J.I. Siepmann
Chem 8561
103
CHEMICAL EQUILIBRIUM
• using
our
knowledge
of
the
canonical
partition
functions for the ideal monatomic, diatomic, and polyatomic gases, we are now
able to predict the equilibrium constants of chemical reactions in the gas phase
• the combination of statistical-mechanical partition functions and spectroscopic input often offers a more convenient (and accurate) way to determine equilibrium
constants then their direct measurements
• the statistical-mechanical basis of the second law of thermodynamics requires that
in equilibrium
microcanonical ensemble:
S → maximum
canonical enemble:
A → minimum
isobaric-isothermal enemble:
G → minimum
J.I. Siepmann
Chem 8561
104
• in an arbitrary multicomponent system of different species A, B, . . . we can solely
focus on only the species which are involved in the chemical reaction, consider as
an example the following homogeneous gas phase reaction
νA A + νB B ⇀
↽ νC C + νD D
(205)
which can be transformed into a more convenient mathematical form
νC C + νD D − νA A − νB B = 0
(206)
• considering a reaction vessel at fixed volume and temperature, the Helmholtz free
energy is (see eqn. 29)
dA = −S dT − p dV +
X
µi dNi
i
=
X
i
µi dNi
(207)
J.I. Siepmann
Chem 8561
105
• now define a variable λ (extent of reaction), such that
dNi = vi′ dλ
where vi′ = νi for products and vi′ = −νi for reactants
dA =
X
i
vi′ µi
!
dλ
(208)
• since eqn. 208 has to be zero (second law) irrespective of the value of dλ, we conclude
X
vi′ µi = νC µC + νD µD − νA µA − νB µB = 0
(209)
i
• this result is a general thermodynamic fact and is valid for all kinds of chemical
processes (i.e. it is not restricted to ideal gas mixtures)
J.I. Siepmann
Chem 8561
106
• eqn. 209 can also be derived using the isobaric-isothermal ensemble and the following approach
• the Gibbs free energy of our chemical system (see eqn. 205) is
G = NA µA + NB µB + NC µC + ND µD + . . .
(210)
• at equilibrium infinitesimal amounts νA dλ of A and νB dλ of B will react to form
νC dλ of C and νD dλ of D; under the assumption that during this infinitesimal
process the intensive variable like the chemical potentials of the individual species
will remain constant, the change in Gibbs free energy is
dG = (νC µC + νD µD − νA µA − νB µB ) dλ = 0
(211)
• since dλ is arbitrary, the term in brackets must be equal to 0 and we again obtain
eqn. 209
J.I. Siepmann
Chem 8561
107
• in case our chemical system is a mixture of ideal gases, i.e. the different species are
independent of each other, but distinguishable, the canonical partition function of
the entire system is
D
Y
qi (V, T )Ni
Q(NA , NB , NC , ND , V, T ) =
Ni !
(212)
i=A
• thus using eqn. 33 the chemical potential of each species is given by
µi = −kB T
= −kB T ln
∂ ln Q
∂Ni
T,V,Nj6=i
qi (V, T )
Ni
(213)
• by substitution of eqn. 213 into eqn. 209, and subsequent multiplication by −kB T ,
we obtain
qD
qA
qB
qC
+ νD ln
= νA ln
+ νB ln
(214)
νC ln
NC
ND
NA
NB
taking the exponentials and regrouping, we get
NC νC ND νD
qC νC qD νD
=
NA νA NB νB
qA νA qB νB
(215)
J.I. Siepmann
Chem 8561
108
• recognizing that q/V is a function of temperature only (see definition of the molecular translational partition function), we can define the equilibrium constant Kc of
the reaction as follows
(qC /V )νC (qD /V )νD
ρ C νC ρ D νD
=
Kc (T ) =
ρ A νA ρ B νB
(qA /V )νA (qB /V )νB
(216)
• the equilibrium constant can also be expressed in terms of partial pressures instead
of densities using the relation pi = ρi kB T
pC νC pD νD
pA νA pB νB
= (kB T )νC +νD −νA −νB Kc (T )
Kp (T ) =
(217)
(218)
J.I. Siepmann
Chem 8561
109
• eqn.
215
can
also
be
derived
from
purely
statisticalmechanical arguments (i.e. without the use of the thermodynamic eqns. 207 and
210)
• consider the simpler reaction
A⇀
↽ 2B
(219)
with the corresponding partition function
qA NA qB NB
Q(NA , NB , V, T ) =
NA ! NB !
(220)
• since the Helmholtz free energy is the characteristic function of the canonical ensemble, we can determine the minimum in A by finding the maximum in Q (or
ln Q) using the constraint
N = 2NA + NB
(221)
which follows from eqn. 219 and the fact that the total amount of material (mass)
in the system remains constant
J.I. Siepmann
Chem 8561
110
• using the constraint, the NB ’s can be eliminated from eqn. 220
ln Q =NA ln qA + (N − 2NA ) ln qB − NA ln NA
− (N − 2NA ) ln(N − 2NA ) + N − NA
(222)
and the maximum in Q can be determined from its partial derivative
∂ ln Q
∂NA
N,V,T
qA (N − 2NA∗ )2
= 0 = ln
qB 2 NA∗
(223)
and by resubstituting NB we obtain the equivalent of eqn. 215 for the binary reaction considered here
(NB∗ )2
qB 2
(224)
=
NA∗
qA