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							Reaction Equilibrium and Chemical Potential
Criteria for chemical equilibrium are:

                    
                    i
                         i   i   0
                                                                13.8
                     (G T ) 
                              0
                      T , P
These criteria are not in a useable form.
    Recall our definition of fugacity which applies to any species
     in any phase (vapour, liquid, solid)
                                             ˆ
                        i  Gi (T )  RT ln fi

Recall that Gi(T) is a temperature-dependent constant.
Also recall that enthalpy, internal energy and Gibb’s energy are
always specified an tabulated using a reference state.



CHEE 311                                                    1
Standard States
For reaction equilibrium calculations, the Gibbs energy at standard
conditions is the usual reference state.
                    Gio  Gi (T )  RT ln fio
These standard conditions are:
    pure component i
    at the reaction temperature
    in a user-defined phase (gas, liquid or solid)
    at a user-defined pressure (often 1 bar)

A great deal of thermodynamic data are published as the standard
properties of formation at STP (Table C.4 of the text)
    DGfo is standard Gibbs energy of formation per mole of the
      compound when formed from its elements in its standard
      state at 25oC.
        » Gases: pure, ideal gas at 1 bar
        » Liquids: pure substance at 1 bar

CHEE 311                                                    2
Chemical Potential and Activity
Subtracting this expression for the standard Gibbs energy (Gio) gets
rid of Gi(T):
                                        ˆ
                                        fi
                   i    Gio    RT ln o                           13.9
                                       fi

We define a new parameter, activity, to simplify this expression:
                     i  Gio  RT ln ai
                                      ˆ
where,
                          a ˆ
                          ˆ i  fi fio
The activity of a component is the ratio of its mixture fugacity to its
pure component fugacity at the standard state.



CHEE 311                                                        3
Reaction Equilibrium and Activity
When a reactive system reaches an equilibrium state, we know that
the equilibrium criterion is satisfied:

                          ii  0
                          i
where i is the stoichiometric coefficient of component i and i is
the chemical potential of component i at the given P,T, and
composition.


Substituting for i in terms of activity gives:


                              i
                                  
                ii   i Gi  RT ln ai  0
                i
                              o
                                        ˆ         
Or,

                                  iiGio
                     i ln ai 
                            ˆ
                    i              RT
CHEE 311                                                      4
The Equilibrium Constant
Our equilibrium expression for reactive systems can be expressed
concisely in the form:
                            i      i iGio
                       ˆ
                  ln  ai                                      13.10
                     i               RT
How does the sum  turn in to a product P ?


The right hand side of equation 13.10 is a function of pure
component properties alone, and is therefore constant at a given
temperature.
    The equilibrium constant, K, for the reaction is defined as:
                         i iGio
                K  exp                ˆ
                                     ai i                     13.11
                          RT         i

K is calculated from the standard Gibbs energies of the pure
components and the stoichiometric coefficients of the reaction.
CHEE 311                                                    5
Standard Gibbs Energy Change of Reaction
The conventional means of writing the equilibrium constant uses DGo,
the standard Gibbs energy change the reaction.
                   DGo  ii Gio

Using this notation, our equilibrium constant assumes the usual form:
                                 DGo                        13.11
                    K  exp
                                RT
When calculating an equilibrium constant (or interpreting a literature
value), pay attention to standard state conditions.
     Each Gio must represent the pure component at the
      temperature of interest and in the state of interest.


                          DGo
                                ln K
                          RT
CHEE 311                                                     6
Temp. Dependence of Reaction Equilibrium

                          DGo
                                ln K
                          RT
K depends on temperature because DG/RT depends on T
               d (DG o / RT )  DH o
                                        Gibbs-Helmholtz eqn.
                   dT          RT 2
From which we can derive the temperature dependence of K:

                      d ln K DHo                              13.14
                            
                       dT     RT2
If we assume that DHo is independent of temperature, we can integrate
13.14 directly to yield:
                                                            13.15
                       K    DHo  1 1 
                     ln         
                       K1    R  T T1 
CHEE 311                                                  7
  K vs Temperature
Equation 12.15 predicts that ln K
versus 1/T is linear. This is based on
the assumption that DHo is only a
weak function of temperature over the
range of interest.
     This is true for the
      reactions, in Figure13.2

     Why is DHo a function of T?

     What do we do if we are
      worried about DHo(T) when
      computing K?




  CHEE 311                               8
Equilibrium State of a Reactive System
Let’s use the equilibrium constant to determine concentrations at
eq’m.
                     i iGio
                exp                    ˆ
                                K   ai i
                      RT             i


Consider the gas phase reaction:
               CH4  H2O             CO  3H2
The equilibrium constant gives us:

                              ˆ ˆ3
                              aCOaH2
                       K
                              ˆ    ˆ
                              aCH4 aH2O
Or

                        ˆ       o    ˆ      o
                      ( fCO / fCO )( fH2 / fH2 )3
                K
                     ˆ         o     ˆ        o
                   ( fCH / fCH )( fH O / fH O )
                          4      4     2     2
CHEE 311                                                   9