18
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:
i k
i k
d(nG)  (n V  n V )dP  (n S  n S )dT    dn    li dnli
l
l
v
v
l
l
v
v
i 1
i k
i 1
v
i
v
i
i 1
i k
 (nV )dP  (nS)dT    dn    li dnli
v
i
v
i
(B+C)
i 1
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 (dT=dP=0), this relation simplifies to:
i k
i k
d(nG) T,P    dn    li dnli
i 1
CHEE 311
v
i
v
i
i 1
J.S. Parent
19
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
d(nG ) T,P  0
Applying this criterion to Equation (B+C):
(D)
d(nG) T,P   iv dniv   li dnli  0
Note that for the two-phase system, conservation of mass requires that
any matter lost from one phase is gained by the other:
dniv  dnli
Therefore, Equation D becomes:
(E)
d(nG) T,P   (iv  li ) dniv  0
CHEE 311
J.S. Parent
20
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
( v   l )  0
i
Or in other terms,
i
iv  li
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:
i  i  ...  ip
(i  1,2,..., k )
10.6
Chemical equilibrium calculations require expressions for i as a function
of T,P, and composition
CHEE 311
J.S. Parent
21