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Kinetics III Lecture 16

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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

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∆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:

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Diffusion

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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.

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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:

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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:

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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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