Kinetics III
Lecture 16
Derivation of 5.67
• Begin with
Ânet = Â+ (1- e∆ G/RT )
• Assume ∆G/RT is small so that e∆G/RT = 1+∆G/RT, then
Ânet = -Â+ ∆ G / RT
• Near equilibrium for constant ∆H, ∆S, ∆G = -(T-Teq)∆S
Ânet = -Â+
-∆ S(T - Teq )
∆ S(T - Teq )
∆G
= -Â+
= Â+
RTeq
RTeq
RTeq
• Equation 5.67 should read:
Ânet =
o no negative, no square
 + ∆ S(T - Teq )
RTeq
∆G & Complex Reactions
• Our equation:
Ânet = Â+ (1- e∆ G/RT )
• was derived for and applies only to elementary
reactions.
• However, a more general form of this equation also
applies to overall reactions:
Ânet = Â+ (1- en∆ G/RT )
• where n can be any real number. So a general form
would be:
Ânet = A+ e- E+ /RT (1- en∆ G/RT )anAA anBB ACnC ...
Diffusion
Importance of Diffusion
• As we saw in the example of the N˚ + O2 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:
v=
8kT
pm
• 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:
¶c
J = -D
¶x
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 3dimensional form (e.g., non-isotropic lattice) is:
J = -DÑC
æ D11
ç
D = ç D21
ç D
è 31
D12
D22
D32
D13 ö
÷
D23 ÷
D33 ÷ø
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 n1 and at the second is n2.
• 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 νn1/6 atoms that jump to
the second plane (there are 6 possible jump directions).
At the second plane there will be νn2/6 atoms that jump
to first plane. The net flux from the first plane to the
second is then:
J=
n n1 / 6 - n n2 / 6
dx 2
=
n (n1 - n2 )
6
dx 2
Deriving Fick’s Law
J=
n n1 / 6 - n n2 / 6
dx 2
=
n (n1 - n2 )
6
dx 2
• We’ll define concentration, c, as the number of
atoms/unit volume n/x3, so:
J ==
n (c1 - c2 )dx 3
6
dx 2
n
= (c1 - c2 )dx
6
• Letting dc = -(c1 - c2) and multiplying by dx/dx
J=-
• Letting
D=
n dx 2
6
n dx 2 dc
6 dx
then
dc
J = -D
dx