Microscopic Treatment of the Equilibrium Constant Lecture
The chemical potential The chemical potential can be expressed in terms of the partition function: μ = RT ln Q j j N j To see this we first expand lnq j, starting with the fact that Q = Nj j q j /N j!, ln Q j = N j ln q j ln N j!=n j ln q j N j ln N j + N j and then take the derivative with respect to N j, ln Q q j j = ln q N j ln N j 1 + 1 = ln j N j
Consideration in a theoretical This result indicates dcatesthat For the reaction we have or chemical reaction μ j = RT ln q j N N j ν A A + ν B B ν Y Y + ν Z Z q Y N + Zln q Z N Aln q A N Bln q B ν N = Y ln ν Z ν A ν B 0 NY NZ NA NB ν Y ν Z N YY N Z ν N A ν A N B B ν q ν = q Y Y qz Z q A ν AqB ν B
Relation to the equilibrium constant We can express the concentration dependence in the equilibrium constant as where ρ j is the number density of species j, ρ j = N j /V. K c T = ρ ν YρZ ν Z Y ν ν ρ A A ρ B B which shows that the equilibrium constant can be expressed in terms of molecular partition functions. N Y /V ν Y K c T = N A /V ν A N Z /V ν Z N B /V ν B q Y /V ν Y q Z /V ν Z = q A /V ν A q B /V ν B
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 q q q trans rot vib elec the molecular motions, translation, rotation and vibration are kinematic contributions to the available energy space. The electronic partition function is somewhat different since it represents the binding energy of a molecule with respect to constituent atoms.
Until now we have simply stated that q elec = g elec the electronic degeneracy. This is equivalent to ignoring the energetic contributions to chemical lbonds and treating molecules l as translating, rotating and vibrating collections of nuclei lih that are held hldtogether by bonds. However, when we deal with chemical reactions there are, by dfiii definition changes in bonding. We can accommodate this by writing q elec = g elec e D 0 /RT where D 0 represents the binding energy.
The binding energy D 0 is equal to the 0 equilibrium energy D e shown in the figure minus the zero point energy. D 0 = D e 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 D e is 96,545 cm 1. Thus, the zero point energy is typically a small correction.
By making this subtraction we also remove the zero point energy from the vibrational partition function. The vibrational partition function that we have considered up to now is: q vib = e hω/2k B T 1 e hω/k B T Theterm in the numerator represents the zero point energy. When this term is incorporated into q elec the vibration partition function becomes: q vib = 1 1 e hω/k B T In this form we can state that for a singly degenerate vibration q vib = 1 at T = 0 K and the magnitude of the vibrational i partition ii function increases with 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 function, on the other hand, represents the energy of stabilization of the molecule with respect to its constituent atoms. In fact q elec approaches infinity it as T 0 K. What sense does this make? Well, if we consider q elec 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 thermal tendency for the molecule to dissociate (even though this is small at most temperatures of interest to chemists). In this sense q elec is quite different from the kinematic partition functions.
Vibrational temperature To simplify the writing of the partition function we can define a vibrational temperature, Θ vib. We define Θ vib = hω/k B q = 1 vib 1 e Θ vib /T 1 Θ /T Thus, the vibrational partitionc function has a simple form. Note that the vibrational temperature represents the at significant population of higher vibrational levels occur.
Rotational partition function and rotational temperature In the high temperature limit the rotational partition function nctionisis q rot 8π2 IkT = kt h 2 B where B is called the rotational constant. The rotational spectrum line spacing is 2B. Thus, we can define a rotational temperature, Θ rot, such that: Θ rot = 8π2 Ik h 2 q rot T Θ rot
Symmetry number For molecules with an axis of symmetry there are fewer unique rotational states accessible. The partition function is therefore reduced by the symmetrynumber 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: V q trans = V Λ 3 where Λ is the thermal wavelength: Λ = h 2 2πmkT 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 created or destroyed in the chemical reaction.