Microscopic Treatment of the Equilibrium Constant
Lecture
The chemical potential
The chemical potential
The chemical potential can be expressed in terms of the partition function:
f h
f
∂ ln Q j
μ j = – RT
∂N j
To see this we first expand lnQ j, starting with the fact that Q j = q
fact that Q
qjNj/Nj!,
ln Q j = N jln q j – ln N j! = N jln q j – N jln N j + N j
and then take the derivative with respect to Nj,
∂ln Q j
qj
= ln q j – ln N j – 1 + 1 = ln
∂N j
Nj
Consideration in a theoretical chemical reaction
This result indicates that s esu t d cates t at
qj
μ j = – RT ln
Nj
For the reaction
ν AA + ν BB ↔ ν YY + ν ZZ
we have
or
qY
qZ
qA
qB
ν Yln
+ ν Zln
– ν Aln
– ν Bln
=0
NY
NZ
NA
NB
NYν YNZν Z q Yν Yq Zν Z
= ν ν
νA νB
q AAq BB
N A NB
Relation to the equilibrium constant
Relation to the equilibrium constant
We can express the concentration dependence in the equilibrium constant as where ρj is the number density of species j, ρ
y
p
j, ρj = Nj//V.
ρ νYYρ νZZ
Kc T = ν ν
ρ AAρ BB
which shows that the equilibrium constant can be expressed in terms of molecular partition
be expressed in terms of molecular partition functions. N /V ν Y N /V ν Z
νY
νZ
Kc T =
Y
N A/V
Z
νA
N B/V
νB
=
q Y/V
q A/V
q Z/V
νA
q B/V
νB
In considering the molecular partition function we must consider the kinematic contributions and electronic contributions. In other words if we write the molecular partition function as
q = q transq rotq vibibq elec
l
the molecular motions, translation, rotation and vibration are kinematic contributions to the available energy space. The electronic partition function is somewhat different since it f
ff
represents the binding energy of a molecule with respect to constituent atoms. h
Until now we have simply stated that qelec = gelec
the electronic degeneracy. This is equivalent to h l
i d
hi i
i l
ignoring the energetic contributions to chemical bonds and treating molecules as h i lb d
d
i
l l
translating, rotating and vibrating collections of nuclei that are held together by bonds. l i h
h ld
h b b d
However, when we deal with chemical reactions there are, by definition changes in i
h
b d fi i i
h
i
bonding. We can accommodate this by writing
q elec = g elece
D 0/RT
where D
h
D0 represents the binding energy.
h bi di
The binding energy D0 is equal to the equilibrium energy De shown in the figure minus the zero point energy.
D0 = De – hν/2
The zero point energy is shown as the red stripe at the bottom. For CO this energy is calculated to be 1067 cm‐1
and De is –96,545 cm‐1. Thus, the zero point energy is typically a small correction.
By making this subtraction we also “remove” the zero p
point energy from the vibrational partition function. gy
p
The vibrational partition function that we have considered up to now is:
–hω/2k BT
e
q vib =
1 – e –hω/k BT
The term in the numerator represents the zero point The
term in the numerator represents the zero point
energy. When this term is incorporated into qelec the vibration partition function becomes:
vibration partition function becomes:
1
q vib =
1 – e –hω/k BT
In this form we can state that for a singly degenerate vibration qvib = 1 at T = 0 K and the magnitude of the vibrational partition function increases with ib i
l
ii f
i i
ih
temperature. Using this separation the significance of the kinematic partition functions is clear. These represent the temperature dependence of occupation of various levels, translation, rotation and vibration, respectively. The electronic partition
rotation and vibration, respectively. The electronic partition function, on the other hand, represents the energy of stabilization of the molecule with respect to its constituent atoms. In fact q
t
I f t elec approaches infinity as T Æ
h i fi it
T Æ 0 K. What sense 0 K Wh t
does this make? Well, if we consider qelec as a contribution to an equilibrium constant we can think of temperature as a parameter that determines the relative stability of the molecule. The bound state of the molecule will be most favored at T = 0 K. As temperature increases there is some favored at T 0 K. As temperature increases there is some
thermal tendency for the molecule to dissociate (even though this is small at most temperatures of interest to chemists). In this sense qelec is quite different from the kinematic partition this sense q
is quite different from the kinematic partition
functions.
Vibrational temperature
To simplify the writing of the partition function we can define a vibrational temperature, Θvib. We define Θ vib = hω/k B
1
q vib =
1 – e –ΘΘ vib /T
Thus, the vibrational partitionc function has a simple form. Note that the vibrational temperature represents the at
Note that the vibrational temperature represents the at significant population of higher vibrational levels occur.
Rotational partition function and rotational temperature
rotational temperature
In the high temperature limit the rotational partition f nction is
function is 2
IkT = kT
8π
q rot ≈
2
B
h
where B is called the rotational constant. The rotational spectrum line spacing is 2B. Thus, we can define a rotational temperature, Θrot, such that:
2
8π
Ik
Θ rot =
2
h
q rot ≈ T
Θ rot
Symmetry number
For molecules with an axis of symmetry there are fewer unique rotational states accessible The
fewer unique rotational states accessible. The partition function is therefore reduced by the symmetry number σ, which corresponds to the
symmetry number σ, which corresponds to the multiplicity of the symmetry axis. For example, for a diatomic molecule the symmetry number is 2. For the rotation about the axis of symmetry in ammonia is 3.
q rot ≈
T
σΘ rot
The translational partition function
As discussed previously, the translational partition function can be written as:
q trans = V
Λ
3
where Λ is the thermal wavelength:
Λ=
2
h
2πmkT
Notice that the V in the translational partition function cancels Notice
that the V in the translational partition function cancels
the V in the number density. Thus, the contribution of the translational partition function to equilibrium is the thermal wavelength. This can still be significant if moles of gas are l
h h
ll b
f
f
l
f
created or destroyed in the chemical reaction.