# Kinetics III Lecture 16. Derivation of 5.67 Begin with Assume ∆G/RT is small so that e ∆G/RT = 1+∆G/RT, then Near equilibrium for constant ∆H, ∆S, ∆G.

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Kinetics III Lecture 16

Derivation of 5.67 Begin with Assume ∆G/RT is small so that e ∆G/RT = 1+∆G/RT, then Near equilibrium for constant ∆H, ∆S, ∆G = -(T-T eq )∆S Equation 5.67 should read: o no negative, no square

∆G & Complex Reactions Our equation: was derived for and applies only to elementary reactions. However, a more general form of this equation also applies to overall reactions: where n can be any real number. So a general form would be:

Diffusion

Importance of Diffusion As we saw in the example of the N˚ + O 2 reaction in a previous lecture, the first step in a reaction is bringing the reactants together. In a gas, ave. molecular velocities can be calculated from the Maxwell-Boltzmann equation: which works out to ~650 m/sec for the atmosphere Bottom line: in a gas phase, reactants can come together easily. In liquids, and even more so for solids, bringing the reactants together occurs through diffusion and can be the rate limiting step.

Fick’s First Law Written for 1 component and 1 dimension, Fick’s first Law is: o where J is the diffusion flux (mass or concentration per unit time per unit area) o ∂c/∂x is the concentration gradient and D is the diffusion coefficient that depends on, among other things, the nature of the medium and the component. Fick’s Law says that the diffusion flux is proportional to the concentration gradient. A more general 3- dimensional form (e.g., non-isotropic lattice) is:

Deriving Fick’s Law On a microscopic scale, the mechanism of diffusion is the random motion of atoms. Consider two adjacent lattice planes in a crystal spaced a distance dx apart. The number of atoms (of interest) at the first plane is n 1 and at the second is n 2. We assume that atoms can randomly jump to an adjacent plane and that this occurs with an average frequency ν (i.e., 1 jump of distance dx every 1/ν sec) and that a jump in any direction has equal probability. At the first plane there will be νn 1 /6 atoms that jump to the second plane (there are 6 possible jump directions). At the second plane there will be νn 2 /6 atoms that jump to first plane. The net flux from the first plane to the second is then:

Deriving Fick’s Law We’ll define concentration, c, as the number of atoms/unit volume n/x 3, so: Letting dc = -(c 1 - c 2 ) and multiplying by dx/dx Lettingthen

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