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Equilibrium Constant
Lecture
The chemical potential The chemical potential can be expressed in terms f h fo t e partition unction: μ j = – RT ∂ ln Qj ∂N To see this we first expand lnQj, starting with the fact that Qj = qjNj/Nj!, j ln Qj = Njln qj – ln Nj! = Njln qj – Njln Nj + Nj and then take the derivative with respect to Nj, ∂ln Qj = ln q ln N 1 + 1 = ln qj ∂Nj j – j – Nj 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 ibq l the molecular motions, translation, rotation trans rot v e ec and vibration are kinematic contributions to the available energy space. The electronic f ffpartition unction is somewhat di erent since it represents the binding energy of a molecule hwit respect to constituent atoms. Until now we have simply stated that qelec = gelec h l i d hi i i lt e e ectron c egeneracy. T s s equ va ent to ignoring the energetic contributions to h i l b d d i l lc em ca on s an treat ng mo ecu es as translating, rotating and vibrating collections of l i h h ld h b b dnuc e t at are e toget er y on s. However, when we deal with chemical i h b d fi i i h ireact ons t ere are, y e n t on c anges n bonding. We can accommodate this by writing h D h bi di qelec = geleceD0/RT w ere 0 represents t e n ng energy. 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. Vibrational temperature To simplify the writing of the partition function we can define a vibrational temperature, Θvib. We define Θvib = hω/k B qvib = 11 Θ /T Thus, the vibrational partitionc function has a simple form. Note that the vibrational temperature represents the at – e– vib significant population of higher vibrational levels occur. Rotational partition function and rotational temperature In the high temperature limit the rotational partition f nction is u qrot ≈ 8π2IkT h2 = kTB where B is called the rotational constant. The rotational spectrum line spacing is 2B. Thus, we can define a rotational temperature, Θrot, such that: Θ rot = 8π2Ik h2 qrot ≈ TΘ 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 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. qrot ≈ TσΘ rot
